# Principal spectral curves for Lane-Emden fully nonlinear type systems and
applications

Ederson Moreira dos<EMAIL_ADDRESS>partially supported by CNPq
grant 309006/2019-8. Instituto de Ciências Matemáticas e de Computação,
Universidade de São Paulo, Av. Trabalhador São-carlense 400, 13566-590 Centro,
São Carlos SP, Brazil Gabrielle<EMAIL_ADDRESS>supported
by FAPESP grants 2018/04000-9 and 2019/031019-9, São Paulo Research
Foundation. Departamento de Ingeniería Matemática, Universidad de Chile,
Beauchef 851, Torre Norte, Santiago, Chile  Delia
<EMAIL_ADDRESS>supported by project Vain-Hopes within
the program VALERE: VAnviteLli pEr la RicErca. Dipartimento di Matematica e
Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale A.
Lincoln 5, 81100 Caserta, Italy  Hugo
<EMAIL_ADDRESS>partially supported by the
Portuguese government through FCT-Fundação para a Ciência e a Tecnologia,
I.P., under the projects PTDC/MAT-PUR/28686/2017 and UID/MAT/04459/2020.
CAMGSD and Mathematics Department, Instituto Superior Técnico, Universidade de
Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

(August 27, 2024)

Abstract. In this paper we exploit the phenomenon of two principal half
eigenvalues in the context of fully nonlinear Lane-Emden type systems with
possibly unbounded coefficients and weights. We show that this gives rise to
the existence of two principal spectral curves on the plane. We also construct
a possible third spectral curve related to a second eigenvalue and an anti-
maximum principle, which are novelties even for Lane-Emden systems involving
linear operators. As applications, we derive a maximum principle in small
domains for these systems, as well as existence and uniqueness of positive
solutions in the sublinear regime. Most of our results are new even in the
scalar case, in particular for a class of Isaac’s operators with unbounded
coefficients, whose $W^{2,\varrho}$ regularity estimates we also prove.

Keywords. First eigenvalue; Lane-Emden elliptic systems; Fully nonlinear
operators; Solvability.

MSC2020. 35D35, 35D40, 35J47, 35P30, 35B50.

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Assumptions on the operators
    2. 1.2 Statement of the main results
    3. 1.3 Examples and applications
    4. 1.4 Structure of the paper
  2. 2 Preliminaries
    1. 2.1 Some known results
    2. 2.2 Principal scalar eigenvalues for proper operators
    3. 2.3 Definition of principal eigenvalues for systems
    4. 2.4 Scaling and asymptotic behavior
  3. 3 The scalar case with unbounded coefficients
  4. 4 Auxiliary results for systems
  5. 5 The first eigenvalue problem
    1. 5.1 Existence and simplicity
    2. 5.2 Maximum principles
    3. 5.3 The Dirichlet problem for $\lambda<m_{1}$
    4. 5.4 Local isolation
  6. 6 The anti-maximum principle
  7. 7 The second eigenvalue problem
  8. 8 The signed Dirichlet problem
  9. 9 An application: Isaac’s operators

##  1 Introduction and main results

In this paper we study existence and uniqueness properties of the Dirichlet
problem for partial differential fully nonlinear systems of Lane-Emden nature
with weights, such as

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}(x,u,Du,D^{2}u)+\lambda\tau_{1}(x)|v|^{q-1}v&=&f_{1}(x)&\mbox{in}&\;\Omega\\\
F_{2}(x,v,Dv,D^{2}v)+\mu\tau_{2}(x)|u|^{p-1}u&=&f_{2}(x)&\mbox{in}&\;\Omega\\\
u\,\;=\,\;v&=&0&\mbox{on}&\;\partial\Omega,\end{array}\right.$ (1.4)

in the viscosity sense. Here $\Omega$ is a $C^{1,1}$ bounded domain in
${\mathbb{R}^{N}}$ with $N\geq 1$, $\lambda,\mu\in\mathbb{R}$ and $p,q>0$ are
constants, $F_{i}$ is a uniformly elliptic fully nonlinear operator in
nondivergence form, $f_{i}\in L^{\varrho}(\Omega)$ for some $\varrho>N$,
$i=1,2$, and the respective weights satisfy

$\displaystyle\textrm{$\tau_{i}\in L^{\varrho}(\Omega)$\; with \;
$\tau_{i}\gneqq 0$ in $\Omega$, \;$i=1,2$,
\quad}|\textrm{supp}\tau_{1}\cap\textrm{supp}\tau_{2}|>0.$ (1.5)

Here $\tau_{i}\gneqq 0$ means that $\tau_{i}\geq 0$ a.e. in $\Omega$ and
$\tau_{i}\not\equiv 0$. When $p=q=1$, $\tau_{1}=\tau_{2}=:\tau$,
$F_{1}=F_{2}=:F$ and $f_{1}=f_{2}=:f$, we recover the scalar case

$\displaystyle F(x,u,Du,D^{2}u)+\lambda\tau(x)u=f(x)\;\;\textrm{ in
}\,\Omega,\quad u=0\quad\textrm{ on }\,\partial\Omega,$ (1.6)

for which we also present new results.

Spectral properties of uniformly elliptic PDEs in nonvariational form have
long been recognized since the seminal work [7]. Its fully nonlinear scalar
theory in terms of viscosity solutions was developed in [42], for convex
operators with bounded coefficients, and unveils the phenomenon of two half
eigenvalues corresponding to both positive and negative eigenfunctions. The
case of nonconvex operators (again with bounded coefficients) was analyzed in
[4] under additional continuity restrictions on the data and on the operators.

In general, problems involving systems may be much more involved than their
scalar counterpart, specially in the strongly coupled case – for instance we
mention the so called Lane-Emden conjecture, see [40, 47], a long standing
open problem for which only partial results are known. As far as spectral
properties are concerned, in [43] the authors extended their article [42] to
gradient-like systems. Our systems, instead, have a strongly coupled nature,
whose prototype is also called Hamiltonian. Spectral properties for related
cooperative systems with linear operators in nondivergence form $F_{i}=L_{i}$
have been extensively investigated, see [9] for $p=q=1$ and references
therein. When more general power-like nonlinearities are taken into account,
still for lineal operators, in [32] a spectral curve was constructed for
(1.10) when $pq=1$; more recently, related comparison principles appear in
[30]. Both [32] and [30] deal with linear operators with bounded drift and
unbounded weights.

Our main goal here is to understand the phenomenon of two principal half
eigenvalues induced by the fully nonlinear operators $F_{1},F_{2}$ in light of
[42], under the framework of Lane-Emden systems in the regime $pq=1$,
including nonconvex operators with possibly unbounded coefficients and
weights. In this sense, we show that the homogeneous version of (1.4), i.e.

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}(x,u,Du,D^{2}u)+\lambda\tau_{1}(x)|v|^{q-1}v&=&0&\mbox{in}&\;\Omega\\\
F_{2}(x,v,Dv,D^{2}v)+\mu\tau_{2}(x)|u|^{p-1}u&=&0&\mbox{in}&\;\Omega\\\
u\,\;=\,\;v&=&0&\mbox{on}&\;\partial\Omega,\end{array}\right.$ (1.10)

gives rise to the existence of two principal spectral curves to (1.10) in the
plane $(\lambda,\mu)$. We stress that principal eigenvalues are related to the
solvability of (1.4), and to the validity of maximum principles, which we also
study. Moreover, we construct a possible third spectral curve and an anti-
maximum principle, which are novelties even for Lane-Emden systems involving
linear operators. All our results are valid also for a class of Isaac’s
operators with unbounded coefficients (1.18) (see Example 1.15 ahead), and
therefore are new and improve results in the literature even in the scalar
case. In this context, we mention that in [37] it was started a spectral
analysis involving a class of proper operators with unbounded drift and weight
in the scalar case, but only in what concerns existence of eigenvalues. Here
we complement that study, by giving a full characterization of the first
scalar eigenvalues in terms of validity of maximum principles, solvability of
the Dirichlet problem, and more generally the validity of Alexandrov-Bakelman-
Pucci (ABP) inequality for nonproper and possible nonconvex operators, under
improved assumptions. Observe that, once ABP is proved, for any solution
$(u,v)$ we have $uv>0$ in $\Omega$ whenever $u$ is signed in $\Omega$.

This problem brings about several applications. For instance, one may view the
pair $(\lambda,\mu)$ as risk-sensitive averages of the weights $\tau_{1}$ and
$\tau_{2}$, respectively, over the diffusions $F_{1},F_{2}$, see [2, 23].
Besides, it characterizes the range of solvability for equations with
superlinear gradient growth, as well as existence and uniqueness of positive
solutions for (1.10) in the sublinear regime $pq<1$, which we also prove.

For the Laplacian operator, the study of the problem with $pq=1$ involves
basic and important questions in the theory of Harmonic Analysis. As a matter
of fact, it is known that the standard Fourier series of an
$L^{r}(0,1)$-function $f$ converges to $f$ in $L^{r}(0,1)$, for any
$1<r<\infty$. This information was essential to treat the problem

$-u^{\prime\prime}=\lambda|v|^{q-1}v\ \
\text{and}-v^{\prime\prime}=\mu|u|^{\frac{1}{q}-1}u\ \ \text{in $(0,1)$}\ \
\text{with}\ \ u(0)=v(0)=u(1)=v(1)=0,$

see [11], where the asymptotic growth of the eigenvalues for the general case
$pq=1$ was controlled through the eigenfunctions for $p=q=1$. However, the
same question is much more challenging in higher dimensions, see [17, 18], and
indeed it is false in general since the “ball summation” for the double
Fourier series does not work; see [28, Section 3.3 and Theorem 3.5.6]. In this
case, when $\Omega\subset\mathbb{R}^{2}$ is a square, the Fourier functions
(product of sines) do not form a Schauder basis in $L^{r}(\Omega)$, for $r\neq
2$. For systems with nondivergence operators we cannot expect such explicit
formulas for eigenfunctions, and the problem is by far more delicate.

###  1.1 Assumptions on the operators

Next we list our hypotheses on the operators $F_{1}$ and $F_{2}$. We denote by
$\mathbb{S}^{N}$ the space of $N\times N$–symmetric matrices. Let us bear in
mind the following general structural hypothesis on a fully nonlinear operator
$F:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\times\mathbb{S}^{N}\to\mathbb{R}$
given by

$\displaystyle\mathcal{L}^{-}(x,r-s,\xi-\eta,X-Y)\leq
F(x,r,\xi,X)-F(x,s,\eta,Y)\leq\mathcal{L}^{+}(x,r-s,\xi-\eta,X-Y),$ (H1)

for all $X,Y\in\mathbb{S}^{N}$, $\eta,\xi\in\mathbb{R}^{N}$,
$r,s\in\mathbb{R}$, and $x\in\Omega$, where $F(\cdot,0,0,0)\equiv 0$, and

$\displaystyle\mathcal{L}^{\pm}(x,r,\xi,X):=\mathcal{L}_{0}^{\pm}(x,\xi,X)\pm\vartheta(x)|r|,\;\;\textrm{
for }\;\mathcal{L}^{\pm}_{0}(x,\xi,X):=\mathcal{M}^{\pm}(X)\pm\gamma(x)|\xi|,$
(1.11)

for $\gamma,\vartheta\in L^{\varrho}(\Omega)$, $\varrho>N$, with $\gamma\geq
0$ and $\vartheta\gneqq 0$ a.e. in $\Omega$. Also,
$\mathcal{M}^{\pm}=\mathcal{M}^{\pm}_{\alpha,\beta}$ are the Pucci’s extremal
operators with ellipticity constants $0<\alpha\leq\beta$, see (2.1) ahead.

Note that (H1) corresponds to a uniform bound for all operators satisfying a
prescribed ellipticity. In order to measure how far a particular fully
nonlinear operator $F$ is from a linear one, in the spirit of [4, 24, 42], we
may construct from (H1) a more accurate structure:

$F_{*}(x,r-s,\xi-\eta,X-Y)\leq F(x,r,\xi,X)-F(x,s,\eta,Y)\leq
F^{*}(x,r-s,\xi-\eta,X-Y),$

where

$\displaystyle
F^{*}(x,r,\xi,X):=\sup_{r^{\prime},\xi^{\prime},X^{\prime}}\,\\{F(x,r+r^{\prime},\xi+\xi^{\prime},X+X^{\prime})-F(x,r^{\prime},\xi^{\prime},X^{\prime})\\},$
$\displaystyle
F_{*}(x,r,\xi,X):=\inf_{r^{\prime},\xi^{\prime},X^{\prime}}\,\\{F(x,r+r^{\prime},\xi+\xi^{\prime},X+X^{\prime})-F(x,r^{\prime},\xi^{\prime},X^{\prime})\\},$

for all $x\in\Omega$, $r\in\mathbb{R}$, $\xi\in\mathbb{R}^{N}$,
$X\in\mathbb{S}^{N}$. Assume that $F$ satisfies (H1). Then both $F^{*},F_{*}$
satisfy (H1) as well; $F^{*}$ is convex and $F_{*}$ is concave in $(r,\xi,X)$;
$F=F^{*}$ if and only if $F$ is convex, $F=F_{*}$ if and only if $F$ is
concave. Also, the following identity holds

$F^{*}(x,r,\xi,X)=-F_{*}(x,-r,-\xi,-X)$;

see [4, Proposition 4.2]. Finally, we have the ordering $\mathcal{L}^{-}\leq
F_{*}\leq F\leq F^{*}\leq\mathcal{L}^{+}$.

######  Definition 1.1.

For a function $w$ and an operator $F$, we consider the following notations:

  1. (a)

$F[w]:=F(x,w,Dw,D^{2}w)$;

  2. (b)

$\lambda_{1}^{+}(F(\vartheta))$ is the principal weighted eigenvalue
associated to a positive eigenfunction of the scalar Dirichlet problem
$F[u]+\lambda\vartheta(x)u=0$ in $\Omega$, $u=0$ on $\partial\Omega$, see
Section 2.1;

  3. (c)

viscosity solutions are meant in the $L^{N}$-viscosity sense, see Section 2.3;

  4. (d)

we say that $F$ enjoys $W^{2,\varrho}$ regularity if any viscosity solution
$u$ of $F[u]=f(x)$ in $\Omega$ with $f\in L^{\varrho}(\Omega)$ belongs to
$W^{2,\varrho}_{\text{l}oc}(\Omega)$, and in addition $u\in
W^{2,\varrho}(\Omega)$ if $u=\psi$ on $\partial\Omega$, $\psi\in
W^{2,\varrho}(\Omega)$.

Having in mind the existence of eigenvalues, another condition we ask on an
operator $F$ is that it satisfies a positive homogeneity of order one, namely

$\displaystyle F(x,tr,t\xi,tX)=tF(x,r,\xi,X)\;\text{ for all }t\geq 0,\textrm{
for any $X\in\mathbb{S}^{N}$, $\xi\in\mathbb{R}^{N}$, $r\in\mathbb{R}$, and
$x\in\Omega$.}$ (H2)

Moreover, we consider the following control of oscillation in the $x$-entry:

$\displaystyle\forall\theta_{0}>0,\quad\exists
r_{0}>0:\quad\|\beta_{F}(x,\cdot)\|_{L^{\varrho}(B_{r}(x))}\leq\theta_{0}\,r^{{N}/{\varrho}}\;\textrm{
for all }r\leq r_{0},\;x\in\overline{\Omega},$ (1.12)

where
$\beta_{F}(x,y):=\sup_{X\in\mathbb{S}^{N}}{|F(x,0,0,X)-F(y,0,0,X)|}{\|X\|^{-1}}$
for $x,y\in\overline{\Omega}$. It holds for instance when $F$ satisfies
$|F(x,0,0,X)-F(y,0,0,X)|\leq\omega(|x-y|)\|X\|$, for all
$x,y\in\overline{\Omega}$, $X\in\mathbb{S}^{N}$.

Finally, we assume that the Dirichlet problems associated to $F_{i}^{*}$ and
$(F_{i})_{*}$ are uniquely solvable (see Section 3) in the scalar sense
together with regularity of solutions in a suitable Sobolev sense. In other
words, in terms of Definition 1.1 (b),(d), we ask that $F_{i}$ satisfy the
following

$\displaystyle\lambda_{1}^{+}(F^{*}(\vartheta))>0,\qquad F,F^{*}\,\textrm{
satisfy }\eqref{H continuity},$ (H3) $F$ enjoys $W^{2,\varrho}$ regularity.
(H4)

Here and onward in the text, the drift $\gamma$ and the zero order term
$\vartheta$ in (H1) may be unbounded, and this is an advantage of our paper
over the usual literature [4, 42], even in the scalar case. We observe that,
in the system, the structure of each $F_{i}$ with respect to the zero order
term in (H1) could be taken in terms of functions $\varrho_{i}$, which gives
the possibility of prescribing different weights in (H3), see also Remark
2.10. We avoid including so many indexes in order to make the presentation
cleaner.

Existence and positiveness in (H3) are verified if $F^{*}$ is a proper
operator (nonincreasing in $r$) by [37], and we will see this extends for
nonproper operators as well, check Lemma 3.1 ahead. Meanwhile, (H4) will hold
true if $F$ is a convex (or concave) operator in $X$ and satisfy (1.12), see
Lemma 3.5 (consequently, under (1.12), $F^{*}$ fulfills (H4)). However, (H4)
also covers some nonconvex operators, for instance the asymptotic recession
profiles in [39] for which a $W^{2,\varrho}$ theory is available. In
particular, our results are valid for a class of Isaac’s operators which are
sufficiently close to a Bellman operator that has good regularity-estimates,
see Example 1.15. Indeed, in Section 9 we prove that they verify (H4) under
(9.1), even in the presence of unbounded coefficients.

It is worthwhile to mention that hypothesis (H4) is not overly restrective and
in fact it is a natural condition when dealing with comparison principles for
$L^{p}$-viscosity solutions, see [37, 44]. Instead in the universe of
$C$-viscosity solutions it is possible to skip it by the price of asking
stronger continuity assumptions on the coefficients which is not our intention
here, see [4]. Fully nonlinear equations with measurable ingredients were
introduced in [13] for which a more general $L^{p}$-viscosity notion of
solution is required. It is a modern theory which still develops, and results
in such direction with unbounded coefficients are rather involved and
delicate, see [26, 37, 46].

In this paper we treat the optimality of scalar spectral properties of fully
nonlinear operators, and we also exploit the differences arising in the case
of systems; both on a scenario with possible unbounded drift and weights.

######  Remark 1.2.

Note that hypothesis (H3) allows us to treat nonproper operators. This is
equivalent to ask $\lambda_{1}^{+}(F(\vartheta))>0$ when $F$ is a linear
operator. For systems, this condition on the operators $F_{1}$, $F_{2}$ is
somehow required in [30, 32] in terms of MP’s validity, see Definition 1.6.
Instead, in [43, Theorem 1] the coupling proposed does not fall upon the
nonlinearities but on the operators. On the other hand, the regularity
assumption (H4) for $F_{1}$, $F_{2}$ seems to be optimal in the fully
nonlinear case, with respect to the previous scalar works [4, 42], since there
is no need to assume neither convexity nor continuity on the data.

###  1.2 Statement of the main results

Let us consider the space $E_{\varrho}=W^{2,\varrho}(\Omega)\cap
C(\overline{\Omega})$. Our main results are in the sequel.

######  Theorem 1.3 (Existence, simplicity, and asymptotics).

Let $\Omega\subset{\mathbb{R}^{N}}$ be a bounded $C^{1,1}$ domain. Assume
$pq=1$, $\tau_{1},\tau_{2}$ satisfy (1.5), and $F_{1},F_{2}$ satisfy
(H1)–(H4). Then there exist two spectral curves

$\Lambda^{\pm}_{1}\,(\lambda)\,=\,(\,\lambda,\,\mu_{1}^{\pm}(\lambda)\,)\in\mathbb{R}^{2},\;\;\text{
for all }\lambda>0,$

in the first quadrant, corresponding to signed eigenfunctions
$\varphi_{1}^{\pm},\psi_{1}^{\pm}\in E_{\varrho}$ such that both the pairs
$\varphi_{1}^{+},\psi_{1}^{+}>0$ and $\varphi_{1}^{-},\psi_{1}^{-}<0$ satisfy
(1.10) in the strong sense.

The eigenfunctions $(\varphi_{1}^{+},\psi_{1}^{+})$ and
$(\varphi_{1}^{-},\psi_{1}^{-})$ are unique in the sense that any other
eigenfunction $(u^{\pm},v^{\pm})$ corresponding to
$\Lambda_{1}^{\pm}(\lambda)$ satisfies $u^{\pm}\equiv t\varphi^{\pm}$ and
$v^{\pm}\equiv t^{p}\psi^{\pm}$ for a suitable $t\in\mathbb{R}^{+}$.
Furthermore, if $(u,v)$ is a signed solution of (1.10), then necessarily
$(\lambda,\mu)\in\Lambda_{1}^{\pm}$.

Moreover, $\mu_{1}^{\pm}$ is continuous and strictly decreasing with
$\lambda$, and the following asymptotic behavior holds

$\mu_{1}^{\pm}(\lambda)\to\infty$ as $\lambda\to 0$,
$\mu_{1}^{\pm}(\lambda)\to 0$ as $\lambda\to\infty$. (1.13)

######  Remark 1.4.

For the explicit shape of $\Lambda_{1}^{\pm}$, see (2.12) ahead.

In what follows we deal with geometric properties of the first spectral curves
and their characterization. In what follows, for a parametrized curve
$\Sigma=\Sigma(\lambda)=(\lambda,\sigma(\lambda))$, where $\lambda>0$ and
$\sigma(\lambda)$ continous, we say that $\Sigma$ is above $\Lambda_{1}^{\pm}$
if $\mu_{1}^{\pm}(\lambda)<\sigma(\lambda)$ for every $\lambda>0$.

######  Theorem 1.5 (Local isolation).

Assume $pq=1$, $\tau_{1},\tau_{2}$ satisfy (1.5), $F_{1},F_{2}$ satisfy
(H1)–(H4), and take $\Lambda_{1}^{\pm}$ from Theorem 1.3. Then there exists a
curve $\Sigma=(\lambda,\sigma(\lambda))$, strictly above $\Lambda_{1}^{\pm}$,
such that $\sigma$ is strictly decreasing, $\sigma(\lambda)\to\infty$ as
$\lambda\to 0$, $\sigma(\lambda)\to 0$ as $\lambda\to\infty$, with the
property that: if $(\lambda,\mu)$ is an eigenvalue of (1.10) in the region

$\\{\,(\lambda,\mu)\in\mathbb{R}^{2}:\
\lambda>0,\;\,0<\mu<\sigma(\lambda)\,\\},$

then necessarily $(\lambda,\mu)\in\Lambda_{1}^{+}\cup\Lambda_{1}^{-}$. In
other words, in the first quadrant, below and slightly above
$\Lambda_{1}^{\pm}$ there are no other eigenvalues of (1.10).

Next we see how the region below each curve $\Lambda_{1}^{\pm}$ gives a
complete characterization of the plane $\mathbb{R}^{2}$ in terms of maximum
and minimum principles, and in terms of the solvability of the associated
Dirichlet problem. This extends [30] to viscosity solutions, and plays the
role of the condition $\lambda<\lambda_{1}^{\pm}$ in the scalar case.

######  Definition 1.6 (MP and mP).

We say that the maximum principle (MP) holds for (1.10) if any viscosity
subsolution of (1.10), that is, any solution pair $u,v\in
C(\overline{\Omega})$ of

$\displaystyle F_{1}[u]+\lambda\tau_{1}(x)|v|^{q-1}v\geq
0,\;\;\;F_{2}[v]+\mu\tau_{2}(x)|u|^{p-1}u\geq 0\;\;\textrm{ in
}\Omega,\quad\textrm{$u,v\leq 0$ on $\partial\Omega$}$ (1.14)

satisfies $u,v\leq 0$ in $\Omega$. Likewise, we say that the minimum principle
(mP) holds for (1.10) if $u,v\geq 0$ in $\Omega$ for any viscosity
supersolution pair $u,v\in C(\overline{\Omega})$ of

$\displaystyle F_{1}[u]+\lambda\tau_{1}(x)|v|^{q-1}v\leq
0,\;\;\;F_{2}[v]+\mu\tau_{2}(x)|u|^{p-1}u\leq 0\;\;\textrm{ in
}\Omega,\quad\textrm{$u,v\geq 0$ on $\partial\Omega$}.$ (1.15)

Let $\mathcal{C}_{1}^{+}$ be the open region in the first quadrant below
$\Lambda_{1}^{+}$, and similarly for $\mathcal{C}_{1}^{-}$ associated to
$\Lambda_{1}^{-}$, namely:

$\mathcal{C}_{1}^{\pm}:=\\{\,(\lambda,\mu)\in\mathbb{R}^{2}:\
\lambda>0,\;\,0<\mu<\mu_{1}^{\pm}(\lambda)\,\\}.$ (1.16)

######  Theorem 1.7 (Characterization of $\Lambda_{1}^{\pm}$).

Assume $pq=1$, $\tau_{1},\tau_{2}$ satisfy (1.5), $F_{1},F_{2}$ satisfy
(H1)–(H4), and let $\Lambda_{1}^{\pm}$ be as in Theorem 1.3. Then:

  1. (i)

$(\lambda,\mu)\in\overline{\mathcal{C}_{1}^{+}}\setminus\Lambda_{1}^{+}$ if,
and only if, MP holds for (1.10);

  2. (ii)

$(\lambda,\mu)\in\overline{\mathcal{C}_{1}^{-}}\setminus\Lambda_{1}^{-}$ if,
and only if, mP holds for (1.10).

Let us now consider the Dirichlet problem (1.4) in the viscosity sense, for
functions $f_{1},f_{2}\in L^{\varrho}(\Omega)$, $\varrho>N$, with $u,v\in
C(\overline{\Omega})$.

######  Theorem 1.8 (Solvability of the Dirichlet problem).

Assume $F_{1},F_{2}$ satisfy (H1)–(H4), $pq=1$, $\tau_{1},\tau_{2}$ satisfy
(1.5), and let $f_{1},f_{2}\in L^{\varrho}(\Omega)$.

  1. (i)

If $(\lambda,\mu)\in\mathcal{C}_{1}^{+}\cap\mathcal{C}_{1}^{-}$, then (1.4) is
solvable among viscosity solutions.

  2. (ii)

If $f_{1},f_{2}\leq 0$ a.e. and $(\lambda,\mu)\in\mathcal{C}_{1}^{+}$, then
(1.4) has a unique nonnegative solution pair in $E_{\varrho}$.

  3. (iii)

If $f_{1},f_{2}\geq 0$ a.e. and $(\lambda,\mu)\in\mathcal{C}_{1}^{-}$, then
(1.4) has a unique nonpositive solution pair in $E_{\varrho}$.

In other words, as long as we are in the region
$\mathcal{C}_{1}^{+}\cap\mathcal{C}_{1}^{-}$ we obtain the complete standard
solvability of the Dirichlet problem (1.4). The solvability up to
$\mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{-}$ may not hold in general (e.g.
[42, Theorem 1.8]), in contrast to the case of linear eigenvalues in [30].
However, if the pair $f_{1},f_{2}$ has the “good” sign, we do obtain
solvability in this larger region.

######  Remark 1.9.

In particular, Theorem 1.8 applied to $F_{1}=F_{2}=\mathcal{L}^{+}$,
$\tau_{1}=\tau_{2}=\tau$, $p=q=1$ (scalar case) gives the optimal range of
solvability in [44, Proposition 3.4]; see also Remark p.595 therein where the
problem of spectral properties for these operators under unbounded
coefficients was left open. See also our Theorem 3.4 in Section 3 for a priori
bounds of ABP type to solutions produced by Theorem 1.8 in the scalar case, as
well as necessary conditions which characterize their validity.

In what concerns the optimality of Theorem 1.8, we show that an anti-maximum
principle occurs when we move a little bit above the region
$\mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{-}$. This type of result is
essential, for instance, in bifurcation and resonance phenomena [3, 22, 23]. A
classical reference for it in the linear scalar case is [16] (see also [8]),
while its fully nonlinear scalar counterpart can be found in [4, Theorem 2.5].
Here we extend [4] to fully nonlinear Lane-Emden systems as follows.

######  Theorem 1.10 (Anti-maximum principle).

Let $F_{1},F_{2}$ satisfy (H1)–(H4), $pq=1$, $\tau_{1},\tau_{2}$ satisfy
(1.5), and $f_{i}\in L^{\varrho}(\Omega)$ with $f_{i}\not\equiv 0$, $i=1,2$,
$\varrho>N$. Then there exists a curve
$\Gamma=(\lambda,\bar{\gamma}(\lambda))$, depending on $f_{1},f_{2}$, which is
above $\Lambda_{1}^{\pm}$, where $\bar{\gamma}$ is stricly decreasing,
$\bar{\gamma}(\lambda)\to 0$ as $\lambda\to\infty$, and
$\bar{\gamma}(\lambda)\to\infty$ as $\lambda\to 0$, such that

  1. (i)

if $f_{1},f_{2}\leq 0$ a.e., $\Lambda_{1}^{+}$ is below or coincides with
$\Lambda_{1}^{-}$, and $(\lambda,\mu)$ is a pair between $\Lambda_{1}^{-}$ and
$\Gamma$, then any solution pair $u,v\in C(\overline{\Omega})$ of (1.4)
satisfies $u,v<0$ in $\Omega$;

  2. (ii)

if $f_{1},f_{2}\geq 0$ a.e., $\Lambda_{1}^{-}$ is below or coincides with
$\Lambda_{1}^{+}$, and $(\lambda,\mu)$ is a pair between $\Lambda_{1}^{+}$ and
$\Gamma$, then any solution pair $u,v\in C(\overline{\Omega})$ of (1.4)
verifies $u,v>0$ in $\Omega$.

We highlight that Theorem 1.10 is new even in the case of the standard Lane-
Emden system involving the Laplacian operator, i.e. when $F_{1}=F_{2}=\Delta$.
Up to our knowledge, this is the first result on anti-maximum principle
regarding strongly coupled systems.

We also prove an existence result for the region above $\Lambda_{1}^{+}$ and
$\Lambda_{1}^{-}$ when $p=q=1$. Let us consider the pairs
$(\lambda^{\pm}_{1},\lambda^{\pm}_{1})=(\lambda_{1}^{\pm}(F_{1},F_{2}),\lambda_{1}^{\pm}(F_{1},F_{2}))$
in the intersection of the curve $\Lambda_{1}^{\pm}$ with the line
$\lambda=\mu$ (cf. Sections 2.3 and 2.4). Then we define the following
quantity:

$\lambda_{2}=\lambda_{2}(F_{1},F_{2},\Omega):=\inf\\{\lambda>\max\\{\lambda_{1}^{+}(F_{1},F_{2}),\lambda_{1}^{-}(F_{1},F_{2})\\}:\,(\lambda,\lambda)\text{
is an eigenvalue of \eqref{LE}}\\},$

which could be infinite. Denote by $K$ the first quadrant on the plane
$(\lambda,\mu)$.

######  Theorem 1.11 (The second spectral curve and the Dirichlet problem).

Let $F_{1},F_{2}$ satisfy (H1)–(H4), $\tau_{1},\tau_{2}$ satisfy (1.5), and
$pq=1$.

  1. (i)

If $\lambda_{2}<\infty$, then there exists a curve
$\Lambda_{2}=(\lambda,\mu_{2}(\lambda))$ lying in the region
$K\setminus\overline{\mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{-}}$. Moreover,
$\Lambda_{2}$ is such that each point $(\lambda,\mu)$ on $\Lambda_{2}$ is an
eigenvalue for (1.10). Also, $\mu_{2}(\lambda)$ is continuous, stricly
decreasing, and satisfies

$\mu_{2}(\lambda)\to\infty,\text{ as }\lambda\to 0,\quad\mu_{2}(\lambda)\to
0\text{ as }\lambda\to\infty.$

  2. (ii)

Further, if $p=q=1$, the Dirichlet problem (1.4) is solvable for
$f_{1},f_{2}\in L^{\varrho}(\Omega)$, $\varrho>N$, when $(\lambda,\mu)$
belongs to the region

$\left(\mathcal{C}_{2}\setminus\overline{\mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{-}}\,\right)$

where $\mathcal{C}_{2}$ the region below $\Lambda_{2}$ in $K$ if
$\lambda_{2}<\infty$, while $\mathcal{C}_{2}=K$ if $\lambda_{2}=\infty$.

######  Remark 1.12.

The explicit parametrization of $\Lambda_{2}$ is given in (7.3) ahead.

This result is an extension for systems of [4, Theorem 2.4]. We mention that
one may have $\lambda_{2}=+\infty$ if for instance $F_{1},F_{2}$ are not
symmetric, see [4]. Meanwhile, $\lambda_{2}<+\infty$ in the scalar case when
$F_{1}=F_{2}$ is a Pucci’s radial operator and $\tau_{1}=\tau_{2}=1$, see
[12]. Note that if $\tau_{1}=\tau_{2}=1$ then
$\lambda_{2}(\Delta,\Delta)=\lambda_{2}(\Delta)$ (the second eigenvalue of the
Laplacian operator). In general, finding higher eigenvalues for systems is a
difficult issue and it seems that only particular cases involving the
Laplacian operator are available. We quote a one dimensional picture displayed
in [11, Section 3] for $pq=1,$ and [15] for a higher dimensional scenario when
$p=q=1$; both explore the method of reduction by inversion which transforms
the second order system into single equation of higher order. Here instead we
use a degree-theoretical approach which allows us to deal with fully nonlinear
operators in any dimension when $p=q=1$; the general case $pq=1$ is still
open.

###  1.3 Examples and applications

We start by highlighting that the curves $\Lambda_{1}^{+}$ and
$\Lambda_{1}^{-}$ obtained in Theorem 1.3 can be different when the operator
is not linear, as shows the following example.

######  Example 1.13.

In light of [12], one may consider the Fucik-like spectrum

$Lu+\lambda u^{+}-\frac{\lambda}{\kappa}\,u^{-}=0$,

associated to the linear operator $Lu:=\mathrm{tr}(A(x)D^{2}u)+\gamma(x)\cdot
Du$, where $\kappa>0$ is fixed, which can be viewed as the spectrum of a
nonlinear convex or concave operator given by

$F_{1}[u]:=\max\\{Lu,\kappa Lu\\}=-\lambda u$ if $\kappa\geq 1$,
$F_{2}[u]:=\min\\{Lu,\kappa Lu\\}=-\lambda u$ if $\kappa\leq 1$.

Now let us fix $\kappa>1$ and $p,q>0$ such that $pq=1$. From [32], there
exists a first positive eigenvalue-parameter $\sigma$ (see Section 2.4) and an
eigenfunction pair $(\varphi,\psi)$ so that

$L\varphi+\sigma\psi^{q}=0$, $L\psi+\sigma\varphi^{p}=0$, $\varphi,\psi>0$ in
$\Omega$, $\varphi,\psi=0$ on $\partial\Omega$. (1.17)

Then the pair $\varphi_{1}^{+}:=\varphi$, $\psi_{1}^{+}:=t\psi$, with $t>0$ to
be chosen, solves

$F_{1}[\varphi^{+}_{1}]=\max\\{L\varphi,\,\kappa
L\varphi\\}=-\sigma\psi^{q}=-\sigma t^{-q}(\psi^{+}_{1})^{q}$,

$F_{2}[\psi^{+}_{1}]=\min\\{tL\psi,\,\kappa tL\psi\\}=-\kappa
t\sigma\varphi^{p}=-\kappa t\sigma(\varphi^{+}_{1})^{p}$,

so it is a positive eigenfunction pair if one choses
$t:=\kappa^{-\frac{1}{q+1}}$. Moreover, this eigenpair is unique up to scaling
by our Theorem 1.3, and we conclude
$\lambda_{1}^{+}(F_{1},F_{2})=\kappa^{\frac{q}{q+1}}\sigma$.

Analogously, the pair given by $\varphi_{1}^{-}:=-\varphi$,
$\psi_{1}^{-}:=-s\psi$ solves

$F_{1}[\varphi^{-}_{1}]=\max\\{-L\varphi,-\kappa
L\varphi\\}=\kappa\sigma\psi^{q}=-\kappa\sigma
s^{-q}\,|\psi^{-}_{1}|^{q-1}\psi^{-}_{1}$,

$F_{2}[\psi^{-}_{1}]=\min\\{-sL\psi,-\kappa
sL\psi\\}=s\sigma\varphi^{p}=-s\sigma\,|\varphi^{-}_{1}|^{p-1}\varphi_{1}^{-}$,

and becomes a negative eigenfunction pair when $s:=\kappa^{\frac{1}{q+1}}$ –
again unique up to scaling by Theorem 1.3, and
$\lambda_{1}^{-}(F_{1},F_{2})=\kappa^{\frac{1}{q+1}}\sigma$, compare with the
scalar case in [4, Example 3.10].

Now, since $\kappa>1$, then one always has
$\lambda_{1}^{+}(F_{1},F_{2})\neq\lambda_{1}^{-}(F_{1},F_{2})$, for $q\neq 1$,
and so by scaling one recovers that the two parallel curves $\Lambda^{+}_{1}$
and $\Lambda^{-}_{1}$ are different, see Section 2.4. Furthermore,
$\Lambda_{1}^{+}$ stays below $\Lambda_{1}^{-}$ if $q<1$; while
$\Lambda_{1}^{+}$ lies above $\Lambda_{1}^{-}$ if $q>1$.

On the other hand, is also simple to verify that for $\kappa>1$,
$\lambda_{1}^{+}(F_{1},F_{1})=\sigma<\kappa\sigma=\lambda_{1}^{-}(F_{1},F_{1})$
and
$\lambda_{1}^{-}(F_{2},F_{2})=\sigma<\kappa\sigma=\lambda_{1}^{+}(F_{2},F_{2})$,
for all $p,q>0$ with $pq=1$.

More generally, one can also consider different operators $L_{1},L_{2}$ in
(1.17).

The next examples comprise important classes of fully nonlinear operators for
which all our results apply, being novelties even in the scalar case in the
context of unbounded drift and weight.

######  Example 1.14.

Simple prototypes we may have in mind are extremal operators involving
Pucci’s, for instance $F_{1}=F_{1}^{*}=\mathcal{L}^{+}$,
$F_{2}=(F_{2})_{*}=\mathcal{L}^{-}$, with $\vartheta=\ell\theta$, for some
$\ell>0$ and $\theta\in L^{\varrho}(\Omega)$ satisfying $\theta\gneqq 0$ a.e.
in $\Omega$. They obviously fulfill (H1), (H2), (1.12), and (H4). Moreover,
recall $\lambda_{1}^{+}(\mathcal{L}^{+}_{0}(\theta))>0$ by [37] (see also our
Proposition 2.9), and
$\lambda_{1}^{+}(\mathcal{L}^{-}(\vartheta))=\lambda_{1}^{-}(\mathcal{L}^{+}(\vartheta))\geq\lambda_{1}^{+}(\mathcal{L}^{+}(\vartheta))$.
Thus (H3) is verified for $F_{1}$, $F_{2}$ if one chooses
$\ell<\lambda_{1}^{+}(\mathcal{L}^{+}_{0}(\theta))$ as in (3.3), see Lemma
3.1.

######  Example 1.15.

A bit more sophisticated model case arising from control theory are Hamilton-
Jacobi-Bellman-Isaac’s type operators [4, 23, 42, 38], with unbounded
coefficients, such as

$\displaystyle\textstyle
F_{1}[w]=\sup_{s\in\mathbb{N}}\inf_{t\in\mathbb{N}}L_{s,t}(w)\,,\qquad
F_{2}[w]=\inf_{s\in\mathbb{N}}\sup_{t\in\mathbb{N}}L_{s,t}(w),$ (1.18)

wher $L_{s,t}$ for $s,t\in\mathbb{N}$ is a linear operator in the form

$\displaystyle L_{s,t}[w]=\mathrm{tr}(A_{s,t}(x)D^{2}w)+\gamma_{s,t}(x)\cdot
Dw+\ell\vartheta_{s,t}(x)w,\quad\ell<\lambda_{1}^{+}(\mathcal{L}_{0}^{+}(\vartheta)),\qquad$
(1.19)
$\displaystyle|\gamma_{s,t}|\leq\gamma,\;|\vartheta_{s,t}|\leq\vartheta,\;\gamma,\vartheta\in
L^{\varrho}_{+}(\Omega),\;\alpha I\leq A_{s,t}\leq\beta I,\;A_{s,t}\in
C(\overline{\Omega})\textrm{ uniformly in }s,t\in\mathbb{N},$

for $0<\alpha\leq\beta$. For instance, when $L_{s,t}\equiv L_{0,t}$ for all
$s\in\mathbb{N}$ then $F_{1}$ and $F_{2}$ are called Bellman operators, which
are concave and convex, respectively. In the general case, $F_{1}$ and $F_{2}$
are neither convex nor concave, and are called Isaac’s operators. Again, (H3)
holds for $F_{1}$, $F_{2}$ under (1.19) as in (3.3), see Lemma 3.1; while we
show (H4) for (1.18)–(1.19) under (9.1) in Section 9.

#### Results in the superlinear and sublinear regimes

As a byproduct of our arguments, we complement a study on maximum principles
in small domains for Lane-Emden systems. Besides being of independent interest
[7], they play an important role in symmetry problems [6, 19], aside from
spectral constructions when the domain is not smooth [42]. It also appeared in
[34] as the main tool to derive a Unique Continuation Principle of radial
fully nonlinear type in the case $pq\geq 1$. For $pq=1$, an explicit form was
previously/independently proved in [30, Theorem 1.3, Corollary 1.1], under a
smallness hypothesis on the weights. Here we instead make it an alternative,
by asking either the domain or the weighs to be small. This in particular
extends and unifies [37, Lemma 5.4] (for domains with small measure) and [44,
the maximum principle in Proposition 3.4] (for operators with small weight)
even in the scalar case. Furthermore, in what concerns the system, the result
is valid not only for $pq=1$ but also for $pq\geq 1$.

######  Theorem 1.16.

Let $F_{1},F_{2}$ satisfy (H1). Let $\tau_{1},\tau_{2}\in
L^{\varrho}(\Omega)$, $\varrho>N$, $pq\geq 1$, and $\lambda,\mu\geq 0$. Then
the following MP result holds.

  * (i)

Assume (1.12), (H2) hold for $F_{i}^{*}$ with
$\lambda_{1}^{+}(F^{*}_{i}(\vartheta))>0$, $i=1,2$. Then, there exists
$\varepsilon_{0}>0$, depending on
$N,\varrho,p,q,\alpha,\beta,\lambda,\mu,\|\gamma\|_{L^{\varrho}(\Omega)}$,
$\|\vartheta\|_{L^{\varrho}(\Omega)}$, $\|\tau_{1}\|_{L^{\varrho}(\Omega)}$,
$\|\tau_{2}\|_{L^{\varrho}(\Omega)}$, $\|u\|_{\infty}$, $\|v\|_{\infty}$,
$\mathrm{diam}(\Omega)$, and $\lambda_{1}^{+}(F_{i}^{*}(\vartheta))$, such
that if

either $|\Omega|\leq\varepsilon_{0}$ or
$\min\\{\,\|\tau_{1}\|_{L^{N}(\Omega)}\,\|\tau_{2}\|^{q}_{L^{N}(\Omega)},\,\|\tau_{1}\|^{p}_{L^{N}(\Omega)}\,\|\tau_{2}\|_{L^{N}(\Omega)}\\}\leq\epsilon_{0}$,

then any viscosity subsolution pair $u,v\in C(\overline{\Omega})$ of

$\displaystyle F_{1}[u]+\lambda\tau_{1}(x)|v|^{q-1}v\geq 0,\quad
F_{2}[v]+\mu\tau_{2}(x)|u|^{p-1}u\geq 0\textrm{ in }\Omega,\quad u,v\leq
0\textrm{ on }\partial\Omega,$

satisfies $u,v\leq 0$ in $\Omega$.

  * (ii)

Assume (1.12), (H2) hold for $(F_{i})_{*}$ with
$\lambda_{1}^{+}((F_{i})_{*}(\vartheta))>0$, $i=1,2$. Then, there exists
$\varepsilon_{0}>0$, depending on
$N,\varrho,p,q,\alpha,\beta,\lambda,\mu,\|\gamma\|_{L^{\varrho}(\Omega)}$,
$\|\vartheta\|_{L^{\varrho}(\Omega)}$, $\|\tau_{1}\|_{L^{\varrho}(\Omega)}$,
$\|\tau_{2}\|_{L^{\varrho}(\Omega)}$, $\|u\|_{\infty}$, $\|v\|_{\infty}$,
$\mathrm{diam}(\Omega)$, and $\lambda_{1}^{+}((F_{i})_{*}(\vartheta))$, such
that if

either $|\Omega|\leq\varepsilon_{0}$ or
$\min\\{\,\|\tau_{1}\|_{L^{N}(\Omega)}\,\|\tau_{2}\|^{q}_{L^{N}(\Omega)},\,\|\tau_{1}\|^{p}_{L^{N}(\Omega)}\,\|\tau_{2}\|_{L^{N}(\Omega)}\\}\leq\epsilon_{0}$,

then for any $u,v\in C(\overline{\Omega})$ viscosity supersolution of

$F_{1}[u]+\lambda\tau_{1}(x)|v|^{q-1}v\leq 0,\quad
F_{2}[v]+\mu\tau_{2}(x)|u|^{p-1}u\leq 0\text{ in }\Omega,\quad u,v\geq 0\text{
on }\partial\Omega,$

one has $u,v\geq 0$ in $\Omega$.

If $pq=1$, then $\varepsilon_{0}$ does not depend on a bound from above of the
$L^{\infty}$ norms of $u,v$.

On the other hand, the eigenvalue problem furnishes unique solvability in the
sublinear regime.

######  Theorem 1.17 (Sublinear regime).

Assume $F_{1},F_{2}$ verify (H1)–(H4). Let $f_{i}\in L^{\varrho}(\Omega)$ with
$f_{i}\leq 0$ a.e. in $\Omega$, $i=1,2$, and $p,q>0$ such that $pq<1$. Then
the problem (1.4) is uniquely solvable among positive viscosity solutions for
all $\lambda,\mu>0$.

Problems of this nature, for instance involving the Laplacian operator, have
been studied in [20, Theorem 3] [32], and [10, Theorem 7.1]. We also mention
that uniqueness results imply that solutions inherit all symmetries of the
problem. For example, if the operator and its domain are radially symmetric,
then so is the solution. Furthermore, uniqueness simplifies the dynamics of
evolution problems, and in many cases provides global stability properties of
equilibrium.

###  1.4 Structure of the paper

The rest of the paper is organized as follows. In Section 2 we recall some
preliminary facts and definitions. In particular, we define the notion of
principal eigenvalues for the system (1.10).

Section 3 is devoted to the study of the scalar case, namely we prove how
principal eigenvalues relate to maximum principles of ABP type, from which we
obtain Theorem 1.16 as a consequence.

In Section 4 we prove some auxiliary results for systems, in particular
sufficient conditions that imply uniqueness of solutions to systems (up to
scaling), and a priori bounds for the first eigenvalue.

Section 5 addresses the main properties of the first eigenvalue problem when
$pq=1$. It contains the proofs of Theorems 1.3, 1.5, 1.7 and 1.8 (i).

In Section 6 we deal with the anti-maximum principle, proving Theorem 1.10.

In Section 7 we treat the second eigenvalue problem, namely Theorem 1.11.

Section 8 is dedicated to solvability of the Dirichlet problem when the
functions $f_{i}$ have a sign. In this spirit we develop a unified proof for
both Theorem 1.8 (ii)–(iii) and Theorem 1.17.

Finally, Section 9 is devoted to $W^{2,\varrho}$ regularity of Isaac’s
operators in Example 1.15.

##  2 Preliminaries

In this section we begin by recalling some of the different notions of
viscosity solutions and their equivalence under regularity of the data. We
then recall important results such as the Alexandrov-Bakelman-Pucci (ABP for
short) maximum principle, the strong maximum principle and Hopf’s lemma. In
the second part we introduce the notion of principal eigenvalues and comment
on the scaling properties of the system (1.4). All these results are used
throughout the paper.

###  2.1 Some known results

Let us start by recalling the definition of Pucci’s operators

$\displaystyle\textstyle\mathcal{M}^{+}_{\alpha,\beta}(X):=\sup_{\alpha I\leq
A\leq\beta
I}\mathrm{tr}(AX)\,,\quad\mathcal{M}^{-}_{\alpha,\beta}(X):=\inf_{\alpha I\leq
A\leq\beta I}\mathrm{tr}(AX),$ (2.1)

and of viscosity and strong solutions in what follows.

######  Definition 2.1.

Let $f\in L^{\varsigma}_{\textrm{loc}}(\Omega)$ for some $\varsigma\geq N$,
and $F$ an operator satisfying (H1). We say that $u\in C(\Omega)$ is an
$L^{\varsigma}$-viscosity subsolution $($resp. supersolution$)$ of $F[u]=f(x)$
in $\Omega$ if whenever $\phi\in W^{2,\varsigma}_{\mathrm{loc}}(\Omega)$,
$\varepsilon>0$, and $\mathcal{O}\subset\Omega$ open are such that

$\displaystyle
F(x,u(x),D\phi(x),D^{2}\phi(x))-f(x)\leq-\varepsilon\qquad(F(x,u(x),D\phi(x),D^{2}\phi(x))-f(x)\geq\varepsilon)$

for a.e. $x\in\mathcal{O}$, then $u-\phi$ cannot have a local maximum
$($minimum$)$ in $\mathcal{O}$. In this case we also say that $u$ is a
_viscosity solution of the inequality_ $F[u]\geq f(x)$ (resp. $F[u]\leq f(x)$)
in $\Omega$.

A strong subsolution (resp. supersolution) belongs to
$W^{2,\varsigma}_{\mathrm{loc}}(\Omega)$ and satisfies the inequality
$F[u]\geq f(x)$ (resp. $F[u]\leq f(x)$) at almost every point $x\in\Omega$.

In each situation, a solution $u\in C(\Omega)$ is meant to be both subsolution
and supersolution.

We now comment on the equivalence of these definitions.

  * •

The notions of $L^{\varrho}$-viscosity and strong solutions are equivalent
whenever the solution belongs to the space
$W^{2,\varrho}_{\mathrm{loc}}(\Omega)$, see [25, Theorem 3.1, Proposition
9.1].

  * •

Under (H1), the concepts of $L^{\varrho}$ and $L^{N}$ viscosity solutions are
also equivalent whenever $\varrho>N$ and $f\in
L^{\varrho}_{\mathrm{loc}}(\Omega)$, see [36, Proposition 2.9] (observe that
$L^{\varrho}_{\mathrm{loc}}(\Omega)\subset L^{N}_{\mathrm{loc}}(\Omega)$).

Thus, given $f\in L^{\varrho}_{\mathrm{loc}}(\Omega)$, throughout the text, we
say simply viscosity solution of $F[u]=f$ to mean an $L^{N}$-viscosity
solution. This in turn is equivalent to be a strong solution when hypothesis
(H4) is in force.

######  Remark 2.2.

In order to unify the notation, we always assume $f,\vartheta\in L^{\varrho}$
by means of producing $C^{1,\alpha}$ solutions under (1.12) as in Proposition
2.8 (such strategy was also employed in [4] to treat nonconvex operators),
despite sometimes this integrability can be relaxed to $L^{N}$.

Next we recall the ABP maximum principle for proper operators with unbounded
drift (for a proof see [27, Proposition 2.8]). Recall from (1.11) the notation

$\mathcal{L}_{0}^{\pm}[u]:=\mathcal{L}_{0}^{\pm}(x,Du,D^{2}u)=\mathcal{M}^{\pm}(D^{2}u)\pm\gamma(x)|Du|,\qquad\mathcal{L}^{\pm}[u]:=\mathcal{L}_{0}^{\pm}[u]\pm\vartheta(x)|u|.$

######  Proposition 2.3 (ABP).

Let $f\in L^{N}(\Omega)$, $\gamma\in L_{+}^{\varrho}(\Omega)$ for $\varrho>N$,
and $u\in C(\overline{\Omega})$ be a viscosity solution of
$\mathcal{L}_{0}^{+}[u]\geq f(x)$ in $\Omega^{+}$ (resp.
$\mathcal{L}^{-}_{0}[u]\leq f(x)$ in $\Omega^{-}$), where
$\Omega^{\pm}=\Omega\cap\\{\pm u>0\\}$. Then

$\displaystyle\textstyle\max_{\overline{\Omega}}u\leq\max_{\partial\Omega}u^{+}+C\,\|f^{-}\|_{L^{N}(\Omega)}\quad(\mathrm{resp.}\;\min_{\overline{\Omega}}u\geq\min_{\partial\Omega}(-u^{-})-C\,\|f^{+}\|_{L^{N}(\Omega)}),$
(2.2)

for a universal constant
$C=C(N,\alpha,\beta,\|\gamma\|_{\varrho},\mathrm{diam}(\Omega))>0$, which is
bounded if theses quantities are bounded from above. We denote this constant
by $C_{A}$.

######  Remark 2.4.

Recall that $F$ is called proper if $F(x,r,\xi,X)\leq F(x,s,\xi,X)$ for $r\geq
s$. Therefore, the previous statement can be applied to proper operators,
since for instance

$\mathcal{L}^{+}_{0}[u]\geq F(x,0,Du,D^{2}u)\geq
F(x,u,Du,D^{2}u)=F[u]\;\;\textrm{ in }\Omega^{+}.$

A consequence of ABP is the following result on the stability of viscosity
solutions (see [44, Theorem 4], which is based on [13, Theorem 3.8]).

######  Proposition 2.5 (Stability).

Let $F$, $F_{k}$ be operators satisfying (H1), $f,\,f_{k}\in
L^{\varrho}(\Omega)$. Let $u_{k}\in C(\Omega)$ be a viscosity solution of
$F_{k}[u_{k}]\geq f_{k}(x)$ in $\Omega$ (resp. $\leq f(x)$) for all
$k\in\mathbb{N}$. Suppose $u_{k}\to u$ in $L_{\mathrm{loc}}^{\infty}(\Omega)$
as $k\to\infty$ and that, for each ball $B\subset\subset\Omega$ and
$\varphi\in W^{2,\varrho}(B)$, setting

$\displaystyle
g_{k}(x):=F_{k}(x,u_{k},D\varphi,D^{2}\varphi)-f_{k}(x)\,\,\,\,\,\text{and}\,\,\,\,g(x):=F(x,u,D\varphi,D^{2}\varphi)-f(x),$

we have $\|(g_{k}-g)^{+}\|_{L^{\varrho}(B)}\to 0$ as $k\to\infty.$ (resp.
$(\|(g_{k}-g)^{-}\|_{L^{\varrho}(B)})\to 0$ as $k\to 0$). Then $u$ is a
viscosity solution of $F[u]\geq f(x)$ (resp. $\leq f(x)$) in $\Omega$.

Next we recall some important results concerning the strong maximum principle
and the Hopf lemma from [45]. We often refer to them simply by SMP and Hopf
along the text.

######  Proposition 2.6 (SMP).

Let $\Omega$ be a $C^{1,1}$ domain and $u$ a viscosity solution of
$\mathcal{L}^{-}[u]\leq 0$, $u\geq 0$ in $\Omega$, where $\gamma,\vartheta\in
L^{\varrho}_{+}(\Omega)$. Then either $u>0$ in $\Omega$ or $u\equiv 0$ in
$\Omega$.

######  Proposition 2.7 (Hopf).

Let $\Omega$ be a $C^{1,1}$ domain and $u$ a viscosity solution of
$\mathcal{L}^{-}[u]\leq 0$, $u>0$ in $\Omega$, where $\gamma,\vartheta\in
L^{\varrho}_{+}(\Omega)$. If $u(x_{0})=0$ for some $x_{0}\in\partial\Omega$,
then $\varliminf_{t\to 0^{+}}{u(x_{0}+t\nu)}/{t}>0$, where $\nu$ is the
interior unit normal vector to $\partial\Omega$ at $x_{0}$.

In [45], Propositions 2.6 and 2.7 are proved for $\vartheta\equiv 0$, but the
same proofs there work for any coercive operator, in particular for
$\mathcal{L}^{-}=\mathcal{L}_{0}^{-}-\vartheta(x)u$ since $u\geq 0$; see also
[46].

To conclude we recall $C^{1,\alpha}$ regularity estimates for equations with
unbounded drift from [37]. Since Theorem 1 therein was stated for bounded zero
order terms, we briefly explain how to deduce them also for merely
$L^{\varrho}$-integrable ones.

######  Proposition 2.8.

Assume $F$ satisfies (H1), $f\in L^{\varrho}(\Omega)$, $\varrho>N$, and
$\Omega\subset{\mathbb{R}^{N}}$ is a bounded domain. Let $u$ be a viscosity
solution of $F[u]=f(x)$ in $\Omega$. Then, there exists $\alpha\in(0,1)$ and
$\theta_{0}=\theta_{0}(\alpha)$, depending on
$N,\varrho,\lambda,\Lambda,\|\gamma\|_{L^{\varrho}(\Omega)}$, such that if
(1.12) holds for all $r\leq\min\\{r_{0},\mathrm{dist}(x,\partial\Omega)\\}$,
for some $r_{0}>0$ and for all $x\in\Omega$, this implies that $u\in
C^{1,\alpha}_{\mathrm{loc}}(\Omega)$ and for any subdomain
$\Omega^{\prime}\subset\subset\Omega$,

$\displaystyle\|u\|_{C^{1,\alpha}(\overline{\Omega^{\prime}})}\leq
C\,\\{\,\|u\|_{L^{\infty}(\Omega)}+\|f\|_{L^{p}(\Omega)}\\}$

where $C$ depends on
$\,r_{0},N,\varrho,\lambda,\Lambda,\alpha,\|\gamma\|_{L^{\varrho}(\Omega)},\|\vartheta\|_{L^{\varrho}(\Omega)},\mathrm{diam}(\Omega)$,
$\mathrm{dist}(\Omega^{\prime},\partial\Omega)$.

If in addition, $\partial\Omega\in C^{1,1}$ and $u\in C(\overline{\Omega})\cap
C^{1,\tau}(\partial\Omega)$, then there exists $\alpha\in(0,\tau)$ and
$\theta_{0}=\theta_{0}(\alpha)$, depending on
$N,\varrho,\lambda,\Lambda,\|\gamma\|_{L^{\varrho}(\Omega)}$, so that if
(1.12) holds for some $r_{0}>0$ and for all $x\in\overline{\Omega}$, this
implies that $u\in C^{1,\alpha}(\overline{\Omega})$ and

$\displaystyle\|u\|_{C^{1,\alpha}(\overline{\Omega})}\leq
C\,\\{\,\|u\|_{L^{\infty}(\Omega)}+\|f\|_{L^{p}(\Omega)}+\|u\|_{C^{1,\tau}(\partial\Omega)}\\}$

where $C$ depends on
$\,r_{0},n,p,\lambda,\Lambda,\alpha$,$\|\gamma\|_{L^{\varrho}(\Omega)},\|\vartheta\|_{L^{\varrho}(\Omega)},\mathrm{diam}(\Omega),\partial\Omega$.

###### Proof.

We first observe that our structural hypothesis (H1) takes into account a
Lipschitz modulus of continuity as the zero order term. Moreover, when $\mu=0$
in [37, Theorem 1], one can perform a simpler rescaling of variable $W=N(0)$
as in [37, Remark 3.4], but instead we use directly (1.12) (for $\beta_{F}$ in
place of $\bar{\beta}_{F}$ there) and set $\widetilde{u}(x):=\frac{u(\sigma
x)}{W}$ in [37, Claim 3.2]. This allows us to achieve the regularity desired
for $\vartheta\in L^{\varrho}(\Omega)$, with the estimates depending on
$\|\vartheta\|_{L^{\varrho}(\Omega)}$. ∎

###  2.2 Principal scalar eigenvalues for proper operators

In the scalar case, for an operator $F$ satisfying (H1), (H2), we set

$\lambda_{1}^{\pm}\,(F(\vartheta))=\sup\left\\{\lambda\in\mathbb{R}\,,\,\Phi^{\pm}_{\lambda}\neq\emptyset\right\\},$

where

$\displaystyle\Phi^{+}_{\lambda}=\left\\{\phi:\ \phi>0\textrm{ in
}\Omega,\;F[\phi]+\lambda\vartheta(x)\phi\leq 0\textrm{ in }\Omega\right\\},$
$\displaystyle\Phi^{-}_{\lambda}=\left\\{\phi:\ \phi<0\textrm{ in
}\Omega,\;F[\phi]+\lambda\vartheta(x)\phi\geq 0\textrm{ in }\Omega\right\\}.$

Our goal is to show that these suprema are achieved accordingly to Definition
1.1 (b), i.e. there exist eigenfunctions $u^{\pm}$ such that
$-F[u]=\lambda_{1}^{\pm}u^{\pm}$ in $\Omega$ – for instance when $F=F_{i}$,
$i=1,2$.

The first step is to deduce it, in light of [37], for proper operators $F$
with unbounded drift and weight for which it holds (H1), (H2), (1.12), and
(H4).

######  Proposition 2.9.

Let $\Omega\subset{\mathbb{R}^{N}}$ be a bounded $C^{1,1}$ domain, $\tau\in
L^{\varrho}(\Omega)$, $\tau\gneqq 0$, $\varrho>n$, where $F$ is a proper
operator satisfying (H1), (H2), (1.12), and (H4), for $\gamma,\vartheta\in
L^{\varrho}_{+}(\Omega)$. Then $\lambda_{1}^{\pm}>0$ and $F$ has two signed
eigenfunctions $\varphi_{1}^{\pm}\in C^{1,\alpha}(\overline{\Omega})$ so that

$\displaystyle\textstyle
F[\varphi_{1}^{\pm}]+\lambda_{1}^{\pm}\tau(x)\varphi_{1}^{\pm}=0,\quad\pm\varphi_{1}^{\pm}>0\textrm{
in }\Omega,\quad\varphi_{1}^{\pm}=0\textrm{ on
}\partial\Omega,\quad\max_{\overline{\Omega}}\,(\pm\varphi_{1}^{\pm})=1.$

###### Proof.

Let us first observe that Proposition 2.9 is already proved in [37, Theorem
5.2] when $\vartheta$ is bounded. We stress that $C^{1,\alpha}$ regularity
estimates in Proposition 2.8 hold for unbounded $\vartheta$. On the other
hand, we note that the solvability asked in [37, hypothesis (H) of Theorem
5.2] is now ensured due to (H4), since in this case unique solvability of the
Dirichlet problem comes from [44, Theorem 1 (i), (ii)]. Moreover, the
existence result on first eigenvalues does not require the drift nor the zero
order term to be bounded. Indeed, the bound [37, (5.8)] is replaced by Lemma
5.7 there, with the blow-up argument comprising an unbounded zero order term
as well; see ahead Step 2 in the proof of our Proposition 4.5, by taking
$u=v$, $\tau_{1}=\tau_{2}$. ∎

Hence, for such $F$, the following ordering holds

$\displaystyle\lambda_{1}^{+}(F^{*}(\vartheta))=\lambda_{1}^{-}(F_{*}(\vartheta))\leq\lambda_{1}^{+}(F(\vartheta)),\lambda_{1}^{-}(F(\vartheta))\leq\lambda_{1}^{-}(F^{*}(\vartheta))=\lambda_{1}^{+}(F_{*}(\vartheta)),$
(2.3)

since $F^{*}$ is convex and $F_{*}$ is concave, see [4, Proposition 4.2] and
[42, Lemma 1.1].

######  Remark 2.10.

The following monotonicity property with respect to the weight holds:

if $\vartheta_{1}\leq\vartheta_{2}$ a.e. in $\Omega$ then
$\lambda_{1}^{+}(F(\vartheta_{1}))\geq\lambda_{1}^{+}(F(\vartheta_{2}))$.
(2.4)

They are instrumental in risk-sensitive control and probabilistic arguments,
see [2, 21].

###  2.3 Definition of principal eigenvalues for systems

Inspired by [7, 9, 42], we define the notion of principal eigenvalues for the
system (1.10) as follows:

$\lambda_{1}^{\pm}=\lambda_{1}^{\pm}\,(F_{1},F_{2})=\lambda_{1}^{\pm}\,(F_{1}(\tau_{1}),F_{2}(\tau_{2})):=\sup\left\\{\lambda\in\mathbb{R}\,,\,\Psi^{\pm}_{\lambda}\neq\emptyset\right\\},$

where

$\displaystyle\Psi^{+}_{\lambda}=\left\\{(\varphi,\psi);\;\varphi,\psi>0\textrm{
in }\Omega,\;F_{1}[\varphi]+\lambda\tau_{1}(x)\psi^{q}\leq
0,\;F_{2}[\psi]+\lambda\tau_{2}(x)\varphi^{p}\leq 0\textrm{ in
}\Omega\right\\},$
$\displaystyle\Psi^{-}_{\lambda}=\left\\{(\varphi,\psi);\;\varphi,\psi<0\textrm{
in }\Omega,\;F_{1}[\varphi]+\lambda\tau_{1}(x)|\psi|^{q-1}\psi\geq
0,\;F_{2}[\psi]+\lambda\tau_{2}(x)|\varphi|^{p-1}\varphi\geq 0\textrm{ in
}\Omega\right\\},$

with inequalities holding in the $L^{N}$-viscosity sense, for functions
$\varphi,\psi\in C(\overline{\Omega})$. When necessary, we will also highlight
the dependence of $\lambda_{1}^{\pm}$ on $\Omega$. Observe that

$\displaystyle\lambda_{1}^{\pm}(G_{1},G_{2})=\lambda_{1}^{\mp}(F_{1},F_{2}),\;\textrm{
for }\,G_{i}(x,r,p,X)=-F_{i}(x,-r,-p,-X).$ (2.5)

######  Remark 2.11.

By (2.3) and hypothesis (H3) on $F=F_{i}$, we have
$\lambda_{1}^{\pm}(F_{i}(\vartheta))>0$, $i=1,2$. Then we infer that
$\lambda_{1}^{\pm}(F_{1},F_{2})\geq 0$. Indeed, to fix the ideas let us
consider the $\lambda_{1}^{+}$ case. Taking the positive eigenfunctions
$\varphi_{1}^{+}$ and $\psi_{1}^{+}$ associated to
$\lambda_{1}^{+}(F_{1}(\vartheta))>0$ and
$\lambda_{1}^{+}(F_{2}(\vartheta))>0$ respectively, we have
$(\varphi_{1}^{+},\psi_{1}^{+})\in\Psi_{0}^{+}$, from which the desired bound
follows.

Let us also denote

$\displaystyle
m_{1}=\min\\{\lambda_{1}^{+}(F_{1},F_{2}),\lambda_{1}^{-}(F_{1},F_{2})\\},\;\;\quad
M_{1}=\max\\{\lambda_{1}^{+}(F_{1},F_{2}),\lambda_{1}^{-}(F_{1},F_{2})\\}.$
(2.6)

###  2.4 Scaling and asymptotic behavior

In this section we show some equivalent forms of problem (1.10) obtained by
means of a suitable scaling, and how to build a spectral curve starting from a
scalar-like eigenvalue. These observations are crucial in the proof of our
main results, as often it will be convenient to take $\lambda=\mu$ in (1.10).

Take $p,q>0$ with $pq=1$, and consider:

$\displaystyle-F_{1}[u]=\lambda\tau_{1}(x)|v|^{q-1}v,\quad-
F_{2}[v]=\lambda\tau_{2}(x)|u|^{p-1}u\quad\textrm{ in }\;\Omega,\quad
u=v=0\textrm{ on }\partial\Omega.$ (2.7)

Let us check that to study this system for $\lambda>0$ is, in a way,
equivalent to study (1.10) for $(\lambda,\mu)$ in the first quadrant.

Assume we have an eigenfunction pair $(u_{0},v_{0})$ associated to an
eigenvalue $\lambda_{0}>0$, i.e. $(u_{0},v_{0})\neq(0,0)$ viscosity solution
to

$\displaystyle-F_{1}[u_{0}]=\lambda_{0}\tau_{1}(x)|v_{0}|^{q-1}v_{0},\quad-
F_{2}[v_{0}]=\lambda_{0}\tau_{2}(x)|u_{0}|^{p-1}u_{0}\quad\textrm{ in
}\;\Omega\quad u_{0}=v_{0}=0\textrm{ on }\partial\Omega.$ (2.8)

for some $\lambda_{0}>0$. We infer that this implies the existence of a curve
of eigenvalues of the form
$(\lambda,\mu)\in\mathbb{R}^{+}\times\mathbb{R}^{+}$, with associated
eigenfunctions $u,v$ such that

$\displaystyle-F_{1}[u]=\lambda\tau_{1}(x)|v|^{q-1}v,\quad-
F_{2}[v]=\mu\tau_{2}(x)|u|^{p-1}u\quad\textrm{ in }\;\Omega\quad u=v=0\textrm{
on }\partial\Omega.$ (2.9)

(i.e, they solve (1.10)). Indeed, given $\lambda>0$, set

$\textstyle u=u_{0},\quad
v=\frac{\lambda_{0}^{p}}{\lambda^{p}}v_{0},\quad\text{ and
}\quad\mu=\frac{\lambda_{0}^{p+1}}{\lambda^{p}}.$ (2.10)

By the homogeneity assumption (H2), we see that (2.9) is satisfied. In other
words, given $\lambda_{0}>0$, (2.7) produces a spectral curve
$\Lambda_{\lambda_{0}}$ parametrized by

$\textstyle\Lambda_{\lambda_{0}}(\lambda)=(\lambda,\mu(\lambda)),\quad\text{
where }\;\mu(\lambda)=\frac{\lambda_{0}^{p+1}}{\lambda^{p}},\;\lambda>0.$
(2.11)

Observe that $\lambda\mapsto\mu(\lambda)$ is one-to-one, $\mu(\lambda)\to 0$
as $\lambda\to\infty$, $\mu(\lambda)\to\infty$ as $\lambda\to 0$. Moreover,
$\Lambda_{\lambda_{0}}\cap\Lambda_{\lambda_{0}^{\prime}}$ if
$\lambda_{0}\neq\lambda_{0}^{\prime}$, and
$\mathbb{R^{+}}\times\mathbb{R}^{+}=\cup_{\lambda_{0}}\Lambda_{\lambda_{0}}$.

From these comments one derives that, via the relation (2.10), to study (1.10)
for $(\lambda,\mu)\in\mathbb{R}^{+}\times\mathbb{R}^{+}$ is equivalent to
study (2.7), and that statements in the Introduction can be equivalently
written in terms of these scalings. Since in what follows we are going to
consider almost exclusively the first quadrant of the plane $(\lambda,\mu)$
(with the exception of Theorem 1.7), we will equivalently write (1.10) in the
form (2.7).

In particular, ahead in Section 5.1 we will prove that the principal
eigenvalues $\lambda_{1}^{\pm}=\lambda_{1}^{\pm}(F_{1},F_{2})$, defined in the
previous section, exist and are positive, being associated with positive
solutions of

$-F_{1}[u]=\lambda_{1}^{\pm}\tau_{1}(x)|v|^{q-1}v,\quad-
F_{2}[v]=\lambda_{1}^{\pm}\tau_{2}(x)|u|^{p-1}u\quad\textrm{ in
}\;\Omega,\quad u=v=0\textrm{ on }\partial\Omega.$

From this one builds two spectral curves

$\textstyle\Lambda^{\pm}_{1}\,(\lambda)\,=\,(\,\lambda,\,\mu_{1}^{\pm}(\lambda)\,),\quad\text{
where
}\;\mu^{\pm}_{1}(\lambda)=\frac{(\lambda_{1}^{\pm})^{p+1}}{\lambda^{p}},\;\;\text{
for all }\lambda>0.$ (2.12)

Each $\mu_{1}^{\pm}(\lambda)$ is strictly decreasing as a function of
$\lambda$, satisfying the asymptotic behavior (1.13) stated in Theorem 1.3.
Moreover, in view of the proof of Theorem 1.7, observe that saying that
$(\lambda,\mu)$ lies for instance below the curve $\Lambda_{1}^{+}$ and in the
first quadrant is completely equivalent to saying that

$\textstyle\mu<\frac{(\lambda_{1})^{p+1}}{\lambda^{p}}\;\;\Leftrightarrow\;\;\lambda_{0}:=(\mu\lambda^{p})^{\frac{1}{p+1}}<\lambda_{1}^{+}.$

######  Remark 2.12.

Just to justify other scaling prototypes we might find in the literature,
instead of (2.7) we could also have written either

$\displaystyle-F_{1}[u]=\tau_{1}(x)|v|^{q-1}v,\quad-
F_{2}[v]=\lambda\tau_{2}(x)|u|^{p-1}u\quad\textrm{ in }\;\Omega,\quad
u=v=0\textrm{ on }\partial\Omega;$

or

$\displaystyle-F_{1}[u]=\lambda\tau_{1}(x)|v|^{q-1}v,\quad-
F_{2}[v]=\tau_{2}(x)|u|^{p-1}u\quad\textrm{ in }\;\Omega,\quad u=v=0\textrm{
on }\partial\Omega;$

which are both equivalent to (2.7) whenever we are in the first quadrant.

On the other hand, we point out that we may reparametrize the curve (2.11) as

$\textstyle\Lambda_{\lambda_{0}}(a)=(\lambda(a),\mu(a)),\qquad\text{ where
}\mu(a)=a\lambda(a),\quad\text{ for
}a=\frac{\mu}{\lambda}=\frac{\lambda_{0}^{p+1}}{\lambda^{p+1}}.$

This way one recovers the notation and asymptotic behavior from [32],

$\lambda(a)=\frac{\lambda_{0}}{a^{1/(p+1)}}\to 0$ as $a\to+\infty$,
$\lambda(a)\to+\infty$ as $a\to 0$,

$\mu(a)=a\lambda=a^{\frac{p}{p+1}}\lambda_{0}\to+\infty$ as $a\to+\infty$,
$\mu(a)\to 0$ as $a\to 0$.

##  3 The scalar case with unbounded coefficients

We first recall that, in the case of proper operators with unbounded drift and
weights, existence of positive principal eigenvalues $\lambda_{1}^{\pm}$ with
associated eigenfunctions $\varphi_{1}^{\pm}$ is proved in Proposition 2.9. In
this section we start by extending the existence of eigenvalues for nonproper
operators.

######  Lemma 3.1.

Set $F_{0}[u]:=F[u]-\vartheta(x)u$, where $F$ satisfies (H1), (H2), (1.12),
and (H4). Then the quantity defined by

$\displaystyle\lambda_{1}^{+}(F({\vartheta}))\,:=\,\lambda_{1}^{+}(F_{0}({\vartheta}))-1$
(3.1)

is the first eigenvalue associated to a positive eigenfunction of the scalar
Dirichlet problem $F[u]+\lambda\vartheta(x)u=0$ in $\Omega$, $u=0$ on
$\partial\Omega$. Moreover, if $C_{A}$ is the ABP-constant in Proposition 2.3,
then

$\displaystyle\lambda^{+}_{1}(F(\vartheta))\geq\,\frac{1}{\,C_{A}\,\|\vartheta\|_{L^{N}(\Omega)}}-1.$
(3.2)

In particular, assumption (H3) is verified whenever
$\|\vartheta\|_{L^{N}(\Omega)}<\frac{1}{C_{A}}$.

An analogous result holds for
$\lambda_{1}^{-}(F(\vartheta))=\lambda_{1}^{+}(G(\vartheta))$ by applying it
to $G$, see (2.5).

###### Proof.

Notice that $F_{0}$ is a proper operator, i.e. $F_{0}(x,r,\xi,X)\leq
F_{0}(x,s,\xi,X)$ for $r\geq s$, since

$F_{0}(x,r,\xi,X)-F_{0}(x,s,\xi,X)\leq\vartheta(x)|r-s|-\vartheta(x)(r-s)=2\vartheta(x)(r-s)^{-}$
for $r,s\in\mathbb{R}$.

We evoke the existence and positivity of the first eigenvalue
$\lambda_{1}^{+}(F_{0}(\hat{\vartheta}))$ for the proper fully nonlinear
operator $F_{0}$ with unbounded drift $\gamma$ and weight $\hat{\vartheta}$
from [37]. Hence, by the definition of scalar eigenvalue, one derives the
first statement.

Now, if one writes $\vartheta=\ell\hat{\vartheta}$, for
$\ell=\|\vartheta\|_{L^{N}(\Omega)}$ and
$\|\hat{\vartheta}\|_{L^{N}(\Omega)}=1$, then by definition of
$\lambda^{+}_{1}$, one deduces
$\ell\lambda_{1}^{+}(F_{0}({\vartheta}))=\lambda_{1}^{+}(F_{0}(\hat{\vartheta}))$,
and so

$\displaystyle\lambda_{1}^{+}(F(\vartheta))>0\;\;\Leftrightarrow\;\;\ell<\lambda_{1}^{+}(F_{0}(\hat{\vartheta})).$
(3.3)

Let us check that (3.3) is verified if $\|\vartheta\|_{N}$ is sufficiently
small. We claim that $\lambda_{1}^{+}(F_{0}(\theta))\geq C>0$ uniformly in
$\theta$ whenever $\|\theta\|_{L^{N}(\Omega)}$ is fixed. Indeed, since
$\lambda_{0}=\lambda_{1}^{+}(F_{0}(\theta))$ is well defined and positively
attained by [37], then there exists a positive eigenfunction $\phi_{0}$
related to $\lambda_{0}$ such that

$F_{0}[\phi_{0}]=-\lambda_{0}\,\theta(x)\phi_{0}$, $\phi_{0}>0$ in $\Omega$,
$\phi_{0}=0$ on $\partial\Omega$.

Since $F_{0}$ is a proper operator, then
$\mathcal{L}^{+}_{0}[\phi_{0}]\geq-\lambda_{0}\,\theta(x)\phi_{0}$ in
$\Omega^{+}$ (see Remark 2.4). Therefore, ABP (Proposition 2.3) and
$\phi_{0}>0$ yield the existence of a universal constant $C_{A}$ such that

$\displaystyle\textstyle\sup_{\Omega}\phi_{0}\leq
C_{A}\,\lambda_{0}\,\sup_{\Omega}\phi_{0}\,\|\theta\|_{L^{N}(\Omega)}.$ (3.4)

In particular we achieve (3.3) by taking $\theta=\hat{\vartheta}$ and $\ell
C_{A}<1$. Equivalently, for $\theta=\vartheta$ we derive (3.2) due to relation
(3.1). ∎

######  Remark 3.2.

The bound from below (3.2) extends and improves [44, Proposition 3.3] to the
context of unbounded coefficients. In particular, with our spectral tools, the
proof of [44, Theorem 1] can now be considerably shorten, see also Remark in
p.595 there.

We now show how scalar principal eigenvalues are related to the validity of
the Alexandrov-Bakelman-Pucci estimate in the scalar case, in the following
sense.

######  Definition 3.3.

Let $F$ satisfy (H1). We say that ABP-MP (resp. ABP-mP) holds for $F$ in
$\Omega$ if whenever $f\in L^{N}(\Omega)$ and $u\in C(\overline{\Omega})$
viscosity solution of $F[u]\geq f(x)$ (resp. $F[u]\leq f(x)$) in $\Omega$,
then

$\displaystyle\textstyle\max_{\overline{\Omega}}u\leq
C_{B}\\{\max_{\partial\Omega}u^{+}+\|f^{-}\|_{L^{N}(\Omega)}\\}\;\;(\mathrm{resp.}\;\min_{\overline{\Omega}}u\geq
C_{B}\\{\min_{\partial\Omega}(-u^{-})-\,\|f^{+}\|_{L^{N}(\Omega)}\\}),$ (3.5)

for some positive constant $C_{B}$ not depending on the norm of $u$.

######  Theorem 3.4.

Let $\Omega$ be a bounded $C^{1,1}$ domain. Assume (H1), (1.12) on $F$,
(1.12), (H2) on $F^{*}$.

  1. (i)

If $\lambda_{1}^{+}(F^{*}(\vartheta))=\lambda_{1}^{-}(F_{*}(\vartheta))>0$
then ABP-MP holds for $F$ in $\Omega$, with $C_{B}$ depending on
$N,\varrho,\alpha,\beta,\|\gamma\|_{\varrho},\|\vartheta\|_{\varrho},\mathrm{diam}(\Omega),\lambda_{1}^{+}(F^{*}(\vartheta))$.
On the other hand, if $F$ satisfies (H2), (H4), and
$\lambda_{1}^{+}(F(\vartheta))\leq 0$ then ABP-MP does not hold for $F$;

  2. (ii)

If $\lambda_{1}^{-}(F^{*}(\vartheta))=\lambda_{1}^{+}(F_{*}(\vartheta))>0$
then ABP-mP holds for $F$ in $\Omega$, with $C_{B}$ depending on
$N,\varrho,\alpha,\beta,\|\gamma\|_{\varrho},\|\vartheta\|_{\varrho},\mathrm{diam}(\Omega),\lambda_{1}^{-}(F^{*}(\vartheta))$.
On the other side, if $F$ satisfies (H2), (H4), and
$\lambda_{1}^{-}(F(\vartheta))\leq 0$ then ABP-mP does not hold for $F$.

Note that, in order prove ABP-mP and ABP-MP, we do not need impose $F$
verifying (H2). This is good since, for instance, one may take $F^{*}$ to be
$\mathcal{L}^{+}$, which always satisfies (H2) even when $F$ does not. Of
course notice that, if $F$ satisfies (H2), then this is also the case for
$F^{*}$.

######  Lemma 3.5.

Let $\Omega\in C^{1,1}$ be a bounded domain. If $F$ is either a convex or
concave operator in the $X$-entry, for which it holds (H1) and (1.12), then
$F$ satisfies (H4). In particular, $\lambda_{1}^{+}(F(\vartheta))$ as in Lemma
3.1 is well defined for convex (or concave) operators satisfying only (H1),
(H2), and (1.12).

###### Proof.

By [49, Theorem 5.3] we already know that $F$ enjoys $W^{2,\varrho}$ interior
regularity estimates. Thus it is enough to obtain the global statement. Let
$u$ be a viscosity solution of $F[u]=f(x)$ in $\Omega$, where $f\in
L^{\varrho}(\Omega)$, $u=\psi$ on $\partial\Omega$ for some $\psi\in
W^{2,\varrho}(\Omega)$. Then by the local regularity we know that $u$ is a
strong solution. Thus, the $C^{1,\alpha}$ global regularity in Proposition 2.8
and the proof of Nagumo’s lemma in [37, Lemma 4.4] imply the desired global
regularity and estimates. ∎

###### Proof of Theorem 3.4.

We only show item (i), since (ii) is analogous. Assume $u\in
C(\overline{\Omega})$ is a viscosity solution of $F[u]\geq f(x)$ in $\Omega$.
We first notice that, if $\lambda_{1}^{+}(F(\vartheta))\leq 0$, then ABP-MP is
not satisfied. Indeed, in this situation we obtain

$F[\varphi_{1}^{+}]=-\lambda_{1}^{+}\vartheta(x)\varphi_{1}^{+}\geq 0\textrm{
in }\Omega,$

with $\varphi^{+}_{1}=0$ on $\partial\Omega$, but $\varphi_{1}^{+}>0$ in
$\Omega$. Consequently, ABP-MP does not hold in general.

Now set $\lambda_{1}^{+}:=\lambda_{1}^{+}(F^{*}(\vartheta))>0$. Let us show
that this is a sufficient condition for ABP-MP.

Step 1) Let us check that, if there exists a solution $\psi\in
W^{2,\varrho}(\Omega)\cup C^{1}(\overline{\Omega})$ of $F^{*}[\psi]\leq 0$ in
$\Omega$ so that $\psi>0$ in $\overline{\Omega}$, $\psi=1$ on
$\partial\Omega$, and $\psi\in[a,b]$ for some universal constants $0<a<b$,
then $F$ satisfies ABP-MP in $\Omega$, for a constant that depends also on
$a=\inf_{\Omega}\psi$ and $b=\sup_{\Omega}\psi$.

Set $D:=\|\psi\|_{C^{1}(\overline{\Omega})}$. Note that $v=\frac{u}{\psi}$ is
a viscosity solution of $F_{\psi}[v]\geq f(x)$ in $\Omega$, where

$\displaystyle
F_{\psi}(x,r,\xi,X):=F(x,r\psi,\,rD\psi+\psi\xi,\,rD^{2}\psi+\psi
X+2D\psi\otimes\xi).$ (3.6)

The operator $F_{\psi}$ satisfies (H1) with ellipticity constants $a\alpha$,
$b\beta$, with drift term $\gamma_{\psi}(x)=(2D+b)\gamma(x)$. Furthermore,
$F_{\psi}$ is a proper operator: indeed, by applying (H1) for $F$, and (H2)
for $F^{*}$, one finds

$F_{\psi}(x,r,\xi,X)-F_{\psi}(x,r,\eta,Y)\leq\mathcal{M}^{+}_{a\alpha,\,b\beta}\,(X-Y)+\gamma_{\psi}(x)|\xi-\eta|$,

$F_{\psi}(x,r,\xi,X)-F_{\psi}(x,s,\xi,X)\leq
F^{*}[(r-s)\psi]=(r-s)F^{*}[\psi]\leq 0$ for $r\geq s$.

Now ABP for proper operators with unbounded coefficients (Proposition 2.3 and
subsequent remark) produces the estimate (2.2) for $v=u/\psi$. In addition,
$u\leq 0$ in the set where $v\leq 0$, and $u=v\psi\leq bv$ in the set where
$v>0$, and so $u$ satisfies (3.5).

Step 2) Now we prove that $\lambda_{1}^{+}>0$ yields the existence of a
function $\psi$ as in Step 1.

Note that there exists a neighborhood of $\partial\Omega$ such that
$\varphi_{1}^{+}$ attains its global maximum outside it. Moreover, as in [42,
Lemma 4.5], by the Lipschitz estimate (see [45, Theorem 2.3] for a version of
[42, Proposition 4.9] for unbounded coefficients) we may take this
neighborhood $\mathcal{U}$ depending only on
$N,\lambda,\Lambda,\|\gamma\|_{\varrho},\|\vartheta\|_{N},\lambda_{1}^{\pm}$,
i.e. uniform with respect to the class of equations we consider. Indeed,
$F^{*}[\varphi^{+}_{1}]+\lambda_{1}^{+}\vartheta(x)\varphi_{1}^{+}=0$ in
$\Omega$, and so $\varphi_{1}^{+}$ is a viscosity positive solution of
$\mathcal{L}_{0}^{+}[\varphi^{+}_{1}]\geq-(1+\lambda_{1}^{+})\vartheta(x)\varphi^{+}_{1}$
in $\Omega$, $\varphi^{+}_{1}=0$ on $\partial\Omega$, and by [45, Theorem 2.3]
we get

$\displaystyle\textstyle
1=\varphi^{+}_{1}(x_{0})=\max_{\Omega}\varphi^{+}_{1}\leq
C(1+\lambda_{1}^{+})\|\vartheta\|_{N}\,\mathrm{dist}(x_{0},\partial\Omega)\Rightarrow\,\mathrm{dist}(x_{0},\partial\Omega)\geq(C(1+\lambda_{1}^{+})\|\vartheta\|_{N})^{-1},$

where $C$ is a universal positive constant depending only on
$n,\varrho,\alpha,\beta,\|\gamma\|_{L^{\varrho}(\Omega)},\Omega$. We then take
a compact set $K\subset({\mathbb{R}^{N}}\setminus\mathcal{U})$ such that
$\varphi_{1}^{+}$ attains its maximum equal to $1$ in $K$ and

$|\Omega\setminus
K|\leq\varepsilon:=(2C_{A}\|\vartheta\|_{L^{\varrho}(\Omega)})^{\frac{-N\varrho}{\varrho-N}}$

where $C_{A}$ is the constant in Proposition 2.3. Since $F^{*}$ satisfies
(H1), by [44, Theorem 1.1(ii)] we may consider $w\in C(\overline{\Omega})$ a
viscosity solution of the Dirichlet problem

$F^{*}(x,0,0,D^{2}w)+\gamma(x)|Dw|=f(x)$ in $\Omega$, $w=0$ on
$\partial\Omega$, (3.7)

where $f(x)=-2\vartheta(x)$ in $\Omega\setminus K$ and $f(x)=0$ in $K$. Note
that $w\in W^{2,\varrho}(\Omega)$ by Lemma 3.5.

Next, we apply ABP (Proposition 2.3) and Hölder’s inequality to find

$0<w\leq 2C_{A}\|\vartheta\|_{L^{N}(\Omega\setminus K)}\leq
2C_{A}|\Omega\setminus
K|^{{1}/{N}-{1}/{\varrho}}\,\|\vartheta\|_{L^{\varrho}(\Omega\setminus K)}\leq
2C_{A}\varepsilon^{\frac{\varrho-N}{N\varrho}}\|\vartheta\|_{L^{\varrho}(\Omega)}=1$
in $\Omega$.

Then $w$ also solves, in the strong sense,

$F^{*}[w+1]\leq
F^{*}(x,0,0,D^{2}w)+\gamma(x)|Dw|+\vartheta(x)(w+1)=\vartheta(x)(w-1)\leq 0$
in $\Omega\setminus K$.

Now we infer that Harnack inequality gives us $\varphi_{1}^{+}\geq\eta$ on
$K$, for some $\eta>0$. In fact, since $\varphi^{+}_{1}$ is a positive
solution of the inequalities
$\mathcal{L}_{0}^{+}[\varphi^{+}_{1}]+(1+\lambda_{1}^{+})\vartheta(x)\varphi_{1}^{+}\geq
0$ and $\mathcal{L}^{-}[\varphi^{+}_{1}]\leq 0$ in $\Omega$, this is a
combination of the Local Maximum Principle for the nonproper operator
$\mathcal{L}_{0}^{+}+(1+\lambda_{1}^{+})\vartheta(x)$ with unbounded zero
order term (which is obtained from [45, Theorem 2.5] through a Moser type
argument, see details in [36, proof of Theorem 2.2]) followed by the Weak
Harnack inequality for the proper operator
$\mathcal{L}^{-}=\mathcal{L}^{-}_{0}-\vartheta$ (since $\varphi^{+}_{1}>0$)
with unbounded coercive term, see [46, Theorem 2.1]. This produces a positive
constant $\eta$ depending on
$n,\varrho,\alpha,\beta,\Omega,\|\gamma\|_{L^{\varrho}(\Omega)},\|\vartheta\|_{L^{\varrho}(\Omega)}$
and $\lambda_{1}^{+}$.

Now we set $A\lambda_{1}^{+}\eta=2$ and $\psi=1+w+A\varphi_{1}^{+}$. Thus
$1\leq\psi\leq 2+A=b$ in $\Omega$, $\psi=1$ on $\partial\Omega$, and $\psi$ is
a strong solution of

$\displaystyle F^{*}[\psi]\leq F^{*}[w+1]+F^{*}[A\varphi_{1}^{+}]\leq
f(x)+\vartheta(x)(w+1)-A\lambda_{1}^{+}\vartheta(x)\varphi_{1}^{+}=:h(x)\leq
0\;\textrm{ in }\Omega,$

since $h(x)=\vartheta(x)(w+1-A\lambda_{1}^{+}\eta)\leq 0$ in $K$. Note that
$a=\inf_{\Omega}\psi=1$, and $b=\sup_{\Omega}\psi$ depend only upon the same
constants of ABP (see for instance [27]) since $\psi=1$ on $\partial\Omega$.
In conclusion, $F$ satisfies ABP-MP in $\Omega$, which proves the theorem. ∎

######  Remark 3.6.

If $F^{*}$ is a proper operator then Proposition 3.4 reduces to the usual ABP
(Proposition 2.3). Moreover, in this case $\psi\equiv 1$ verifies the
conditions in Step 1, since $F^{*}[1]\leq F^{*}[0]=0$.

Let us now discuss some applications of Theorem 3.4. In what concerns the
Dirichlet problem, the following solvability in the scalar case will be
essential for the solvability of the system.

######  Theorem 3.7.

Let $\Omega\in C^{1,1}$ be a bounded domain. Assume (H1), (H3) on $F$, and
(H2) on $F^{*}$. Let $f\in L^{\varrho}(\Omega)$, $\varrho>N$. Then there
exists a viscosity solution $u\in C^{1,\alpha}(\overline{\Omega})$ of the
problem

$\displaystyle F(x,u,Du,D^{2}u)=f(x)\text{ in }\Omega,\;\;\quad u=0\text{ on
}\partial\Omega.$ (3.8)

Further, if (3.8) possesses a strong solution $u\in
W^{2,\varrho}_{\mathrm{loc}}(\Omega)$, then $u$ is the unique solution of
(3.8) in the class of viscosity solutions. In particular, (3.8) is uniquely
solvable under (H4).

###### Proof.

By Step 2 in the proof of Theorem 3.4 we know that there exists a function
$\psi\in W^{2,\varrho}(\Omega)$ with $\psi>0$ in $\overline{\Omega}$ such that
$\psi$ solves $F^{*}[\psi]\leq 0$ in $\Omega$ in the strong sense. Next, by
Step 1 in that proof, one may define (3.6), from which we see that solving
(3.8) is equivalent to solve $F_{\psi}[v]=f(x)$ in $\Omega$, where $F_{\psi}$
is a proper operator satisfying (H1). In turn, the existence of a viscosity
solution $v\in C(\overline{\Omega})$ to $F_{\psi}[v]=f(x)$ comes from [44,
Theorem 1(ii), case $\mu=0$]; and the uniqueness in the presence of a strong
solution follows by [44, Theorem 1(iii), case $\mu=0$]. The regularity
assertions are an immediate consequence of the $C^{1,\alpha}$ results in
Proposition 2.8. ∎

As a nontrivial application of Theorem 3.4 to systems, we prove MP and mP for
either domains with small measure or weights with small $L^{N}$-norm.

###### Proof of Theorem 1.16.

We only prove item $(i)$, as $(ii)$ is completely analogous. We start choosing
$A>0$ such that $\lambda\mu^{q}\,C_{B}^{1+q}\leq A,$ where $C_{B}>0$ is the
universal constant in Definition 3.3, which depends upon
$N,\alpha,\beta,\mathrm{diam}(\Omega)$, $\|\gamma\|_{L^{\varrho}(\Omega)}$, in
addition to $\lambda_{1}^{+}(F_{i}^{*}(\vartheta))$.

Note that $(u,v)$ is a pair of viscosity solutions to

$-F_{1}^{*}[u]\leq\lambda\tau_{1}(x)(v^{+})^{q}$,
$-F_{2}^{*}[v]\leq\mu\tau_{2}(x)(u^{+})^{p}$ in $\Omega$.

Hence, Theorem 3.4 just proved, applied to the scalar equation for $u$ and
$v$, yields

$\displaystyle\textstyle\sup_{\Omega}u\leq
C_{B}\lambda\,\|\tau_{1}\|_{L^{N}(\Omega)}\sup_{\Omega}(v^{+})^{q},\quad\sup_{\Omega}v\leq
C_{B}\,\mu\,\|\tau_{2}\|_{L^{N}(\Omega)}\sup_{\Omega}(u^{+})^{p}.$ (3.9)

If we had either $u\leq 0$ or $v\leq 0$ in $\Omega$, then by (3.9) we would
obtain $u,v\leq 0$ in $\Omega$, and the proof is done. We then assume that
both $u$ and $v$ assume their positive maxima in $\Omega$. Then,

$\displaystyle\textstyle\sup_{\Omega}(u^{+})$ $\displaystyle\leq
C_{B}^{1+q}\,\lambda\mu^{q}\,\|\tau_{1}\|_{L^{N}(\Omega)}\,\|\tau_{2}\|_{L^{N}(\Omega)}^{q}\,\sup_{\Omega}(u^{+})^{pq}$
$\displaystyle\textstyle\leq
A\,\|\tau_{1}\|_{L^{N}(\Omega)}\,\|\tau_{2}\|_{L^{N}(\Omega)}^{q}\,\|u\|_{\infty}^{pq-1}\,\sup_{\Omega}(u^{+}).$

For MP with small weights $\tau_{1},\tau_{2}\in L^{N}_{+}(\Omega)$, we choose
$\varepsilon_{0}>0$ with
$\|\tau_{1}\|_{L^{N}(\Omega)}\,\|\tau_{2}\|_{L^{N}(\Omega)}^{q}\leq\varepsilon_{0}$
so that $A\varepsilon_{0}\,\|u\|_{\infty}^{pq-1}\leq 1/2$. Then, $u\leq 0$ in
$\Omega$, and so does $v$ by (3.9). Upon performing the above argument with
$\sup_{\Omega}(v^{+})$, one can assume instead
$\|\tau_{1}\|^{p}_{L^{N}(\Omega)}\,\|\tau_{2}\|_{L^{N}(\Omega)}\leq\varepsilon_{0}$.

On the other hand, for $\tau_{1},\tau_{2}\in L^{\varrho}_{+}(\Omega)$, say
$\|\tau_{1}\|_{L^{\varrho}(\Omega)}\,\|\tau_{2}\|_{L^{\varrho}(\Omega)}^{q}\leq
W$, we apply Holder inequality to obtain

$\displaystyle\textstyle\sup_{\Omega}(u)^{+}\leq
AW\,|\Omega|^{(\frac{1}{N}-\frac{1}{\varrho})(1+q)}\|u\|_{\infty}^{pq-1}\sup_{\Omega}(u^{+}).$

Then we pick $\varepsilon_{0}>0$ with $|\Omega|\leq\varepsilon_{0}$ such that
$AW\varepsilon_{0}^{(\frac{1}{N}-\frac{1}{\varrho})(1+q)}\|u\|_{\infty}^{pq-1}\leq
1/2$, from which we also derive $u,v\leq 0$ in $\Omega$.

The argument for supersolutions is analogous, by considering the negative
parts. In any case, observe that, if $pq=1$, then $\varepsilon_{0}$ can be
chosen independently of the $L^{\infty}$-norm of $u$ and $v$. ∎

##  4 Auxiliary results for systems

In this section we consider some fundamental results which appear throughout
the text. We start with an instrumental proposition to our analysis of
uniqueness results.

######  Proposition 4.1.

Let $F_{1},F_{2}$ satisfy (H1), (H2), (H3). Let $pq=1$, $\lambda,\mu\geq 0$,
and $(u_{1},v_{1})$, $(u_{2},v_{2})$ in $C(\overline{\Omega})$ be viscosity
solutions of

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}[u_{1}]+\,\lambda\tau_{1}(x)v_{1}^{q}&\leq&0&\mbox{in}&\;\Omega\\\
F_{2}[v_{1}]+\,\mu\tau_{2}(x)u_{1}^{p}&\leq&0&\mbox{in}&\;\Omega\\\
u_{1}\,,\;v_{1}&>&0&\mbox{in}&\;\Omega\end{array}\right.\;\;,\;\;\;\left\\{\begin{array}[]{rclcc}F_{1}[u_{2}]+\,\lambda\tau_{1}(x)|v_{2}|^{q-1}v_{2}&\geq&0&\mbox{in}&\;\Omega\\\
F_{2}[v_{2}]+\,\mu\tau_{2}(x)|u_{2}|^{p-1}u_{2}&\geq&0&\mbox{in}&\;\Omega\\\
u_{2}\,,\;v_{2}&\leq&0&\mbox{on}&\;\partial\Omega.\end{array}\right.$

In addition, assume that

either $u_{2}(x_{0})>0$ or $v_{2}(x_{0})>0$, for some $x_{0}\in\Omega$; (4.1)

and that one of the pairs of solutions is in
$E_{\varrho}=W^{2,\varrho}(\Omega)\cap C(\overline{\Omega})$. Then
$u_{1}\equiv tu_{2}$ and $v_{1}\equiv t^{p}v_{2}$ in $\Omega$ for some $t>0$.

Analogously, if $pq=1$, $\lambda,\mu\geq 0$, and $(u_{1},v_{1})$,
$(u_{2},v_{2})$ in $C(\overline{\Omega})$ satisfy

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}[u_{1}]+\,\lambda\tau_{1}(x)|v_{1}|^{q-1}v_{1}&\geq&0&\mbox{in}&\;\Omega\\\
F_{2}[v_{1}]+\,\mu\tau_{2}(x)|u_{1}|^{p-1}u_{1}&\geq&0&\mbox{in}&\;\Omega\\\
u_{1}\,,\;v_{1}&<&0&\mbox{in}&\;\Omega\end{array}\right.\;\;,\;\;\;\left\\{\begin{array}[]{rclcc}F_{1}[u_{2}]+\,\lambda\tau_{1}(x)|v_{2}|^{q-1}v_{2}&\leq&0&\mbox{in}&\;\Omega\\\
F_{2}[v_{2}]+\,\mu\tau_{2}(x)|u_{2}|^{p-1}u_{2}&\leq&0&\mbox{in}&\;\Omega\\\
u_{2}\,,\;v_{2}&\geq&0&\mbox{on}&\;\partial\Omega.\end{array}\right.$

with either $u_{2}(x_{0})<0$ or $v_{2}(x_{0})<0$, for some $x_{0}\in\Omega$;
and that one of the pairs of solutions is in $E_{\varrho}$. Then $u_{1}\equiv
tu_{2}$ and $v_{1}\equiv t^{p}v_{2}$ in $\Omega$ for some $t>0$.

######  Remark 4.2.

The assumption (4.1) on the pair $(u_{2},v_{2})$ means that the system (1.10)
does not satisfy the maximum principle for $(\lambda,\mu)$.

###### Proof.

Let us prove the first statement, since the other one is carried out
similarly.

We observe that (4.1) implies $\sup_{\Omega}u_{2}>0$ and
$\sup_{\Omega}v_{2}>0$. Indeed, say $u_{2}(x_{0})>0$, which yields
$\sup_{\Omega}u_{2}>0$. If we had $v_{2}\leq 0$ in $\Omega$ then
$-F_{1}^{*}[u_{2}]\leq 0$ in $\Omega$, $u_{2}\leq 0$ on $\partial\Omega$, so
we would get $u_{2}\leq 0$ by ABP-MP (since we assume $F_{1}$ satisfies (H3),
the first eigenvalue of $F_{1}^{*}$ is positive and so Theorem 3.4 can be
applied). This yields a contradiction.

Observe that for each compact set $K\subset\Omega$, with $x_{0}\in K$, there
exists $s_{K}$ such that $u_{1}>s_{K}u_{2}$, $v_{1}>s_{K}^{p}v_{2}$ in $K$.
This comes from the fact that $\min u_{1}$, $\min{v_{1}}$, $\max u_{2}$, $\max
v_{2}$ are positive over $K$.

Next we claim that $u_{1}\geq su_{2}$, $v_{1}\geq s^{p}v_{2}$ in a
neighborhood of $\partial\Omega$ for some small $s>0$.

It is enough to prove the first inequality; the second one is analogous.
Notice that $u_{1}-su_{2}\geq 0$ on $\partial\Omega$, for all $s>0$. Fix
$\hat{x}\in\partial\Omega$. If $u_{1}(\hat{x})-su_{2}(\hat{x})>0$, then by
continuity of $u_{1}$ and $u_{2}$ up to the boundary there exists a
neighborhood of $\hat{x}$, namely $\hat{B}$, such that $u_{1}-su_{2}>0$ in
$\Omega\cap\hat{B}$. Assume then $u_{1}(\hat{x})-su_{2}(\hat{x})=0$ for some
$s$. Thus $u_{1}(\hat{x})=u_{2}(\hat{x})=0$. Let us look at the quantities

$A_{i}=\varliminf_{t\to 0^{+}}\frac{u_{i}(\hat{x}+t\nu)-u_{i}(\hat{x})}{t}$,
$i=1,2$,

where $\nu$ is the interior unit normal vector to $\partial\Omega$ at
$\hat{x}$. Hopf’s lemma for viscosity solutions (Proposition 2.7) yields
$A_{1}>0$. If we had $A_{2}\leq 0$, then $A_{1}-sA_{2}>0$ for all $s>0$.
Otherwise, if $A_{2}>0$ then we may pick some small $\hat{s}>0$ such that
$A_{1}-\hat{s}A_{2}>0$. Recall that one of the solutions pair is in
$E_{\varrho}$, then one of the $A_{i}$’s is the normal derivative of $u_{i}$
at $\hat{x}$. Since $u_{1}(\hat{x})-su_{2}(\hat{x})=0$, this is enough to
ensure that $u_{1}-su_{2}>0$ in $\Omega\cap\hat{B}$. A covering argument then
concludes the claim.

Therefore one obtains the existence of some $s_{0}>0$ such that $u_{1}\geq
s_{0}u_{2}$ and $v_{1}\geq s_{0}^{p}v_{2}$ in $\Omega$, for all $s\leq s_{0}$.
In particular, the following set is nonempty,

$S=\\{s>0\,:\,u_{1}>su_{2},\,v_{1}>s^{p}v_{2}\textrm{\, in }\Omega\\}$,

and the quantity $s_{*}=\sup S$ is well defined. We have $s_{*}<+\infty$ by
(4.1).

Notice that $w:=u_{1}-s_{*}u_{2}\geq 0$ and $z:=v_{1}-s_{*}^{p}\,v_{2}\geq 0$
in $\Omega$. Moreover, $s_{*}^{p}\,|u_{2}|^{p-1}u_{2}\leq u_{1}^{p}$ and
$s_{*}|v_{2}|^{q-1}v_{2}\leq v_{1}^{q}$ in $\Omega$. By using also (H1), (H2),
and $\lambda,\mu\geq 0$, one sees that $w,z$ satisfy

$\displaystyle(F_{1})_{*}[w]\leq
F_{1}[u_{1}]-s_{*}\,F_{1}[u_{2}]\leq-\lambda\tau_{1}(x)v_{1}^{q}+\lambda\tau_{1}(x)\,s_{*}\,|v_{2}|^{q-1}v_{2}\leq
0,$ $\displaystyle(F_{2})_{*}[z]\leq
F_{2}[v_{1}]-s_{*}^{p}\,F_{2}[v_{2}]\leq-\mu\tau_{2}(x)u_{1}^{p}+\mu\tau_{2}(x)\,s_{*}^{p}\,|u_{2}|^{p-1}u_{2}\leq
0,$ (4.2)

in the viscosity sense in $\Omega$, with $w,z\geq 0$ on $\partial\Omega$.
Whence, by applying twice Theorem 3.4 and SMP for scalar equations we get
either $w>0$ or $w\equiv 0$ in $\Omega$; and either $z>0$ or $z\equiv 0$ in
$\Omega$.

Notice that $w>0$ is equivalent to $z>0$ by (4). In other words, one has
either $w,z>0$ or $w,z\equiv 0$. Under the latter we are done.

Suppose on the contrary that $w,z>0$ in $\Omega$. Now we may reproduce the
preceding argument with the pair $(u_{2},v_{2})$ replaced by $(w,z)$ in order
to conclude the existence of some small $\varepsilon>0$ such that
$u_{1}>(s_{*}+\varepsilon)u_{2}$ in $\Omega$. But this contradicts the
definition of $s_{*}$ as the supremum of $S$. ∎

######  Corollary 4.3.

Let $F_{1},F_{2}$ satisfy (H1), (H2), (H3). Let $pq=1$, $\lambda\geq 0$, and
let $(u_{1},v_{1})$, $(u_{2},v_{2})$ be viscosity solutions of

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}[u_{1}]+\,\lambda\tau_{1}(x)|v_{1}|^{q-1}v_{1}&=&0&\mbox{in}&\;\Omega\\\
F_{2}[v_{2}]+\,\lambda\tau_{2}(x)|u_{1}|^{p-1}u_{1}&=&0&\mbox{in}&\;\Omega\\\
u_{1}\,,\;v_{1}&=&0&\mbox{on}&\;\partial\Omega\end{array}\right.\;\;,\;\;\;\left\\{\begin{array}[]{rclcc}F_{1}[u_{2}]+\,\Lambda\tau_{1}(x)|v_{2}|^{q-1}v_{2}&=&0&\mbox{in}&\;\Omega\\\
F_{2}[v_{2}]+\,\Lambda\tau_{2}(x)|u_{2}|^{p-1}u_{2}&=&0&\mbox{in}&\;\Omega\\\
u_{2}\,,\;v_{2}&=&0&\mbox{on}&\;\partial\Omega\end{array}\right.$

with one of the solutions pair in $E_{\varrho}$. Suppose that either
$u_{i},v_{i}>0$ in $\Omega$ or $u_{i},v_{i}<0$ in $\Omega$, $i=1,2$. Then
$\lambda=\Lambda$, and $u_{1}\equiv tu_{2}$, $v_{1}\equiv t^{p}v_{2}$ in
$\Omega$ for some $t>0$.

We conclude the section presenting a priori bounds for the first eigenvalue.

######  Lemma 4.4.

Let $F_{1},F_{2}$ satisfy (H1), (H2), (H3) with $\tau_{i}(x)\geq\delta$ a.e.
in $B_{R}$ for $i=1,2$, for some $B_{R}\subset\subset\Omega$. then

$\lambda_{1}^{\pm}(F_{1}(\tau_{1}),F_{2}(\tau_{2}),\Omega)\leq\delta^{-1}\,\lambda_{1}^{\pm}(F_{1}(1),F_{2}(1),B_{R}).$

###### Proof.

We work on the $\lambda^{+}_{1}$ case, since for $\lambda_{1}^{-}$ it is just
a question of replacing $F_{i}$ by $G_{i}(x,r,p,X)=-F_{i}(x,-r,-p,-X)$, recall
(2.5). Observe that
$\lambda_{1}^{\pm}(F_{i},\Omega)\leq\lambda_{1}^{\pm}(F_{i},B_{R})$, by
definition. Also, both quantities are nonnegative by Remark 2.11. Hence, given

$\mathcal{A}:=\\{\lambda\in\mathcal{R},\
\Psi_{\lambda}^{\pm}(\Omega)\neq\emptyset\\},\qquad\mathcal{B}:=\\{\lambda\in\mathcal{R},\
\Psi_{\lambda}^{\pm}(B_{R})\neq\emptyset\\},$

it is enough to see that $\mathcal{A}\cap\\{\lambda\geq
0\\}\subset\mathcal{B}/\delta\cap\\{\lambda\geq 0\\}$, since

$\lambda_{1}^{+}(F_{1}(\tau_{1}),F_{2}(\tau_{2}),\Omega)=\sup_{\mathcal{A}}\lambda=\sup_{\mathcal{A}\cap\\{\lambda\geq
0\\}}\lambda\;,\;\;\lambda_{1}^{+}(F_{1}(1),F_{2}(1),B_{R})=\sup_{\mathcal{B}}\lambda=\sup_{\mathcal{B}\cap\\{\lambda\geq
0\\}}\lambda,$

as settled before in Section 2.3. Let $\lambda\in\mathcal{A}\cap\\{\lambda\geq
0\\}$, then there exist positive functions $\varphi,\psi\in
C(\overline{\Omega})$ solving $F_{1}[\varphi]+\lambda\tau_{1}(x)\psi^{q}\leq
0$, $F_{2}[\psi]+\lambda\tau_{2}(x)\varphi^{p}\leq 0$ in $\Omega$, in the
viscosity sense. Hence, $(\varphi,\psi)$ is a positive viscosity solution of
$F_{1}[\varphi]+\lambda\delta\psi^{q}\leq 0$,
$F_{2}[\psi]+\lambda\delta\varphi^{p}\leq 0$ in $B_{R}$, and so
$\delta\lambda\in\mathcal{B}$. ∎

######  Proposition 4.5.

Suppose $F_{1},F_{2}$ verify (H1)–(H4), $pq=1$, and $\tau_{1}\geq\delta>0$,
$\tau_{2}\geq\delta>0$ a.e. in $B_{R}\subset\subset\Omega$, for some $R\leq
1$. Let $\lambda>0$ and $(u,v)$, with $uv>0$ in $\Omega$, be a solution of

$-F_{1}[u]=\lambda\tau_{1}(x)|v|^{q-1}v,\quad-
F_{2}[v]=\lambda\tau_{2}(x)|u|^{p-1}u\ \textrm{ in }\;\Omega,\quad
u=v=0\textrm{ on }\partial\Omega.$

Then $\lambda\leq C$, for a positive constant $C$ depending on
$N,\alpha,\beta,\Omega,\|\gamma\|_{L^{\varrho}(\Omega)}$,
$\|\vartheta\|_{L^{\varrho}(\Omega)}$, $\delta$, and $R$.

###### Proof.

We use some constructions from [7, 32, 37]. By Lemma 4.4, it is enough to
prove the result for $\tau_{i}\equiv 1$, $i=1,2$. Let us consider $u>0$ and
$v>0$ in $\Omega$; for the case with $u<0$ and $v<0$ in $\Omega$ it is
sufficient to apply the case with positive sign to $G_{i}$ in place of
$F_{i}$, see (2.5).

Step 1) We first consider a bounded drift and zero order term $\gamma\in
L^{\infty}_{+}(\Omega)$, namely $|\gamma(x)|\leq\gamma$ and
$|\vartheta|\leq\vartheta$ for a.e. $x\in\Omega$. Let us prove in what follows
that there exists $C>0$ depending only on $N,\alpha,\beta,R,\gamma$, and
$\vartheta$, such that

$\lambda_{1}^{\pm}(F_{1}(1),F_{2}(1),B_{R})\leq\frac{C}{R^{2}}.$

Moreover, if $\gamma,\vartheta=0$ then the constant does not depend on $R$,
for all $R>0$ such that $B_{R}\subset\Omega$.

Note that the function $U(x)=(R^{2}-|x|^{2})^{2}$ is a positive strong
solution of $F_{1}[U]+\frac{C_{0}}{R^{2}}U\geq 0$ in $B_{R}$, see [7]. Next we
consider the strong solution of

$F_{2}[V]+U^{p}=0$ in $B_{R}$, $V=0$ on $\partial B_{r}$,

given for instance in [42]. Here $V>0$ in $B_{R}$ by Theorem 3.4 and SMP,
since $\tau_{2}>0$ a.e. and $U>0$ in $B_{R}$. Also, $\partial_{\nu}V>0$ on
$\partial\Omega$ by Hopf, where $\nu$ is the interior unit normal. Now, since
$pq=1$, without loss of generality we may assume $q\leq 1$. Thus $U^{1/q}\in
C^{1}(\overline{\Omega})$, and we can pick up some $a>0$ large enough so that

$U^{1/q}\leq aV$ in $B_{R}$.

Therefore $U$ becomes a strong solution of
$F_{1}[U]+\frac{a^{q}C_{0}}{R^{2}}V^{q}\geq 0$ in $B_{R}$.

Set $C=a^{q}C_{0}>1$. Suppose by contradiction that there exists some
$\lambda>\frac{C}{R^{2}}$ such that
$\varphi,\psi\in\Psi^{+}_{\lambda}(B_{R})\neq\emptyset$, i.e. let
$(\varphi,\psi)\in C(\overline{B}_{R})$ be positive viscosity solutions of
$F_{1}[\varphi]+\lambda\psi^{q}\leq 0$ and $F_{2}[\psi]+\lambda\varphi^{p}\leq
0$ in $B_{R}$. They also solve $F_{1}[\varphi]+\frac{C}{R^{2}}\psi^{q}\leq 0$,
$F_{2}[\psi]+\frac{C}{R^{2}}\varphi^{p}\leq 0$ in $B_{R}$.

Now we apply Proposition 4.1 to obtain that $\varphi=tU$ and $\psi=t^{p}V$ in
$B_{R}$ for some $t>0$. However, this is not possible since $\varphi>0$ on
$\partial B_{R}\subset\Omega$, while $U=0$ on $\partial B_{R}\,$.

Step 2) In general case we assume $\gamma,\vartheta\in
L^{\varrho}_{+}(\Omega)$. Let us show that there exists a universal constant
such that $\lambda^{1-\frac{N}{\varrho}}\leq
C\,(\|\gamma\|_{L^{\varrho}(\Omega)}+\|\vartheta\|_{L^{\varrho}(\Omega)})^{2}.$

We argue by contradiction. Suppose there exist sequences
$\gamma_{k},\vartheta_{k}\in L^{\infty}_{+}(\Omega)$ with
$\|\gamma_{k}\|_{L^{\varrho}(\Omega)}+\|\vartheta_{k}\|_{L^{\varrho}(\Omega)}\leq
C$, but
$\|\gamma_{k}\|_{L^{\varrho}(\Omega)}+\|\vartheta_{k}\|_{L^{\infty}(\Omega)}\rightarrow+\infty$,
and let the respective eigenvalue problem

$F_{1}^{k}[u_{k}]+\lambda_{k}v_{k}^{q}=0$,
$F_{2}^{k}[v_{k}]+\lambda_{k}u_{k}^{p}=0$, $u_{k},v_{k}>0$ in $B_{R}$,
$u_{k},v_{k}=0$ on $\partial B_{R}$,

in the viscosity sense, such that $\lambda_{k}\to+\infty$ as $k\to\infty$,
where $F_{i}^{k}$ is a fully nonlinear operator satisfying (H1)–(H4) for the
respective $\gamma_{k}$ and $\vartheta_{k}$. Up to using the rescaling
$(u,v)\longmapsto(tu,t^{p}v)$ with $t=1/\|u\|_{\infty}$, we may assume
$\max_{\overline{\Omega}}\,u_{k}=1$, $\max_{\overline{\Omega}}\,v_{k}=c_{0}$
for all $k\in\mathbb{N}$. Say $\max_{\overline{\Omega}}u_{k}=u_{k}(x_{0}^{k})$
for $x_{0}^{k}\in\Omega$. Then, $x_{0}^{k}\rightarrow
x_{0}\in\overline{\Omega}$ as $k\rightarrow+\infty$, up to a subsequence.

Since $B_{R}$ is a convex domain we know that $x_{0}\in\Omega$. Let
$2\rho=\mathrm{dist}(x_{0},\partial B_{R})>0$, so $x_{0}^{k}\in
B_{\rho}(x_{0})$ for all $k\geq k_{0}$. Set $r_{k}={\lambda_{k}}^{-{1}/{2}}$
and $U_{k}(x)=U_{k}(x_{0}^{k}+r_{k}x)$, $V_{k}(x)=V_{k}(x_{0}^{k}+r_{k}x)$.
Thus, $(U_{k},V_{k})$ is a viscosity solution pair of

$\tilde{F}_{1}^{k}\,[U_{k}]+V_{k}^{q}=0$,
$\tilde{F}_{2}^{k}\,[V_{k}]+U_{k}^{p}=0$, $U_{k},V_{k}>0$ in
$\widetilde{B}_{k}:=B_{\rho/{r_{k}}}(0)$, (4.3)

where
$\widetilde{F}_{i}^{k}(x,r,p,X)=r^{2}_{k}\,F_{i}^{k}(x_{0}^{k}+r_{k}x,r,p/{r_{k}},X/{r_{k}^{2}})$
satisfies (H1)–(H4) for $\tilde{\gamma_{k}}$ and $\tilde{\vartheta}_{k}$,
where $\tilde{\gamma}_{k}(x)=r_{k}\,\gamma_{k}(x_{0}^{k}+r_{k}x)$ and
$\tilde{\vartheta}_{k}=r_{k}^{2}\,\vartheta_{k}(x_{0}^{k}+r_{k}x)$. Then one
has

$\|\tilde{\gamma}_{k}\|_{L^{\varrho}}=r_{k}^{1-\frac{N}{\varrho}}\|\gamma_{k}\|_{L^{\varrho}(\Omega)}\to
0$ and
$\|\tilde{\vartheta}_{k}\|_{L^{\varrho}(\Omega)}=r_{k}^{2-\frac{N}{\varrho}}\|\vartheta_{k}\|_{L^{\varrho}(\Omega)}\to
0$ as $k\to\infty$.

Also, $\sup_{\widetilde{B}_{k}}U_{k}=U_{k}(0)=1$ and
$\sup_{\widetilde{B}_{k}}V_{k}=V_{k}(0)\leq c_{0}$ for all $k\in\mathbb{N}$
with $B_{R}(0)\subset\subset\widetilde{B}_{k}$ for large $k$, for any fixed
$R>0$. By $C^{1,\alpha}$ regularity estimates (Proposition 2.8) we have
$U_{k},V_{k}\in C_{\mathrm{loc}}^{1,\alpha}$ and

$\displaystyle\|U_{k}\|_{C^{1,\alpha}(\overline{B}_{R}(0))},\;\|V_{k}\|_{C^{1,\alpha}(\overline{B}_{R}(0))}\leq
C,$

since the constant depends only on a bound from above on the
$L^{\varrho}$-norm of the coefficient $\gamma_{k}$, which is uniformly
bounded. Hence, by compact embeddings, we have that there exists $U,V\in
C^{1}(\overline{B}_{R}(0))$ such that $U_{k}\to U$, $V_{k}\to V$ as
$k\to+\infty$, up to a subsequence. Doing the same for each ball $B_{R}(0)$,
for every $R>0$, it yields $U_{k}\rightarrow U$, $V_{k}\to V$ in
$L^{\infty}_{\mathrm{loc}}({\mathbb{R}^{N}})$. By applying a stability
argument (Proposition 2.5) in each ball, one gets that $U,V$ is a viscosity
solution of

$J_{1}(x,D^{2}U)+V^{q}=0$, $J_{2}(x,D^{2}V)+U^{p}=0$ in ${\mathbb{R}^{N}}$,

for some measurable operators $J_{1},J_{2}$ satisfying (H1) with $\vartheta$
and $\gamma$ equal to zero. The operators $J_{i}$ are obtained by using
Arzela-Ascoli theorem, since $\widetilde{F}_{i}^{k}$ are
$(\alpha,\beta)$–uniformly elliptic, with zero and first order coefficients
converging to zero. Further, $U(0)=1$ implies $U>0$ in ${\mathbb{R}^{N}}$ by
SMP, and so $V>0$ in ${\mathbb{R}^{N}}$. Thus, by Step 1,

$1\leq\lambda_{1}^{+}(J_{1},J_{2},B_{r})\leq\frac{C_{0}}{r^{2}}$ for all
$r>0$,

where $C_{0}$ does not depend on $r$. One derives a contradiction by letting
$r\rightarrow+\infty$. ∎

##  5 The first eigenvalue problem

In this section we investigate the main properties of the first eigenvalue
problem. We stress once again that we can equivalently consider (1.10), (2.7)
if positive parameters are taken into account, recall Section 2.4. In
particular, we recall that the existence of a principal eigenvalue for problem
(2.7) implies the existence of a spectral curve for problem (1.10). Moreover,
$(\lambda,\mu)$ is below the curve $\Lambda_{1}^{\pm}$ if and only if
$\lambda_{0}<\lambda_{1}^{\pm}(F_{1},F_{2})$, where $\lambda_{0}$ is defined
through the identity $\mu=\frac{\lambda_{0}^{p+1}}{\lambda^{p}}$. Therefore,
in what follows, we will always reduce ourselves to (2.7) by exploiting this
scaling.

###  5.1 Existence and simplicity

We start by recalling a well known result from degree theory, see [5] for the
proof.

######  Proposition 5.1.

Let $(\mathcal{E},\left\|\cdot\right\|)$ be a Banach space. Let
$T\colon\mathbb{R}^{+}_{0}\times\mathcal{E}\to\mathcal{E}$ be a completely
continuous operator such that $T(0,u)=0$ for all $u\in\mathcal{E}$; then there
exists an unbounded, connected component $\mathcal{C}$ of
$\mathbb{R}^{+}\times\mathcal{E}$ of solutions of $u=T(\mu,u)$ and starting
from $(0,0)$.

We first show the following result.

######  Proposition 5.2.

Let $F_{1},F_{2}$ satisfy (H1)–(H4), $pq=1$, and assume (1.5). Then there
exist positive numbers $\sigma_{1}^{\pm}$, and signed functions
$\varphi_{1}^{\pm},\psi_{1}^{\pm}$ in $E_{\varrho}=W^{2,\varrho}(\Omega)\cap
C(\overline{\Omega})$ such that

$\left\\{\begin{array}[]{rclcc}F_{1}[\varphi_{1}^{\pm}]+\sigma_{1}^{\pm}\,\tau_{1}(x)(\psi_{1}^{\pm})^{q}&=&0&\mbox{in}&\;\Omega\\\
F_{2}[\psi_{1}^{\pm}]+\sigma_{1}^{\pm}\,\tau_{2}(x)(\varphi_{1}^{\pm})^{p}&=&0&\mbox{in}&\;\Omega\\\
\pm\varphi_{1}^{\pm}\,,\;\pm\psi_{1}^{\pm}&>&0&\mbox{in}&\;\Omega\\\
\varphi_{1}^{\pm}\,\;=\,\;\psi_{1}^{\pm}&=&0&\mbox{on}&\;\partial\Omega,\end{array}\right.$

###### Proof.

We prove the $\sigma_{1}^{+}$ case; for $\sigma_{1}^{-}$ is analogous by
replacing the operator $F_{i}$ by $G_{i}$, see (2.5).

Step 1) Assume that $\tau_{i}(x)>0$ a.e. in $\Omega$ for $i=1,2$.

In this case we adapt Krein-Rutman’s theorem, exploiting ideas from [5, 41].
Let us consider the Banach space $E=C^{1}_{0}(\bar{\Omega})$. Define
$\mathcal{F}_{i}=-F_{i}^{-1}\circ\tau_{i}$ in $E$, i.e.

$\mathcal{F}_{i}\,u=U\;\;\Leftrightarrow\;\;\,-F_{i}\,[U]=\tau_{i}(x)u$ in
$\Omega$, $U=0$ on $\partial\Omega$, $i=1,2$,

in the viscosity sense. Thus $\mathcal{F}_{i}:E\to E$ is well defined and
completely continuous. Indeed, this comes from (H3), (H4) for $F_{i}$, and
Theorem 3.7.

Consider the closed positive cone $K:=\\{w\in E,\,w\geq 0\\}$. Note that $K$
is solid, that is, it has nonempty interior. Then $\mathcal{F}_{i}$ is
strictly positive with respect to $K$, in the sense that
$\mathcal{F}_{i}\,(K\setminus\\{0\\})\subset\mathrm{int}K$. This is due to
ABP-MP, SMP, and Hopf for scalar equations (see Theorem 3.4, and Propositions
2.6, 2.7), since the weight $\tau_{i}$ is strictly positive in $\Omega$.

We first choose $w_{0}\in K\setminus\\{0\\}$. We can choose $M>0$ such that
$M\mathcal{F}_{2}(w_{0})\geq w_{0}$, $M\mathcal{F}_{2}(w_{0}^{p})\geq
w_{0}^{p}$ and $M\mathcal{F}_{1}(w_{0})\geq w_{0}$. In fact, this choice is
possible because $K$ is solid and each $\mathcal{F}_{i}$ is a strongly
positive operator with respect to $K$. Fix $\varepsilon>0$ and define
$T_{\varepsilon}:\mathbb{R}^{+}\times K\times K\to K\times K$ as

$\textstyle{T_{\varepsilon}(\mu,u,v)=\left(\mu\mathcal{F}_{1}(v^{q})+\varepsilon\mu\mathcal{F}_{2}(w_{0}),\mu\mathcal{F}_{2}(u^{p})+\varepsilon\mu\mathcal{F}_{1}(w_{0})\right).}$

By Proposition 5.1, there exists $\mathcal{C}_{\varepsilon}$, an unbounded
connected component of solutions of $(u,v)=T_{\varepsilon}(\mu,u,v)$ which
contains $(0,0,0)$. We claim that $\mathcal{C}_{\varepsilon}\subset[0,M]\times
K\times K$. Indeed, let $(\mu,u,v)\in\mathcal{C}_{\varepsilon}$. In
particular,

$\textstyle{u=\mu\mathcal{F}_{1}(v^{q})+\varepsilon\mu\mathcal{F}_{2}(w_{0})\quad\textrm{and}\quad
v=\mu\mathcal{F}_{2}(u^{p})+\varepsilon\mu\mathcal{F}_{1}(w_{0}).}$

Then,

$\textstyle{u\geq\mu\varepsilon\mathcal{F}_{2}(w_{0})\geq\frac{\mu}{M}\varepsilon
w_{0},}$

which implies that $u^{p}\geq\frac{\mu^{p}}{M^{p}}\varepsilon^{p}w_{0}^{p}$.
By applying $\mathcal{F}_{2}$, and using the comparison principle, see Theorem
3.4, we have

$\textstyle{\mathcal{F}_{2}(u^{p})\geq\frac{\mu^{p}}{M^{p}}\varepsilon^{p}\mathcal{F}_{2}(w_{0}^{p})\geq\frac{\mu^{p}}{M^{p}}\varepsilon^{p}\frac{1}{M}w_{0}^{p}.}$

Moreover,
$v\geq\mu\mathcal{F}_{2}(u^{p})\geq\frac{\mu^{p+1}}{M^{p+1}}\varepsilon^{p}w_{0}^{p},$
whence $v^{q}\geq\frac{\mu^{q+1}}{M^{q+1}}\varepsilon w_{0}$ since $pq=1$.
Now, applying $\mathcal{F}_{1}$,

$\textstyle{\mathcal{F}_{1}(v^{q})\geq\frac{\mu^{q+1}}{M^{q+1}}\varepsilon\mathcal{F}_{1}(w_{0})\geq\frac{\mu^{q+1}}{M^{q+1}}\varepsilon\frac{1}{M}w_{0},}$

and therefore
$u\geq\mu\mathcal{F}_{1}(v^{q})\geq\left(\frac{\mu}{M}\right)^{q+2}\varepsilon
w_{0}$. Upon iteration, one gets

$\textstyle{u\geq\left(\frac{\mu}{M}\right)^{\alpha_{k}}\varepsilon
w_{0}\;\;\textrm{ for all $k\geq 2$,\quad where\; $\alpha_{k}=k(q+1)+1$. }}$

This gives us $\mu\leq M$, and the claim is proved.

Therefore, there exists
$(\mu_{\varepsilon},u_{\varepsilon},v_{\varepsilon})\in\mathcal{C}_{\varepsilon}$
such that $\left\|(u_{\varepsilon},v_{\varepsilon})\right\|_{\infty}=1$. By
compactness of $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$, Theorem 3.4, and SMP,
by taking $\varepsilon\to 0$ we conclude that there exists $\sigma_{1}>0$, in
addition to $u_{1},v_{1}$ such that $\left\|(u_{1},v_{1})\right\|_{\infty}=1$,

$u_{1}=\sigma_{1}\mathcal{F}_{1}(v_{1}^{q})\quad\textrm{and}\quad
v_{1}=\sigma_{1}\mathcal{F}_{2}(u_{1}^{p}).$

The eigenfunctions belong to the interior of the positive cone $K$, by the
positivity of the operators $\mathcal{F}_{i}$ for $i=1,2$. Then the result
follows in the case of positive weights.

Step 2) In the general case we argue by approximation. Since
$\tau_{1},\tau_{2}>0$ a.e. in a common set of positive measure – assumption
(1.5) – then there exists some $\delta>0$ such that
$|\\{\tau_{1}\geq\delta\\}\cap\\{\tau_{2}\geq\delta\\}|>0$. Say
$\tau_{1},\tau_{2}\geq\delta$ a.e. in some ball $B_{R}\subset\subset\Omega$,
$R\leq 1$. Take $\varepsilon\in(0,1)$ and define
$\tau_{i}^{\varepsilon}:=\tau_{i}+\varepsilon>0$ in $\Omega$. By Step 1 we
obtain the existence of $\sigma_{1}^{\varepsilon}>0$ and
$\varphi_{1}^{\varepsilon},\psi_{1}^{\varepsilon}\in C^{1}(\overline{\Omega})$
such that

$\displaystyle
F_{1}[\varphi_{1}^{\varepsilon}]+\sigma_{1}^{\varepsilon}\,\tau_{1}^{\varepsilon}(x)(\psi_{1}^{\varepsilon})^{q}=0,\quad
F_{2}[\psi_{1}^{\varepsilon}]+\sigma_{1}^{\varepsilon}\,\tau_{2}^{\varepsilon}(x)(\varphi_{1}^{\varepsilon})^{p}=0\;\;\textrm{
in }\Omega$
$\displaystyle\varphi_{1}^{\varepsilon},\psi_{1}^{\varepsilon}>0\;\;\textrm{
in }\Omega,\quad\varphi_{1}^{\varepsilon},\psi_{1}^{\varepsilon}=0\;\;\textrm{
on
}\partial\Omega,\quad\textstyle{\max_{\overline{\Omega}}\,\varphi_{1}^{\varepsilon}=1},\;\;\textstyle{\max_{\overline{\Omega}}\,\psi_{1}^{\varepsilon}=1}.$

Then, by Proposition 4.5, $0<\sigma_{1}^{\varepsilon}\leq C_{0}$ for all
$\varepsilon\in(0,1)$. So
$\sigma_{1}^{\varepsilon}\rightarrow\sigma_{1}\in[0,C_{0}]$ up to a
subsequence. Then, applying $C^{1,\alpha}$ global regularity (Proposition
2.8), yields

$\displaystyle\|\varphi_{1}^{\varepsilon}\|_{C^{1,\alpha}(\overline{\Omega})}\leq
C\,\\{\,\|\varphi_{1}^{\varepsilon}\|_{L^{\infty}(\Omega)}+\sigma_{1}^{\varepsilon}\,\|\tau_{1}^{\varepsilon}\|_{L^{\varrho}(\Omega)}\,\|\varphi_{1}^{\varepsilon}\|^{q}_{\infty}\,\\}\leq
C\,(\|\tau_{1}\|_{L^{\varrho}(\Omega)}+1)\,\\}\leq C,$

and analogously
$\|\psi_{1}^{\varepsilon}\|_{C^{1,\alpha}(\overline{\Omega})}\leq C$. Hence
the compact inclusion $C^{1,\alpha}(\overline{\Omega})\subset
C^{1}(\overline{\Omega})$ yields
$\varphi_{1}^{\varepsilon}\rightarrow\varphi_{1}$ and
$\psi_{1}^{\varepsilon}\rightarrow\psi_{1}$ in $C^{1}(\overline{\Omega})$, up
to a subsequence, with $\max_{\overline{\Omega}}\,\varphi_{1}=1$,
$\max_{\overline{\Omega}}\,\psi_{1}=1$, $\varphi_{1},\psi_{1}\geq 0$ in
$\Omega$, and $\varphi_{1},\psi_{1}=0$ on $\partial\Omega$. Since
$\tau_{i}^{\varepsilon}\rightarrow\tau_{i}$ in $L^{\varrho}(\Omega)$ as
$\varepsilon\rightarrow 0$, by stability of viscosity solutions we derive that
$\varphi_{1}$, $\psi_{1}$ is an $L^{N}$-viscosity solution pair of
$F_{1}[\varphi_{1}]+\sigma_{1}\tau_{1}(x)\psi_{1}^{q}=0$,
$F_{2}[\psi_{1}]+\sigma_{1}\tau_{2}(x)\varphi_{1}^{p}=0$ in $\Omega$, which
allows us to apply $C^{1,\alpha}$ regularity again to obtain that
$\varphi_{1},\psi_{1}\in C^{1,\alpha}(\overline{\Omega})$.

Next, by Theorem 3.4 and SMP, we have that $\varphi_{1}>0$ in $\Omega$;
similarly $\psi_{1}>0$. Moreover, we must have $\sigma_{1}>0$. Indeed,
$\sigma_{1}=0$ produces $F_{1}[\varphi_{1}]=0$ in $\Omega$ which in turn would
give us $\varphi_{1}\leq 0$ in $\Omega$ by MP, since we are assuming
$\lambda_{1}^{+}(F_{1}^{*})>0$. Using the regularity property in (H4),
$\varphi_{1}$ and $\psi_{1}$ turn out to be strong solutions. ∎

###### Proof of Theorem 1.3.

Existence. We preliminarily notice that $\lambda_{1}^{+}$ is finite due to
Proposition 4.5. Take $\sigma_{1}^{\pm}$ as in Proposition 5.2. Let us prove
that $\sigma_{1}^{\pm}=\lambda_{1}^{\pm}$. By definition of $\lambda_{1}^{+}$,
we have $\sigma_{1}^{+}\leq\lambda_{1}^{+}$. Suppose there exists
$\varepsilon>0$ such that $\sigma_{1}^{+}<\lambda_{1}^{+}-\varepsilon$. Then,
by definition of $\lambda_{1}^{+}$, we can take $\varphi,\psi>0$ such that

$F_{1}[\varphi]+(\lambda_{1}^{+}-\varepsilon)\tau_{1}(x)\psi^{q}\leq 0,\quad
F_{2}[\psi]+(\lambda_{1}^{+}-\varepsilon)\tau_{2}(x)\varphi^{p}\leq 0.$

Since
$F_{1}[\varphi_{1}^{+}]+(\lambda_{1}^{+}-\varepsilon)\tau_{1}(x)(\psi_{1}^{+})^{q}>F_{1}[\varphi_{1}^{+}]+\sigma_{1}^{+}\tau_{1}(x)(\psi_{1}^{+})^{q}=0$,
then by Proposition 4.1 (and up to suitable rescaling as in Section 2.4) we
have $\varphi=t\varphi_{1}^{+}$ and $\psi=t^{p}\psi_{1}^{+}$ for a suitable
$t>0$, which is a contradiction. Similarly, we obtain
$\sigma_{1}^{-}=\lambda_{1}^{-}$.

Simplicity. Let us prove the case of $\lambda_{1}^{+}$, since for
$\lambda_{1}^{-}$ is analogous. Assume that $(u,v)$ is another eigenfunction
corresponding to $\lambda_{1}^{+}$, with $u\not\equiv 0$ or $v\not\equiv 0$.

In the case that $u$ or $v$ attains a positive maximum in $\Omega$, then we
consider the positive eigenfunctions pair $(\varphi_{1}^{+},\psi^{+}_{1})$ of
the operators $(F_{1},F_{2})$. Then we apply Proposition 4.1 to both pairs
$u,v$ and $\varphi_{1}^{+},\psi^{+}_{1}$ to obtain that $u\equiv
t\varphi_{1}^{+}$ and $v\equiv t^{p}\psi^{+}_{1}$ in $\Omega$ for a suitable
$t\in\mathbb{R}$.

If, in turn, $u\leq 0$ and $v\leq 0$ in $\Omega$, then $u<0$ and $v<0$ in
$\Omega$ by the strong maximum principle for scalar equations. Notice that if
one between $u$ and $v$ is $\equiv 0$, then also the other one has to be null.
Then we consider the negative eigenfunctions pair
$(\varphi_{1}^{-},\psi_{1}^{-})$ of the operators $(F_{1},F_{2})$. Therefore
we use the fact that $\lambda_{1}^{-}$ is the only positive eigenvalue
corresponding to a negative eigenfunction. In other words, we apply Corollary
4.3 to obtain that $\lambda_{1}^{+}=\lambda_{1}^{-}$, as well as $u\equiv
t\varphi^{-}_{1}$ and $v\equiv t^{p}\psi_{1}^{-}$ in $\Omega$. ∎

###  5.2 Maximum principles

###### Proof of Theorem 1.7.

We prove the statement regarding MP; the case mP is analogous.

Step 1) Let us first assume that
$(\lambda,\mu)\in\mathbb{R}^{+}\times\mathbb{R}^{+}$. Then, via the scaling
(2.10), we reduce the problem to check if MP holds for

$-F_{1}[u]=\lambda_{0}\tau_{1}(x)|v|^{q-1}v_{0},\quad-
F_{2}[v]=\lambda_{0}\tau_{2}(x)|u|^{p-1}u_{0}\quad\textrm{ in }\;\Omega\quad
u=v=0\textrm{ on }\partial\Omega.$ (5.1)

if and only if $\lambda_{0}<\lambda_{1}^{+}$. We first notice that, if
$\lambda_{0}\geq\lambda_{1}^{+}$, then MP is not satisfied. Indeed, when
$\lambda_{0}\geq\lambda_{1}^{+}$ we get

$F_{1}[\varphi_{1}^{+}]+\lambda_{0}\tau_{1}(x)(\psi_{1}^{+})^{q}=-\lambda_{1}^{+}\tau_{1}(x)(\psi_{1}^{+})^{q}+\lambda_{0}\tau_{1}(x)(\psi_{1}^{+})^{q}\geq
0,\quad F_{2}[\psi_{1}^{+}]+\lambda_{0}\tau_{2}(x)(\varphi_{1}^{+})^{p}\geq
0\textrm{ in }\Omega,$

with $\varphi^{+}_{1},\psi_{1}^{+}=0$ on $\partial\Omega$, but
$\varphi_{1}^{+}>0,\,\psi_{1}^{+}>0$ in $\Omega$.

Now we prove that MP holds if $0<\lambda_{0}<\lambda_{1}^{+}$. Let $(u,v)$ be
a viscosity solution of

$F_{1}[u]+\lambda_{0}\tau_{1}(x)|v|^{q-1}v\geq
0,\;\;\;F_{2}[v]+\lambda_{0}\tau_{2}(x)|u|^{p-1}u\geq 0\;\;\textrm{ in
}\Omega,\quad\textrm{$u,v\leq 0$ on $\partial\Omega$}.$

Assume by contradiction that one between $u,v$ has a positive maximum in
$\Omega$. Observe that $\varphi_{1}^{+},\psi_{1}^{+}$ is a strong solution
pair of

$F_{1}[\varphi_{1}^{+}]+\,\lambda_{0}\tau_{1}(x)(\psi_{1}^{+})^{q}\leq 0$,
$F_{2}[\psi_{1}^{+}]+\lambda_{0}\tau_{2}(x)(\varphi_{1}^{+})^{p}\leq 0$ in
$\Omega$, $\varphi_{1}^{+},\psi_{1}^{+}>0$ in $\Omega$.

Then Proposition 4.1 yields $u=t\varphi_{1}^{+}$, $v=t^{p}\psi_{1}^{+}$ in
$\Omega$, which contradicts $\lambda_{0}\neq\lambda_{1}^{+}$. So $u,v\leq 0$
in $\Omega$.

Step 2) Let us now check that, if
$(\lambda,\mu)\notin\mathbb{R}^{+}\times\mathbb{R}^{+}$, then MP does not
hold. If $\lambda=0$ or $\mu=0$, then we reduce ourselves to the scalar case,
and the conclusion follows from [42] in the case of bounded drifts, while for
unbounded ones this is a new contribution of this paper, see Section 3. The
only thing which is left to prove is that if at least one between
$\lambda,\mu$ is negative, then the MP and the mP do not hold. Assume for
instance $\lambda<0$. Consider the eigenvalue problem

$\displaystyle
F_{1}[\tilde{\varphi}]+\lambda_{0}\tau_{1}(x)\tilde{\psi}^{q}=0,\quad(F_{2})_{*}[\tilde{\psi}]+\,\lambda_{0}\tau_{2}(x)\tilde{\varphi}^{p}=0\textrm{
in }\Omega,$

where $\tilde{\varphi},\tilde{\psi}>0$ in $\Omega$,
$\tilde{\varphi},\tilde{\psi}=0$ on $\partial\Omega$,
$\lambda_{0}=\lambda_{1}^{+}(F_{1},(F_{2})_{*})$. Let $\bar{\lambda}=1/s^{q}$,
$\bar{\mu}=s\lambda$. Hence, choosing a suitable $s>0$, we have
$\bar{\lambda}>0$ and $\bar{\mu}<0$ such that $\bar{\mu}\leq-\lambda_{0}$.
Hence,

$\begin{cases}F_{1}[\tilde{\varphi}]+\bar{\lambda}\tau_{1}(x)|\tilde{\psi}|^{q-1}(-\tilde{\psi})=F_{1}[\tilde{\varphi}]-\bar{\lambda}\tau_{1}(x)(\tilde{\psi})^{q}=-\tau_{1}(x)\,(\lambda_{0}+\bar{\lambda})\,(\tilde{\psi})^{q}\leq
0,\\\
F_{2}[-\tilde{\psi}]+\bar{\mu}\tau_{2}(x)(\tilde{\varphi})^{p}\leq-(F_{2})_{*}[\tilde{\psi}]+\bar{\mu}\tau_{2}(x)(\tilde{\varphi})^{p}=\tau_{2}(x)\,(\lambda_{0}+\bar{\mu})\,\varphi)^{p}\leq
0.\end{cases}$

However, $-\tilde{\psi}<0$ in $\Omega$. ∎

###  5.3 The Dirichlet problem for $\lambda<m_{1}$

Now we focus on the nonhomogeneous Dirichlet Lane-Emden problem, proving item
$(i)$ of Theorem 1.8. We postpone the proof of items $(ii)$ and $(iii)$ to
Section 8 ahead, as they turn out to follow from an adaptation of the proof of
Theorem 1.17. Recall $m_{1},M_{1}$ from (2.6). We consider functions $f_{i}\in
L^{\varrho}(\Omega)$. Via the scaling (2.10), we reduce our problem to the
study of system

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}[u]+\lambda\tau_{1}(x)|v|^{q-1}v&=&f_{1}(x)&\mbox{in}&\;\Omega,\\\
F_{2}[v]+\lambda\tau_{2}(x)|u|^{p-1}u&=&f_{2}(x)&\mbox{in}&\;\Omega,\\\
u\,\;=\,\;v&=&0&\mbox{on}&\;\partial\Omega.\end{array}\right.$ (5.5)

and the condition $(\lambda,\mu)\in\mathcal{C}_{1}^{+}\cap\mathcal{C}_{1}^{-}$
for (1.10) translates to $0<\lambda<m_{1}$ for (5.5).

###### Proof of Theorem 1.8 (i).

We sketch the proof in light of [30]. One first obtains the following a priori
bounds.

######  Claim 5.3.

Let $(u,v)$ be a viscosity solution of (5.5). Then

$\left\|u\right\|_{\infty}+\|{v}\|_{\infty}^{p}\leq
C\\{\,\|{f_{1}}\|_{L^{N}(\Omega)}+\|{f_{2}}\|_{L^{N}(\Omega)}\\},$ for all
$0<\lambda<m_{1}$.

In order to see this, let us assume by contradiction that there exist
solutions such that

$\textstyle{\left\|u_{k}\right\|_{\infty}+\left\|v_{k}\right\|^{p}_{\infty}>k\,\\{\,\|{f_{1}^{k}}\|_{\varrho}+\|{f_{2}^{k}}\|_{\varrho}\\}.}$

We normalize as in [30, Section 5] to get new functions
$\tilde{u}_{k},\tilde{v}_{k},\tilde{f}_{1}^{k},\tilde{f}_{2}^{k}$ such that

$\textstyle{\left\|\tilde{u}_{k}\right\|_{\infty}+\left\|\tilde{v}_{k}\right\|^{p}_{\infty}=1,\quad\|{\tilde{f}_{1}^{k}}\|_{\varrho},\;\|{\tilde{f}_{2}^{k}}\|_{\varrho}<1/k.}$

The limit functions $\tilde{u},\tilde{v}$ solve (1.10) for $\lambda=\mu$, and
they are nonzero due to our normalization. Hence, they are an eigenfunction,
however $\lambda<m_{1}$, a contradiction. So Claim 5.3 is true.

Define $H(t,u,v)=(U,V)$ as the viscosity solution to the problem

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}\,[U]+\tau_{1}(x)t\lambda|v|^{q-1}v&=&tf_{1}(x)&\mbox{in}&\;\Omega,\\\
F_{2}\,[V]+\tau_{2}(x)t\lambda|u|^{p-1}u&=&tf_{2}(x)&\mbox{in}&\;\Omega,\\\
U\,\;=\,\;V&=&0&\mbox{on}&\;\partial\Omega,\end{array}\right.$

in the space $C^{1}(\overline{\Omega})$. By (H1)-(H4) for $F_{1},F_{2}$,
together with regularity-estimates $C^{1,\alpha}$ theory of viscosity
solutions, the map $H$ is well defined and completely continuous. Observe that
$t\lambda<\min\\{\lambda_{1}^{-},\lambda_{1}^{+}\\}$ for any $t\in[0,1]$. By
using this and the estimates above, we have that any $(u,v)$ which satisfies
$H(t,u,v)=(u,v)$ is bounded by a constant not depending on $t$. By degree
fixed point theory we get a solution to our problem. ∎

###  5.4 Local isolation

###### Proof of Theorem 1.5.

Once again, via the scaling (2.10), we reduce the problem to the study of
(1.10) with $\lambda=\mu$, that is (2.7). Recall $M_{1},m_{1}$ from (2.6). For
$0<\lambda<M_{1}$ then the system (1.10) with $\mu=\lambda$ satisfies MP or mP
depending on whether $M_{1}=\lambda_{1}^{+}$ or $\lambda_{1}^{-}$. Suppose
that $u,v\in C(\bar{\Omega})$ is a nontrivial eigenfunction pair associated to
the eigenvalue $\lambda$, and let $M_{1}=\lambda_{1}^{-}$. If
$\lambda<\lambda_{1}^{-}$ then (1.10) with $\lambda=\mu$ satisfies mP.
Therefore $u,v\geq 0$ in $\Omega$, and $u,v>0$ in $\Omega$ by Hopf. By
Proposition 4.1 one has $\lambda=m_{1}=\lambda_{1}^{+}$. Analogously we deduce
that if $M_{1}=\lambda^{+}_{1}$ then $\lambda=\lambda^{-}_{1}$. In particular,
no eigenfunctions exist on the left of $M_{1}$, except from the one
corresponding to $m_{1}$.

Let us show that there is a neighborhood on the right of $M_{1}$ where
eigenfunctions do not exist. We assume w.l.g. that $M_{1}=\lambda_{1}^{-}$
(the other case is analogous). Let us take a sequence of positive eigenvalues
$\lambda_{n}=\lambda_{1}^{-}+\varepsilon_{n}$ related to normalized
eigenfunctions $(u_{n},v_{n})$ such that $\varepsilon_{n}\to 0$. Then by
stability of viscosity solutions (Proposition 2.5), $(u_{n},v_{n})\to(u,v)$
eigenfunction related to $\lambda_{1}^{-}$. Then, by Proposition 4.1 one
concludes $u=t\varphi_{1}^{-}$ and $v=t^{p}\psi_{1}^{-}$ for some $t>0$. Note
that by Krein-Rutman theorem $\varphi_{1}^{-},\psi_{1}^{-}$ belong to the
interior of the cone of negative solutions. Then $u_{n},v_{n}<0$ for large
$n$. This implies by Proposition 4.1 that $\lambda_{n}=\lambda_{1}^{-}$, which
is a contradiction. ∎

##  6 The anti-maximum principle

In this section we go along with the validity of the maximum principle, this
time when it fails and takes the form of an anti-maximum principle. We move
towards the proof of Theorem 1.10. Our approach relies on some arguments in
[4, 24]. Once again, we reduce ourselves to one of the equivalent forms of
(1.10), and in particular we will always assume that the system is written in
the form (2.7), see Section 2.4.

We start with an auxiliary nonexistence result. We assume $F_{i}$ satisfies
(H1)–(H3), $i=1,2$.

######  Lemma 6.1.

Let $\tau_{1},\tau_{2}\in L^{\varrho}_{+}(\Omega)$ for some $\varrho>N$, and
$f_{i}\in L^{\varrho}(\Omega)$ such that $f_{i}\not\equiv 0$.

  1. (i)

If $\lambda\geq\lambda_{1}^{+}$ and $f_{i}\leq 0$ then there is no nonnegative
solution $u,v\in C(\overline{\Omega})$ of

$\displaystyle F_{1}[u]+\lambda\tau_{1}(x)|v|^{q-1}v\leq
f_{1},\;\;\;F_{2}[v]+\lambda\tau_{2}(x)|u|^{p-1}u\leq f_{2}\;\;\textrm{ in
}\;\Omega,\quad u,v\geq 0\textrm{ on }\partial\Omega.$ (6.1)

If in addition $\lambda_{1}^{+}\leq\lambda\leq\lambda_{1}^{-}$, the problem
(6.1) does not possess a solution $u,v\in C(\overline{\Omega})$.

  2. (ii)

On the other hand, if $\lambda\geq\lambda_{1}^{-}$ and $f_{i}\geq 0$ there is
no nonpositive solution $u,v\in C(\overline{\Omega})$ of

$F_{1}[u]+\lambda\tau_{1}(x)|v|^{q-1}v\geq
f_{1},\;\;\;F_{2}[v]+\lambda\tau_{2}(x)|u|^{p-1}u\geq f_{2}\;\;\textrm{ in
}\;\Omega,\quad u,v\leq 0\textrm{ on }\partial\Omega.$ (6.2)

Moreover, if $\lambda_{1}^{-}\leq\lambda\leq\lambda_{1}^{+}$, the problem
(6.2) does not possess a solution $u,v\in C(\overline{\Omega})$.

###### Proof.

We prove just $(i)$, since the case $(ii)$ is analogous. Assume by
contradiction that there exists a solution pair $u,v\geq 0$ of (6.1) with
$\lambda\geq\lambda_{1}^{+}$ and $f_{i}\leq 0$. Note that the case $u,v\equiv
0$ is not allowed by the hypothesis $f_{i}\not\equiv 0$. Thus, assume that
$u\not\equiv 0$ or $v\not\equiv 0$. Thus by SMP we have $u,v>0$ in $\Omega$.
The definition of $\lambda^{+}_{1}$ implies $\lambda\leq\lambda^{+}_{1}$, from
which we deduce $\lambda=\lambda^{+}_{1}$. By Proposition 4.1 we obtain that
$u=t\varphi^{+}_{1}$ and $v=t^{p}\psi_{1}^{+}$ for some $t>0$. However, this
leads to $f_{i}\equiv 0$, $i=1,2$, which contradicts the hypoteses.

Now assume in addition that $\lambda\leq\lambda^{-}_{1}$, and on the contrary
that there exists a solution $u,v$ of (6.1). By what we just proved, either
$u$ or $v$ must be negative somewhere in $\Omega$. Applying Proposition 4.1 we
get that $u\equiv t\varphi^{-}_{1}$ and $v\equiv t^{p}\psi^{-}_{1}$ in
$\Omega$ for some $t>0$. This yields $\lambda=\lambda^{-}_{1}$, and so
$f_{i}\equiv 0$, $i=1,2$, again a contradiction. ∎

###### Proof of Theorem 1.10.

We prove only $(i)$; $(ii)$ is similar. In order to get a contradiction,
suppose that there exist values $\lambda_{k}$ above $\lambda_{1}^{-}$, and
$u_{k},v_{k}\in C(\overline{\Omega})$ such that
$\lambda_{k}\to\lambda_{1}^{-}$, and $u_{k},v_{k}\in C(\overline{\Omega})$
satisfying

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}[u_{k}]+\lambda_{k}\tau_{1}(x)|v_{k}|^{q-1}v_{k}&=&f_{1}(x)&\mbox{in}&\;\Omega\\\
F_{2}[v_{k}]+\lambda_{k}\tau_{2}(x)|u_{k}|^{p-1}u_{k}&=&f_{2}(x)&\mbox{in}&\;\Omega\\\
u_{k}\,\;=\,\;v_{k}&=&0&\mbox{on}&\;\partial\Omega\end{array}\right.$ (6.6)

such that at least one between $u_{k},v_{k}$ is nonnegative somewhere in
$\Omega$. It then turns out that $u_{k}$ or $v_{k}$ is nonnegative somewhere
for infinite $k$’s, say $u_{k}$. Thus, take such $x_{k}\in\Omega$ where
$u_{k}$ attains a nonnegative maximum at $x_{k}$. In particular, for this
subsequence one has

$\textstyle{u_{k}(x_{k})\geq 0\quad\mbox{and}\quad Du_{k}(x_{k})=0.}$ (6.7)

By taking a further subsequence we may assume $x_{k}\rightarrow x_{0}$ for
some $x_{0}\in\overline{\Omega}$. We claim that

$\textstyle{\mbox{either}\quad\sup_{k}\|u_{k}\|_{L^{\infty}(\Omega)}=+\infty\quad\mbox{
or }\;\quad\sup_{k}\|v_{k}\|_{L^{\infty}(\Omega)}=+\infty.}$ (6.8)

Otherwise, if
$\|u_{k}\|_{L^{\infty}(\Omega)},\|v_{k}\|_{L^{\infty}(\Omega)}\leq C$ for all
$k$, then by $C^{1,\alpha}$ estimates and compact inclusion we obtain $u,v\in
C^{1}(\overline{\Omega})$ such that $u_{k}\rightarrow u$ and $v_{k}\rightarrow
v$ uniformly in $\overline{\Omega}$. By stability of viscosity solutions we
may pass to limits in (6.6), then $u,v$ become solution of the problem

$\displaystyle F_{1}[u]+\lambda_{1}^{-}\tau_{1}(x)|v|^{q-1}v=f_{1}(x),\quad
F_{2}[v]+\lambda_{1}^{-}\tau_{2}(x)|u|^{p-1}u=f_{2}(x)\;\;\mbox{ in
}\;\Omega,\quad u=v=0\mbox{ on }\;\partial\Omega.$

But this contradicts Lemma 6.1 (i), whence (6.8) is proved.

Let us assume w.l.g. that $\left\|u_{k}\right\|_{\infty}\to\infty$ (up to a
subsequence), and let us define $\theta_{k}=\|u_{k}\|_{\infty}$. We also
consider the rescaling $u_{k}=\theta_{k}U_{k}$ and
$v_{k}=\theta_{k}^{p}\,V_{k}$. Since the operators $F_{1},F_{2}$ are
positively $1$-homogeneous,

$\textstyle-
F_{1}[U_{k}]=-\frac{F_{1}[u_{k}]}{\theta_{k}}=\frac{1}{\theta_{k}}\\{\lambda_{k}\tau_{1}(x)|v_{k}|^{q-1}v_{k}-f_{1}(x)\\}=\lambda_{k}\tau_{1}(x)|V_{k}|^{q-1}V_{k}-\frac{f_{1}(x)}{\theta_{k}},$

and

$\textstyle-
F_{2}[V_{k}]=-\frac{F_{2}[v_{k}]}{\theta_{k}^{p}}=\frac{1}{\theta_{k}^{p}}\\{\lambda_{k}\tau_{2}(x)|u_{k}|^{p-1}u_{k}-f_{2}(x)\\}={\lambda_{k}\tau_{2}(x)}|U_{k}|^{p-1}U_{k}-\frac{f_{2}(x)}{\theta_{k}^{p}},$

with $U_{k}=V_{k}=0$ on $\partial\Omega$. Again by $C^{1,\alpha}$ estimates,
our construction, and taking a subsequence if necessary, we may assume
$U_{k}\rightarrow U$, $V_{k}\rightarrow V$ in $C^{1}(\overline{\Omega})$ for
some $U,V\in C^{1}(\overline{\Omega})$. Notice that $\left\|U_{k}\right\|=1$,
and hence the RHS in the last equality is uniformly bounded, therefore by
Theorem 3.4 $V_{k}$ is also bounded. Passing to limits via stability of
viscosity solutions, we deduce that $U,V$ is a solution of

$\displaystyle F_{1}[U]+\lambda_{1}^{-}\tau_{1}(x)|V|^{q-1}V=0,\quad
F_{2}[V]+\lambda_{1}^{-}\tau_{2}(x)|U|^{p-1}U=0\;\;\mbox{ in }\;\Omega,\quad
U=V=0\mbox{ on }\;\partial\Omega.$

If instead $\left\|v_{k}\right\|_{\infty}\to\infty$ we argue similarly, by
defining $\theta_{k}=\left\|v_{k}\right\|_{\infty}^{q}$.

Anyway, our construction produces $\|U\|_{\infty}=1$ or $\|V\|_{\infty}=1$.
Notice that if either $V\equiv 0$ or $U\equiv 0$ in $\Omega$, then $U\equiv
V\equiv 0$ in $\Omega$ which produces a contradiction. W.l.g. suppose
$\|U\|_{\infty}=1$ and fix $x_{1}\in\Omega$ such that $U(x_{1})\neq 0$. By
Proposition 4.1 we conclude that $U\equiv t\varphi$ and $V\equiv t^{p}\psi$
for some $t>0$, where $\varphi=\varphi_{1}^{+},\,\psi=\psi_{1}^{+}$ if
$U(x_{1})>0$ (since $\lambda_{1}^{-}\geq\lambda_{1}^{+}$), while
$\varphi=\varphi_{1}^{-},\,\psi=\psi_{1}^{-}$ if $U(x_{1})<0$. Let us finish
the proof by showing that both cases are not admissible.

First, if $U(x_{1})<0$, then $U,V<0$ in $\Omega$. Using (6.7) we deduce that
$U(x_{0})=0$, and $x_{0}\in\partial\Omega$. However, $DU(x_{0})=0$, in
violation of Hopf lemma, and so $U(x_{1})<0$ fails to be true.

Finally, if $U(x_{1})>0$, then $U,V>0$ in $\Omega$. Note that $U_{k},V_{k}>0$
in any given compact set $K\subset\Omega$, for $k$ suitably large. We claim
that for $k$ sufficiently large we have $U_{k},V_{k}\geq 0$ in $\Omega$.
Indeed, we consider $K\subset\Omega$ such that $|\Omega\setminus
K|<\varepsilon_{0}$, where $\varepsilon_{0}$ is the constant of Theorem 1.16.
Thus $U_{k},V_{k}\geq 0$ in $\partial(\Omega\setminus K)$, and so in
$\Omega\setminus K$ by Theorem 1.16 since
$\lambda_{1}^{-}(F_{1}^{*}(\vartheta)),\,\lambda_{1}^{-}(F_{2}^{*}(\vartheta))>0$
and $f_{1},f_{2}\leq 0$. Hence we derive a contradiction with Lemma 6.1, from
where $U(x_{1})>0$ is also impossible. ∎

##  7 The second eigenvalue problem

This section is dedicated to the proof of Theorem 1.11. Throughout this
section, we will assume $F_{i}$ both satisfy (H1)–(H4). We start recalling

$\lambda_{2}=\lambda_{2}(F_{1},F_{2},\Omega)=\inf\\{\lambda>M_{1}:\,\lambda\text{
is an eigenvalue of \eqref{hyperbola mu=lambda}}\\},$

with
$M_{1}=\max\\{\lambda_{1}^{+}(F_{1},F_{2}),\lambda_{1}^{-}(F_{1},F_{2})\\}$,
as in (2.6). By Theorem 1.5 we know that $\lambda_{2}>M_{1}$. Note that one
may have $\lambda_{2}=+\infty$, for instance if $F_{1},F_{2}$ are not
symmetric, see [4]. However, when $\lambda_{2}$ is finite then it is in fact
an eigenvalue, as it is shown below.

######  Lemma 7.1.

If $\lambda_{2}<+\infty$, then there is a nontrivial solution pair
$\varphi_{2},\psi_{2}\in E_{\varrho}$ of

$F_{1}[\varphi_{2}]+\lambda_{2}\tau_{1}(x)|\psi_{2}|^{q-1}\psi_{2}=0,\;F_{2}[\psi_{2}]+\lambda_{2}\tau_{2}(x)|\varphi_{2}|^{p-1}\varphi_{2}=0\textrm{
in }\Omega\;\;\varphi_{2},\psi_{2}=0\textrm{ in }\partial\Omega.$ (7.1)

###### Proof.

Take a sequence $\lambda_{k}\to\lambda_{2}$ of eigenvalues, with corresponding
eigenfunctions $u_{k},v_{k}\in E_{\varrho}$, with $\|u_{k}\|_{\infty}=1$, and
solving, in the viscosity sense,

$F_{1}[u_{k}]+\lambda_{k}\tau_{1}(x)|v_{k}|^{q-1}v_{k}=0,\;F_{2}[v_{k}]+\lambda_{k}\tau_{2}(x)|u_{k}|^{p-1}u_{k}=0\textrm{
in }\Omega\;\;u_{k},v_{k}=0\textrm{ in }\partial\Omega.$ (7.2)

By $C^{1,\alpha}$ regularity-estimates for scalar equations (Proposition 2.8)
we have
$\|u_{k}\|_{C^{1,\alpha}(\overline{\Omega})},\|v_{k}\|_{C^{1,\alpha}(\overline{\Omega})}\leq
C.$ Thus, up to a subsequence, $u_{k},v_{k}$ converge to functions
$\varphi_{2},\psi_{2}\in C^{1}(\overline{\Omega})$. Since
$\|\varphi_{2}\|_{\infty}=1$ we may pass to the limit in (7.2) via stability
of viscosity solutions. Then $\varphi_{2},\psi_{2}$ is a nontrivial pair of
solutions to the problem (7.1). Observe that $\varphi_{2},\psi_{2}\in
E_{\varrho}$ by hypothesis (H4). ∎

###### Proof of Theorem 1.11-(i).

From the previous lemma and by scaling as in Section 2.4, a second spectral
curve $\Lambda_{2}$ is produced in the first quadrant if
$\lambda_{2}<+\infty$:

$\textstyle\Lambda_{2}\,(\lambda)\,=\,(\,\lambda,\,\mu_{2}(\lambda)\,),\quad\text{
where }\;\mu_{2}(\lambda)=\frac{(\lambda_{2})^{p+1}}{\lambda^{p}},\;\;\text{
for all }\lambda>0.$ (7.3)

Notice that $\Lambda_{2}$ and the curve originating from $M_{1}$ cannot
intersect. This is a consequence of $M_{1}<\lambda_{2}$ together with the
definition of the curves, given by (2.12) and (7.3). ∎

Now we turn to the Dirichlet problem (5.5) for $\lambda\in(M_{1},\lambda_{2})$
in the case $p=q=1$, proving Theorem 1.11-(ii).

We closely follow [4, §5], highlighting only the main differences. We define a
homotopy between (1.4) and the corresponding Dirichlet problem for the
Laplacian with constant weights. For each $0\leq s\leq 1$, define the fully
nonlinear operators

$F^{s}_{i}(x,r,\eta,X)=\beta
s\,\mathrm{tr}(X)+(1-s)F_{i}(x,r,\eta,X),\;\;i=1,2.$ (7.4)

It is easy to verify that $F^{s}_{i}$ satisfies (H1)-(H4). Note that

$\lambda^{\pm}_{1,s}:=\lambda^{\pm}_{1}(F_{1}^{s}(\tau_{1}^{s}),F^{s}_{2}(\tau_{2}^{s}),\Omega)<\infty,$

where $\tau_{i}^{s}=s+(1-s)\tau_{i}$ due to our Proposition 4.5. Analogously
to [4, Lemmas 5.4 & 5.5], one sees that the maps $s\mapsto\lambda^{+}_{1,s}$
and $s\mapsto\lambda^{-}_{1,s}$ are continuous on $[0,1]$, while the map

$s\mapsto\lambda_{2,s}=\lambda_{2}(F^{s}_{1}(\tau_{1}^{s}),F^{s}_{2}(\tau_{2}^{s}),\Omega)$

is lower semi-continuous on $[0,1]$. We also have the following analog of [4,
Proposition 5.6].

######  Lemma 7.2.

Let $M_{1}<\lambda<\lambda_{2}.$ Then there is a continuous function
$\mu:[0,1]\to\mathbb{R}$ such that $\mu_{0}=\lambda$ and
$\max\\{\lambda_{1,s}^{-},\lambda_{1,s}^{+}\\}<\mu_{s}<\lambda_{2,s}$ for all
$s\in[0,1].$ Furthermore, for every constant $M>0$, there is a constant $C>0$
such that for any $f_{1},f_{2}\in L^{\varrho}(\Omega)$ satisfying
$\|f_{1}\|_{\varrho},\|f_{2}\|_{\varrho}\leq M$, for all $0\leq s\leq 1$, and
for any solution $u,v\in C(\overline{\Omega})$ of the Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}^{s}\,[u]+\mu_{s}\tau_{1}^{s}(x)v&=&f_{1}(x)&\mbox{in}&\;\Omega,\\\
F_{2}^{s}\,[v]+\mu_{s}\tau_{2}^{s}(x)u&=&f_{2}(x)&\mbox{in}&\;\Omega,\\\
u\,\;=\,\;v&=&0&\mbox{on}&\;\partial\Omega,\end{array}\right.$ (7.8)

we have the estimate
$\|u\|_{C^{1,\alpha}(\overline{\Omega})},\|v\|_{C^{1,\alpha}(\overline{\Omega})}<C(1+\max\\{\|f_{1}\|_{\varrho},\|f_{2}\|_{\varrho}\\}).$

###### Proof.

The existence of $\mu_{s}$ follows from the continuity statements that precede
the lemma. As for the $C^{1,\alpha}$ estimates, they follow once we obtain
$L^{\infty}$ bounds. Assuming the conclusion is false, we have the existence
of $M>0$ and sequences $0\leq s_{k}\leq 1$, $f_{1k},f_{2k}\in
L^{\varrho}(\Omega)$ with $\|f_{1k}\|_{\varrho},\|f_{2k}\|_{\varrho}\leq M$,
and $u_{k},v_{k}\in C(\overline{\Omega})$ solutions of

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}^{s}\,[u_{k}]+\mu_{s_{k}}\tau_{1}^{s_{k}}(x)v_{k}&=&f_{1k}(x)&\mbox{in}&\;\Omega,\\\
F_{2}^{s}\,[v_{k}]+\mu_{s_{k}}\tau_{2}^{s_{k}}(x)u_{k}&=&f_{2k}(x)&\mbox{in}&\;\Omega,\\\
u_{k}\,\;=\,\;v_{k}&=&0&\mbox{on}&\;\partial\Omega,\end{array}\right.$

such that (without loss of generality)

$\frac{\|v_{k}\|_{\infty}}{1+\max\\{\|f_{1k}\|_{\varrho},\|f_{2k}\|_{\varrho}\\}}\leq\frac{\|u_{k}\|_{\infty}}{1+\max\\{\|f_{1k}\|_{\varrho},\|f_{2k}\|_{\varrho}\\}}\to+\infty\qquad\text{
as }k\to+\infty.$

By letting

$\tilde{u}_{k}:=\frac{u_{k}}{\|u_{k}\|_{\infty}},\quad\tilde{v}_{k}:=\frac{v_{k}}{\|u_{k}\|_{\infty}},$

we have $\|\tilde{u}_{k}\|_{\infty}=1$, $\|\tilde{v}_{k}\|_{\infty}\leq 1$ and

$F_{1}^{s}[\tilde{u}_{k}]+\mu_{s_{k}}\tau_{1}^{s_{k}}\tilde{v}_{k}=\frac{f_{1k}}{\|u_{k}\|_{\infty}},\quad
F_{2}^{s}[\tilde{v}_{k}]+\mu_{s_{k}}\tau_{1}^{s_{k}}\tilde{u}_{k}=\frac{f_{2k}}{\|u_{k}\|_{\infty}}\text{
in }\Omega.$

Observing that the right hand sides converge to $0$ in $L^{\varrho}$, by
$C^{1,\alpha}$ estimates we can pass to the limit an obtain the existence of
$\bar{s}\in[0,1]$ for which $\mu_{\bar{s}}$ is an eigenvalue, a contradiction.
∎

Let us fix $f_{i}\in L^{\varrho}(\Omega)$, and set
$E=C^{1}(\overline{\Omega})^{2}$. We define a map
$\mathcal{A}_{s}=\mathcal{A}_{f_{1},f_{2},s}:E\times[0,1]\to E$ by
$\mathcal{A}_{s}(u,v)=(U,V)$, where $(U,V)\in E$ is the unique strong solution
of the Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}^{s}\,[U]+\mu_{s}\tau_{1}^{s}(x)v&=&f_{1}(x)&\mbox{in}&\;\Omega,\\\
F_{2}^{s}\,[V]+\mu_{s}\tau_{2}^{s}(x)u&=&f_{2}(x)&\mbox{in}&\;\Omega,\\\
U\,\;=\,\;V&=&0&\mbox{on}&\;\partial\Omega.\end{array}\right.$ (7.12)

Observe that the map is well defined by Theorem 3.7.

######  Lemma 7.3.

The map $\mathcal{A}_{s}$ is a homotopy of completely continuous
transformations on $E$.

###### Proof.

For each $s\in[0,1]$, whenever $(U,V)=\mathcal{A}_{s}(u,v)$

$\textstyle\|U\|_{C^{1,\alpha}(\overline{\Omega})}<C(\max_{s\in[0,1]}|\mu_{s}|\,\|v\|_{L^{\varrho}(\Omega)}+\|f_{1}\|_{L^{\varrho}(\Omega)})\leq
C(1+\|v\|_{L^{\infty}(\Omega)}),$ (7.13)

and also
$\|V\|_{C^{1,\alpha}(\overline{\Omega})}<C(1+\|u\|_{L^{\infty}(\Omega)}).$
Thus the operator $(u,v)\mapsto\mathcal{A}_{s}(u,v)$ is completely continuous
for each fixed $s\in[0,1]$. Let us show that for each constant $R>0$, the map
$(u,v,s)\mapsto\mathcal{A}_{s}(u,v)$ is uniformly continuous on the set
$B^{E}_{R}(0)\times[0,1].$ By contradiction assume there exist
$\varepsilon>0$, and sequences of numbers $s_{k},t_{k}\in[0,1]$ and functions
$u_{k},v_{k}\in C^{1}(\overline{\Omega})$ such that $|s_{k}-t_{k}|\rightarrow
0$, $\|(u_{k},v_{k})\|_{E}<R$ but

$\|(U_{k},V_{k})-(\tilde{U}_{k},\tilde{V}_{k})\|_{E}\geq\varepsilon,$ (7.14)

where $(U_{k},V_{k})=\mathcal{A}_{s}(u_{k},v_{k},s_{k})$ and
$(\tilde{U}_{k},\tilde{V}_{k})=\mathcal{A}_{s}(u_{k},v_{k},t_{k})$. By (7.13)
and up to a subsequence, we can find $s\in[0,1]$, functions $(u,v)\in E$ and
$(U,V),(\tilde{U},\tilde{V})\in E$ such that $s_{k}\rightarrow s$,
$t_{k}\rightarrow s$, $v_{k}\rightarrow v$ uniformly on $\overline{\Omega}$,
$(U_{k},V_{k})\rightarrow(U,V)$ in $E$,
$(\tilde{U}_{k},\tilde{V}_{k})\rightarrow(\tilde{U},\tilde{V})$ in $E$.
Passing to limits, we deduce that $U,V$ and $\tilde{U},\tilde{V}$ are both
solutions of the problem

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}^{s}\,[U]+\mu_{s}\tau_{1}^{s}(x)v&=&f_{1}(x)&\mbox{in}&\;\Omega,\\\
F_{2}^{s}\,[V]+\mu_{s}\tau_{2}^{s}(x)u&=&f_{2}(x)&\mbox{in}&\;\Omega,\\\
U\,\;=\,\;V&=&0&\mbox{on}&\;\partial\Omega.\end{array}\right.$

By the uniqueness properties of the operators $F_{1}^{s},F_{2}^{s}$ (Theorem
3.7) one has $U=\tilde{U}$, $V=\tilde{V}$, which contradicts (7.14). This
completes the proof. ∎

Now we look at the following operator
$\mathcal{B}_{s}=\mathcal{B}_{f_{1},f_{2},s}:E\to E$ by
$\mathcal{B}_{s}(u,v)=(u,v)-\mathcal{A}_{s}(u,v).$ For $(u,v)\in E$, set
$(w,z):=\mathcal{A}_{s}(u,v)$, then $(U,V)=\mathcal{B}_{s}(u,v)$ is equivalent
to $w=u-U$ and $z=v-V$ solving the Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{rclcc}F_{1}^{s}\,[w]+\mu_{s}\tau_{1}^{s}(x)v&=&f_{1}(x)&\mbox{in}&\;\Omega,\\\
F_{2}^{s}\,[z]+\mu_{s}\tau_{2}^{s}(x)u&=&f_{2}(x)&\mbox{in}&\;\Omega,\\\
w\,\;=\,\;z&=&0&\mbox{on}&\;\partial\Omega.\end{array}\right.$ (7.18)

Our goal is to show the existence of a solution $(u,v)\in E$ of the equation
$\mathcal{B}_{0}(u,v)=0,$ where $0=(0,0)\in E$. This will be accomplished by
showing that $\deg(\mathcal{B}_{0},B_{R}^{E},0)\neq 0$ for some ball
$B_{R}^{E}\subseteq E$, and then appealing Leray-Schauder degree theory.

######  Lemma 7.4.

Let $R:=1+C(1+\max\\{\|f_{1}\|_{\varrho},\|f_{2}\|_{\varrho}\\})$, for $C$ as
in Lemma 7.2. Then

$\deg(\mathcal{B}_{1},B_{R}^{E}(0),0)=\pm 1.$

###### Proof.

We will show that $\mathcal{B}_{1}=I-\mathcal{A}_{1}$ is bijective. This is
equivalent to prove that for any $(U,V)\in E$ there exists a unique solution
$(w,z)$ to the following

$\beta\Delta w+\mu_{1}z+\mu_{1}V=f_{1},\quad\beta\Delta
z+\mu_{1}w+\mu_{1}U=f_{2}\;\textrm{ in }\;\Omega,\quad w\,=\,z=0\;\textrm{ on
}\;\partial\Omega.$

namely to the following

$\beta\Delta w+\mu_{1}z=g_{1},\quad\beta\Delta z+\mu_{1}w=g_{2}\;\textrm{ in
}\;\Omega,\quad w\,=\,z=0\;\textrm{ on }\;\partial\Omega,$ (7.19)

for any $g_{1},g_{2}$ functions in $L^{\varrho}(\Omega)$. Recall that
$\mu_{1}$ satisfies
$\lambda_{1}(\beta\Delta)=\lambda_{1}(\beta\Delta,\beta\Delta)<\mu_{1}<\lambda_{2}(\beta\Delta,\beta\Delta)=\lambda_{2}(\beta\Delta)$.
We first consider the case $g_{1},g_{2}\in C_{c}^{\infty}(\Omega)$. Take a
basis for $L^{2}$ given by positive eigenvectors $\varphi_{i}$ of the operator
$\beta\Delta$ with related eigenvalues $\lambda_{i}$, for $i=1,\dots,N$. Then
we can write

$\textstyle w=\sum_{i}a_{i}\varphi_{i},\quad z=\sum_{i}b_{i}\varphi_{i}$