# Multiple solutions for the fractional $p$-Laplacian
with jumping reactions

Silvia Frassu, Antonio Iannizzotto  Department of Mathematics and Computer
Science
University of Cagliari
Via Ospedale 72, 09124 Cagliari, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study a nonlinear elliptic equation driven by the degenerate fractional
$p$-Laplacian, with Dirichlet type condition and a jumping reaction, i.e.,
$(p-1)$-linear both at infinity and at zero but with different slopes crossing
the principal eigenvalue. Under two different sets of hypotheses, entailing
different types of asymmetry, we prove the existence of at least two
nontrivial solutions. Our method is based on degree theory for monotone
operators and nonlinear fractional spectral theory.

###### Key words and phrases:

Fractional $p$-Laplacian, Jumping reactions, Degree theory.

###### 2010 Mathematics Subject Classification:

35P30, 35R11, 47H11.

Version of August 27, 2024

##  1\. Introduction

Nonlinear elliptic equations with jumping (a.k.a. asymmetric, or crossing)
reactions represent a classical subject of investigation in nonlinear
analysis. Such equations can be written in the following general form:

$-L_{p}u=f(x,u)\quad\text{in $\Omega$,}$

coupled with some boundary conditions. Here $\Omega$ is some domain, $L_{p}$
is an elliptic operator, which is $(p-1)$-homogeneous for some $p>1$ (linear
if $p=2$), and $f:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a Carathéodory
function s.t. the quotient

$t\mapsto\frac{f(x,t)}{|t|^{p-2}t}$

has different finite limits for $|t|\to\infty$ and/or $t\to 0$. The study of
such problems goes back to [3], with relevant contributions from [16] (where
the term ’jumping’ was also introduced), [32], and [9] (where a general
abstract formulation of the problem was given). All the cited works deal with
the semilinear case. In the quasilinear case, we recall the results of [1, 20,
28] (dealing with the Dirichlet $p$-Laplacian), [2] (dealing with the Neumann
$p$-Laplacian). In the nonlocal framework, we recall [25] (dealing with the
fractional Laplacian). Since the reaction is asymptotically $(p-1)$-linear at
both $\pm\infty$ and $0$, the study of such problems is naturally related to
that of the eigenvalue problem for $L_{p}$. In general, nontrivial solutions
appear as soon as the limits above ’jump’ over the principal eigenvalue of
$L_{p}$. Existence results can be proved via either variational methods
(critical point theory and Morse theory), or topological methods (degree
theory). In this paper we study the following fractional order nonlinear
equation with Dirichlet condition:

(1.1) $\begin{cases}(-\Delta)_{p}^{s}\,u=f(x,u)&\text{in $\Omega$}\\\
u=0&\text{in $\Omega^{c}$.}\end{cases}$

Here $\Omega\subset{\mathbb{R}}^{N}$ ($N\geq 2$) is a bounded domain with
$C^{1,1}$ boundary, $p\geqslant 2$, $s\in(0,1)$ s.t. $N>ps$, the leading
operator is the fractional $p$-Laplacian, defined for all
$u:{\mathbb{R}}^{N}\to{\mathbb{R}}$ smooth enough and all
$x\in{\mathbb{R}}^{N}$ by

$(-\Delta)_{p}^{s}\,u(x)=2\lim_{\varepsilon\to
0^{+}}\int_{B_{\varepsilon}^{c}(x)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\,dy,$

and $f:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a Carathéodory mapping with
$(p-1)$-linear growth both at $0$ and at $\pm\infty$, with different slopes
(jumping reaction). The operator $(-\Delta)_{p}^{s}\,$ is both a nonlocal and
a nonlinear one, which for $p=2$ reduces to the well-known fractional
Laplacian. The corresponding eigenvalue problem can be stated as follows:

(1.2) $\begin{cases}(-\Delta)_{p}^{s}\,u=\lambda|u|^{p-2}u&\text{in
$\Omega$}\\\ u=0&\text{in $\Omega^{c}$.}\end{cases}$

The eigenvalue problem (1.2) has been studied, for instance, in [26, 27],
leading to the existence of a diverging sequence of variational (Lusternik-
Schnirelmann) eigenvalues

$0<\lambda_{1}<\lambda_{2}\leqslant\ldots\leqslant\lambda_{k}\leqslant\ldots,$

with properties analogous to those of the classical $p$-Laplacian. The
nonlinear problem (1.1) (or variants of it) was studied in [10, 12, 14, 15,
21, 22, 23], where asymptotic comparison between $f(x,\cdot)$ and some
eigenvalue of the sequence above is often used as a means to the end of
proving existence of nontrivial solutions. Most of the cited works use
variational methods. In particular, asymmetric reactions ($(p-1)$-superlinear
at $\infty$, $(p-1)$-sublinear at $-\infty$) are considered in [22]. Our
approach is topological, based on Browder’s topological degree for
$(S)_{+}$-maps, and follows [1, 2]. We prove multiplicity results for problem
(1.1) with jumping reactions, under two different sets of hypotheses:

  * $(a)$

if the quotient $f(x,t)/(|t|^{p-2}t)$ is asymptotically bounded below
$\lambda_{1}$ for $|t|\to\infty$, and between $\lambda_{1}$ and $\lambda_{2}$
for $t\to 0$, then (1.1) has at least two nontrivial solutions (Theorem 4.5);

  * $(b)$

if the quotient $f(x,t)/(|t|^{p-2}t)$ is asymptotically bounded below
$\lambda_{1}$ for $t\to\infty$, above $\lambda_{1}$ for both
$t\to-\infty,0^{+}$, and tends to $0$ for $t\to 0^{-}$, then (1.1) has at
least two nontrivial solutions, one of which positive (Theorem 5.5).

In both cases we do not assume that the limits exist. The proofs are based on
a comparison between the operator driving problem (1.1) and the one arising
from convenient weighted eigenvalue problems, which preserves Browder’s degree
by homotopy invariance. In this comparison we use an index formula for
$(-\Delta)_{p}^{s}\,$ proved in [15], and monotonicity properties of weighted
eigenvalues proved in the forthcoming paper [6]. The paper has the following
structure: in Section 2 we recall the basic notions of the degree theory for
demicontinuous $(S)_{+}$-maps; in Section 3 we recall the functional-analytic
framework and some well-known results about fractional $p$-Laplacian problems,
including weighted eigenvalue problems; in Section 4 we deal with case $(a)$;
and in Section 5 we deal with case $(b)$. Notation: Throughout the paper, for
any $A\subset{\mathbb{R}}^{N}$ we shall set $A^{c}={\mathbb{R}}^{N}\setminus
A$. For any two measurable functions $f,g:\Omega\to{\mathbb{R}}$, $f\leqslant
g$ in $\Omega$ will mean that $f(x)\leqslant g(x)$ for a.e. $x\in\Omega$ (and
similar expressions). The positive (resp., negative) part of $f$ is denoted
$f^{+}$ (resp., $f^{-}$). If $X$ is an ordered Banach space, then $X_{+}$ will
denote its non-negative order cone. For all $r\in[1,\infty]$, $\|\cdot\|_{r}$
denotes the standard norm of $L^{r}(\Omega)$ (or $L^{r}({\mathbb{R}}^{N})$,
which will be clear from the context). Every function $u$ defined in $\Omega$
will be identified with its $0$-extension to ${\mathbb{R}}^{N}$. Moreover, $C$
will denote a positive constant (whose value may change case by case).

##  2\. Degree theory for $(S)_{+}$-maps

Topological degree theory for $(S)_{+}$-mappings from a Banach space into its
dual was introduced by Browder in [7] and subsequent papers, as an infinite-
dimensional extension of Brouwer’s degree theory, and then generalized in [19,
1] to set-valued mappings. We recall here some basic features of such theory,
following the general approach of [29, Section 4.3]. Let $(X,\|\cdot\|)$ be a
separable reflexive Banach space with dual $(X^{*},\|\cdot\|_{*})$. We say
that $A:X\to X^{*}$ is a $(S)_{+}$-map, if for any sequence $(u_{n})$ in $X$,
$u_{n}\rightharpoonup u$ in $X$ and

$\limsup_{n\to\infty}\langle A(u_{n}),u_{n}-u\rangle\leq 0$

imply $u_{n}\to u$ (strongly). By Troyanskij’s renorming theorem, we can
assume that both $X$ and $X^{*}$ are locally uniformly convex. So, there is a
(single-valued) duality map $\mathcal{F}:X\to X^{*}$ s.t. for all $u\in X$

$\|\mathcal{F}(u)\|^{2}=\|u\|^{2}=\langle\mathcal{F}(u),u\rangle.$

Such $\mathcal{F}$ is a $(S)_{+}$-homeomorphism between $X$ and $X^{*}$. Also,
we remark that if $A:X\to X^{*}$ is a demicontinuous (i.e., strong to weak∗
continuous) $(S)_{+}$-map and $B:X\to X^{*}$ is a completely continuous map,
then $A+B$ is a demicontinuous $(S)_{+}$-map. We will now define a degree for
a triple $(A,U,u^{*})$, where $U\subseteq X$ is a bounded open set,
$A:\overline{U}\to X^{*}$ is a demicontinuous $(S)_{+}$-map, and $u^{*}\in
X^{*}\setminus A(\partial U)$. First we introduce a Galerkin type
approximation. Since $X$ is separable, there exists an increasing sequence
$(X_{n})$ of finite-dimensional subspaces of $X$ s.t.

$\overline{\bigcup_{n=1}^{\infty}X_{n}}=X.$

For all $n\in{\mathbb{N}}$ we denote $U_{n}=U\cap X_{n}$ and define
$A_{n}:\overline{U}_{n}\to X_{n}^{*}$ ($\overline{U}_{n}$ denotes the closure
of $U_{n}$ in $X_{n}$) by setting for all $u\in\overline{U}_{n}$, $v\in X_{n}$

$\langle A_{n}(u),v\rangle=\langle A(u),v\rangle_{n}$

($\langle\cdot,\cdot\rangle_{n}$ denotes the duality between $X^{*}_{n}$ and
$X_{n}$). By [29, Proposition 4.38], the Brouwer degree of $A_{n}$ eventually
stabilizes as $n\to\infty$, i.e., there exists $n_{0}\in{\mathbb{N}}$ s.t. for
all $n\geqslant n_{0}$ we have $u^{*}\notin A_{n}(\partial U_{n})$ and

$\deg_{B}(A_{n},U_{n},u^{*})=\deg_{B}(A_{n_{0}},U_{n_{0}},u^{*}).$

So, we can define the degree for the triple $(A,U,u^{*})$ as

$\deg_{(S)_{+}}(A,U,u^{*})=\deg_{B}(A_{n_{0}},U_{n_{0}},u^{*}).$

The integer-valued map $\deg_{(S_{+})}$ inherits the main properties of
Brouwer’s degree. In particular, it is invariant with respect to a special
class of homotopies. We say that $h:[0,1]\times\overline{U}\to X^{*}$ is a
$(S)_{+}$-homotopy, if $t_{n}\to t$ in $[0,1]$, $u_{n}\rightharpoonup u$ in
$X$, and

$\limsup_{n\to\infty}\langle h(t_{n},u_{n}),u_{n}-u\rangle\leq 0$

imply $u_{n}\to u$ in $X$ and $h(t_{n},u_{n})\rightharpoonup h(t,u)$ in
$X^{*}$. For instance, if $A,B:\overline{U}\to X^{*}$ are demicontinuous
$(S)_{+}$-maps, then

$h(t,u)=(1-t)A(u)+tB(u)$

defines a $(S)_{+}$-homotopy [29, Proposition 4.41]. For the reader’s
convenience, we summarize the properties of $\deg_{(S)_{+}}$:

######  Proposition 2.1.

[29, Theorem 4.42] Let $U\subset X$ be a bounded open set, $A:\overline{U}\to
X^{*}$ be a demicontinuous $(S)_{+}$-map, $u^{*}\notin A(\partial U)$. Then:

  1. $(i)$

(normalization) if $u^{*}\in\mathcal{F}(U)$, then
$\deg_{(S)_{+}}(\mathcal{F},U,u^{*})=1$;

  2. $(ii)$

(domain additivity) if $U=U_{1}\cup U_{2}$, with $U_{1},U_{2}\subset X$
nonempty open sets s.t. $U_{1}\cap U_{2}=\emptyset$ and $u^{*}\notin
A(\partial U_{1}\cup\partial U_{2})$, then

$\deg_{(S)_{+}}(A,U,u^{*})=\deg_{(S)_{+}}(A,U_{1},u^{*})+\deg_{(S)_{+}}(A,U_{2},u^{*});$

  3. $(iii)$

(excision) if $C\subset\overline{U}$ is closed s.t. $u^{*}\notin A(C)$, then

$\deg_{(S)_{+}}(A,U\setminus C,u^{*})=\deg_{(S)_{+}}(A,U,u^{*});$

  4. $(iv)$

(homotopy invariance) if $h:[0,1]\times\overline{U}\to X^{*}$ is a
$(S)_{+}$-homotopy s.t. $u^{*}\notin h(t,\partial U)$ for all $t\in[0,1]$,
then

$t\mapsto\deg_{(S)_{+}}(h(t,\cdot),U,u^{*})$

is constant in $[0,1]$;

  5. $(v)$

(solution) if $\deg_{(S)_{+}}(A,U,u^{*})\neq 0$, then there exists $u\in U$
s.t. $A(u)=u^{*}$;

  6. $(vi)$

(boundary dependence) if $B:\overline{U}\to X^{*}$ is a demicontinuous
$(S)_{+}$-map s.t. $A(u)=B(u)$ for all $u\in\partial U$, then

$\deg_{(S)_{+}}(A,U,u^{*})=\deg_{(S)_{+}}(B,U,u^{*}).$

We conclude this section by recalling a result on the degree of a potential
operator, originally established by Rabinowitz [31] for the Leray-Schauder
degree:

######  Proposition 2.2.

[29, Corollary 4.49] Let $\Phi\in C^{1}(X)$ be a functional s.t.
$\Phi^{\prime}:X\to X^{*}$ is a demicontinuous $(S)_{+}$-map, $u_{0}\in X$ be
a local minimizer and an isolated critical point of $\Phi$. Then, there exists
$\rho_{0}>0$ s.t. for all $\rho\in(0,\rho_{0}]$

$\deg_{(S)_{+}}(\Phi^{\prime},B_{\rho}(u_{0}),0)=1.$

##  3\. General Dirichlet problems and weighted eigenvalue problems

In this section we collect some useful results related to the fractional
$p$-Laplacian, which we shall exploit in our analysis of problem (1.1). First
we fix a functional-analytical framework, following [13, 21]. First, for all
measurable $u:{\mathbb{R}}^{N}\to{\mathbb{R}}$ we set

$[u]_{s,p}^{p}=\iint_{{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dxdy.$

Then we define the following fractional Sobolev spaces:

$W^{s,p}({\mathbb{R}}^{N})=\big{\\{}u\in
L^{p}({\mathbb{R}}^{N}):\,[u]_{s,p}<\infty\big{\\}},$
$W^{s,p}_{0}(\Omega)=\big{\\{}u\in W^{s,p}({\mathbb{R}}^{N}):\,u(x)=0\
\text{in $\Omega^{c}$}\big{\\}},$

the latter being a uniformly convex, separable Banach space with norm
$\|u\|=[u]_{s,p}$ and dual space $W^{-s,p^{\prime}}(\Omega)$ (with norm
$\|\cdot\|_{-s,p^{\prime}})$. Set $p_{s}^{*}=Np/(N-ps)$, then the embedding
$W^{s,p}_{0}(\Omega)\hookrightarrow L^{q}(\Omega)$ is continuous for all
$q\in[1,p^{*}_{s}]$ and compact for all $q\in[1,p^{*}_{s})$. On the reaction
of (1.1) we make the following general assumption:

  * ${\bf H}_{0}$

$f:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a Carathéodory function, and
there exist $c_{0}>0$, $q\in(1,p^{*}_{s})$ s.t. for a.e. $x\in\Omega$ and all
$t\in{\mathbb{R}}$

$|f(x,t)|\leq c_{0}(1+|t|^{q-1}).$

Under such hypothesis, the following definition is well posed. We say that
$u\in W^{s,p}_{0}(\Omega)$ is a (weak) solution of (1.1), if for all $v\in
W^{s,p}_{0}(\Omega)$

$\iint_{{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}\,dx\,dy=\int_{\Omega}f(x,u)v\,dx.$

We have the following a priori estimate for the solutions:

######  Proposition 3.1.

[8, Theorem 3.3] Let ${\bf H}_{0}$ hold, $u\in W^{s,p}_{0}(\Omega)$ be a
solution of (1.1). Then, $u\in L^{\infty}(\Omega)$ with
$\|u\|_{\infty}\leqslant C$, for some $C=C(\|u\|)>0$.

Regularity theory for nonlinear, nonlocal operators is still developing. A
major role in such theory is played by the following weighted Hölder spaces,
with weight ${\rm d}_{\Omega}^{s}(x)=\mathrm{dist}(x,\Omega^{c})^{s}$. Set

$C_{s}^{0}(\overline{\Omega})=\Big{\\{}u\in
C^{0}(\overline{\Omega}):\frac{u}{{\rm d}_{\Omega}^{s}}\ \text{has a
continuous extension to}\
\overline{\Omega}\Big{\\}},\quad\|u\|_{0,s}=\Big{\|}\frac{u}{{\rm
d}_{\Omega}^{s}}\Big{\|}_{\infty},$

and for all $\alpha\in(0,1)$

$C_{s}^{\alpha}(\overline{\Omega})=\Big{\\{}u\in
C^{0}(\overline{\Omega}):\frac{u}{{\rm d}_{\Omega}^{s}}\ \text{has a
$\alpha$-H\"{o}lder extension to}\ \overline{\Omega}\Big{\\}},\
\|u\|_{\alpha,s}=\|u\|_{0,s}+\sup_{x\neq y}\frac{|u(x)/{\rm
d}_{\Omega}^{s}(x)-u(y)/{\rm d}_{\Omega}^{s}(y)|}{|x-y|^{\alpha}}.$

The embedding $C_{s}^{\alpha}(\overline{\Omega})\hookrightarrow
C_{s}^{0}(\overline{\Omega})$ is compact for all $\alpha\in(0,1)$. By [21,
Lemma 5.1], the positive cone $C_{s}^{0}(\overline{\Omega})_{+}$ of
$C_{s}^{0}(\overline{\Omega})$ has a nonempty interior given by

${\rm int}(C_{s}^{0}(\overline{\Omega})_{+})=\Big{\\{}u\in
C_{s}^{0}(\overline{\Omega}):\,\frac{u}{{\rm d}_{\Omega}^{s}}>0\text{ in
}\overline{\Omega}\Big{\\}}.$

Combining Proposition 3.1 and [24, Theorem 1.1], we have the following global
regularity result for the degenerate case $p\geqslant 2$:

######  Proposition 3.2.

Let ${\bf H}_{0}$ hold, $u\in W^{s,p}_{0}(\Omega)$ be a solution of (1.1).
Then, $u\in C_{s}^{\alpha}(\overline{\Omega})$ for some $\alpha\in(0,s]$.

We define the operators driving (1.1). For all $u,v\in W^{s,p}_{0}(\Omega)$ we
set

$\langle
A(u),v\rangle=\iint_{{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}\,dxdy.$

It is easily seen that $A:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ is
a demicontinuous $(S)_{+}$-map (see [14, Lemma 2.1], [15, Lemma 3.2]). We
recall from [22, Lemma 2.1] the following inequality, which holds for all
$u\in W^{s,p}_{0}(\Omega)$:

(3.1) $\|u^{\pm}\|^{p}\leqslant\langle A(u),\pm u^{\pm}\rangle.$

We also define the Nemytskij operator

$\langle N_{f}(u),v\rangle=\int_{\Omega}f(x,u)v\,dx.$

By ${\bf H}_{0}$, $N_{f}:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ is
a completely continuous map. Thus, $A-N_{f}$ is a demicontinuous
$(S)_{+}$-map. Clearly, any $u\in W^{s,p}_{0}(\Omega)$ is a (weak) solution
iff in $W^{-s,p^{\prime}}(\Omega)$ we have

$A(u)-N_{f}(u)=0.$

Sometimes we will deal with problem (1.1) variationally. Let us define an
energy functional by setting for all $(x,t)\in\Omega\times{\mathbb{R}}$

$F(x,t)=\int_{0}^{t}f(x,\tau)\,d\tau,$

and for all $u\in W^{s,p}_{0}(\Omega)$

$\Phi(u)=\frac{\|u\|^{p}}{p}-\int_{\Omega}F(x,u)\,dx.$

By ${\bf H}_{0}$, it is easily seen that $\Phi\in C^{1}(W^{s,p}_{0}(\Omega))$
with derivative given for all $u\in W^{s,p}_{0}(\Omega)$ by

$\Phi^{\prime}(u)=A(u)-N_{f}(u),$

so the solutions of (1.1) coincide with the critical points of $\Phi$. By
using Proposition 3.2 above, we get the following useful result about
equivalence of local minimizers of $\Phi$ in the topologies of
$W^{s,p}_{0}(\Omega)$ and $C_{s}^{0}(\overline{\Omega})$, respectively:

######  Proposition 3.3.

[23, Theorem 1.1] Let ${\bf H}_{0}$ hold, $u\in W^{s,p}_{0}(\Omega)$. Then,
the following are equivalent:

  1. $(i)$

there exists $\sigma>0$ s.t. $\Phi(u+v)\geqslant\Phi(u)$ for all $v\in
W^{s,p}_{0}(\Omega)\cap C_{s}^{0}(\overline{\Omega})$,
$\|v\|_{0,s}\leqslant\sigma$;

  2. $(ii)$

there exists $\rho>0$ s.t. $\Phi(u+v)\geqslant\Phi(u)$ for all $v\in
W^{s,p}_{0}(\Omega)$, $\|v\|\leqslant\rho$.

Finally, we recall a strong maximum principle and Hopf’s lemma:

######  Proposition 3.4.

[11, Theorems 1.2, 1.5] Let ${\bf H}_{0}$ hold, and $c_{1}>0$ be s.t. for a.e.
$x\in\Omega$ and all $t\geq 0$

$f(x,t)\geq-c_{1}t^{p-1}.$

Then, for all $u\in W^{s,p}_{0}(\Omega)_{+}\setminus\\{0\\}$ solution of (1.1)
we have $u\in\mathrm{int}(C_{s}^{0}(\overline{\Omega})_{+})$.

The rest of this section is devoted to the following weighted eigenvalue
problem with $m\in L^{\infty}(\Omega)_{+}\setminus\\{0\\}$ and
$\lambda\in{\mathbb{R}}$:

(3.2) $\begin{cases}(-\Delta)_{p}^{s}\,u=\lambda m(x)|u|^{p-2}u&\text{in
$\Omega$}\\\ u=0&\text{in $\Omega^{c}$.}\end{cases}$

This reduces to (1.2) for $m\equiv 1$, and it falls into the general problem
studied in [18] and in the forthcoming paper [6] (with possibly singular
weights). Set for all $u,v\in W^{s,p}_{0}(\Omega)$

$\langle K_{m}(u),v\rangle=\int_{\Omega}m(x)|u|^{p-2}uv\,dx.$

By [15, Lemma 3.2], $K_{m}:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$
is a completely continuous map. So, $A-\lambda K_{m}$ is a demicontinuous
$(S)_{+}$-map for all $\lambda\in{\mathbb{R}}$. According to the general
definition above, we say that $u\in W^{s,p}_{0}(\Omega)$ is a (weak) solution
of (3.2) if in $W^{-s,p^{\prime}}(\Omega)$ we have

$A(u)-\lambda K_{m}(u)=0.$

So, $\lambda\in{\mathbb{R}}$ is an eigenvalue of $(-\Delta)_{p}^{s}\,$, with
weight $m$, if there exists $u\in W^{s,p}_{0}(\Omega)\setminus\\{0\\}$
solution of (3.2), which is then an eigenfunction associated to $\lambda$.
Arguing as in [26] and following the general scheme of [30], we set

$\mathcal{S}_{p}(m)=\Big{\\{}u\in
W^{s,p}_{0}(\Omega):\,\int_{\Omega}m(x)|u|^{p}\,dx=1\Big{\\}}.$

For all $k\in{\mathbb{N}}$ we set

$\mathcal{F}_{k}=\\{S\subseteq\mathcal{S}_{p}(m):S=-S\text{ closed, }i(S)\geq
k\\},$

where $i(\cdot)$ denotes the Fadell-Rabinowitz cohomological index, and

(3.3) $\lambda_{k}(m)=\inf_{S\subset\mathcal{F}_{k}}\sup_{u\in S}\|u\|^{p}.$

By means of (3.3) we define a sequence of variational (Lusternik-Schnirelmann)
weighted eigenvalues

$0<\lambda_{1}(m)<\lambda_{2}(m)\leq\cdots\leq\lambda_{k}(m)\leq\cdots\to\infty.$

If $m\equiv 1$, then we set $\lambda_{k}=\lambda_{k}(1)$ (eigenvalues of
(1.2)). The following proposition, proved in [18, Subsection 3.1] and [15,
Proposition 3.4], summarizes the properties of the principal weighted
eigenvalue $\lambda_{1}(m)$, including a strong monotonicity property with
respect to $m$ (see also [10]):

######  Proposition 3.5.

Let $m\in L^{\infty}(\Omega)_{+}\setminus\\{0\\}$. Then, $\lambda_{1}(m)>0$ is
the smallest eigenvalue of $(-\Delta)_{p}^{s}\,$ with weight $m$ and has the
following variational characterization:

$\lambda_{1}(m)=\inf_{u\in
W^{s,p}_{0}(\Omega)\setminus\\{0\\}}\frac{\|u\|^{p}}{\int_{\Omega}m(x)|u|^{p}\,dx},$

the infimum being attained at a unique eigenfunction
$\hat{u}_{1,m}\in\mathcal{S}_{p}(m)\cap{\rm
int}(C^{0}_{s}(\overline{\Omega})_{+})$. Besides, any eigenfunction associated
to an eigenvalue $\lambda>\lambda_{1}(m)$ is nodal. Finally, if $m^{\prime}\in
L^{\infty}(\Omega)_{+}\setminus\\{0\\}$ is s.t. $m\leqslant m^{\prime}$ in
$\Omega$ and $m\not\equiv m^{\prime}$, then
$\lambda_{1}(m)>\lambda_{1}(m^{\prime})$.

Regarding the second eigenvalue $\lambda_{2}(m)$, from [18, Theorem 3.7] and
[6] we have the following properties, including a weaker monotonicity property
(analogous to [4, Proposition 3] for the $p$-Laplacian):

######  Proposition 3.6.

Let $m\in L^{\infty}(\Omega)_{+}\setminus\\{0\\}$. Then, $\lambda_{2}(m)$ is
the smallest eigenvalue of $(-\Delta)_{p}^{s}\,$ with weight $m$, greater than
$\lambda_{1}(m)$. Besides, if $m^{\prime}\in
L^{\infty}(\Omega)_{+}\setminus\\{0\\}$ is s.t. $m<m^{\prime}$ in $\Omega$,
then $\lambda_{2}(m)>\lambda_{2}(m^{\prime})$.

For the demicontinuous $(S)_{+}$-map $A-\lambda K_{m}$ there holds the
following index formula (see [2, Lemma 2, Theorem 2] for the Neumann
$p$-Laplacian):

######  Proposition 3.7.

[15, Theorem 3.5] Let $m\in L^{\infty}(\Omega)_{+}\setminus\\{0\\}$, $r>0$.
Then

  1. $(i)$

$\deg_{(S)_{+}}(A-\lambda K_{m},B_{r}(0),0)=1$ for all
$\lambda\in(0,\lambda_{1}(m))$;

  2. $(ii)$

$\deg_{(S)_{+}}(A-\lambda K_{m},B_{r}(0),0)=-1$ for all
$\lambda\in(\lambda_{1}(m),\lambda_{2}(m))$.

We also recall the following technical property:

######  Proposition 3.8.

[22, Lemma 2.7] Let $\theta\in L^{\infty}(\Omega)$ be s.t.
$\theta\leqslant\lambda_{1}$ in $\Omega$, $\theta\not\equiv\lambda_{1}$. Then,
there exists $\sigma>0$ s.t. for all $u\in W^{s,p}_{0}(\Omega)$

$\|u\|^{p}-\int_{\Omega}\theta(x)|u|^{p}\,dx\geqslant\sigma\|u\|^{p}.$

Finally, we consider problem (3.2) with a bounded perturbation $\beta\in
L^{\infty}(\Omega)_{+}\setminus\\{0\\}$:

(3.4) $\begin{cases}(-\Delta)_{p}^{s}\,u=\lambda
m(x)|u|^{p-2}u+\beta(x)&\text{in $\Omega$}\\\ u=0&\text{in
$\Omega^{c}$.}\end{cases}$

The following result, which will be useful in our study, is analogous to [12,
Lemma 4.1], dealing with supersolutions for $m=1$ (see [17, Proposition 4.1]
for the $p$-Laplacian case):

######  Lemma 3.9.

Let $m,\beta\in L^{\infty}(\Omega)_{+}\setminus\\{0\\}$,
$\lambda\geqslant\lambda_{1}(m)$, and $u\in W^{s,p}_{0}(\Omega)$ be a solution
of (3.4). Then, $u^{-}\not\equiv 0$.

###### Proof.

Since $\beta\not\equiv 0$, we clearly have $u\not\equiv 0$. We argue by
contradiction, assuming $u\in W^{s,p}_{0}(\Omega)_{+}\setminus\\{0\\}$. By
Proposition 3.4, then we have $u\in{\rm
int}(C^{0}_{s}(\overline{\Omega})_{+})$. For all $n\in{\mathbb{N}}$ set

$u_{n}=u+\frac{1}{n}.$

Let $\hat{u}_{1,m}\in{\rm int}(C^{0}_{s}(\overline{\Omega})_{+})$ be as in
Proposition 3.5, and for all $n\in{\mathbb{N}}$, $x\in{\mathbb{R}}^{N}$ set

$v_{n}(x)=\frac{\hat{u}_{1,m}^{p}(x)}{u_{n}^{p-1}(x)}.$

It is easily seen that $v_{n}\in L^{p}(\Omega)_{+}$ and $v_{n}=0$ in
$\Omega^{c}$. Moreover, by applying Lagrange’s theorem and
$u_{n}\geqslant\frac{1}{n}$ in ${\mathbb{R}}^{N}$, for all
$x,y\in{\mathbb{R}}^{N}$ we have

$\displaystyle|v_{n}(x)-v_{n}(y)|$
$\displaystyle=\Big{|}\frac{\hat{u}_{1,m}^{p}(x)-\hat{u}_{1,m}^{p}(y)}{u_{n}^{p-1}(x)}+\hat{u}_{1,m}^{p}(y)\frac{u_{n}^{p-1}(y)-u_{n}^{p-1}(x)}{u_{n}^{p-1}(x)u_{n}^{p-1}(y)}\Big{|}$
$\displaystyle\leqslant
n^{p-1}|\hat{u}_{1,m}^{p}(x)-\hat{u}_{1,m}^{p}(y)|+\|\hat{u}_{1,m}\|_{\infty}^{p}\frac{|u_{n}^{p-1}(y)-u_{n}^{p-1}(x)|}{u_{n}^{p-1}(x)u_{n}^{p-1}(y)}$
$\displaystyle\leqslant
pn^{p-1}(\hat{u}_{1,m}^{p-1}(x)+\hat{u}_{1,m}^{p-1}(y))|\hat{u}_{1,m}(x)-\hat{u}_{1,m}(y)|$
$\displaystyle+(p-1)n^{p-1}\|\hat{u}_{1,m}\|_{\infty}^{p}\Big{(}\frac{1}{u_{n}(x)}+\frac{1}{u_{n}(y)}\Big{)}|u(x)-u(y)|$
$\displaystyle\leqslant
C_{n}\big{(}|\hat{u}_{1,m}(x)-\hat{u}_{1,m}(y)|+|u(x)-u(y)|\big{)},$

for some $C_{n}>0$ depending on $n$. So,

$\iint_{{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}}\frac{|v_{n}(x)-v_{n}(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy\leqslant
C\big{(}\|\hat{u}_{1,m}\|^{p}+\|u\|^{p}),$

which implies $v_{n}\in W^{s,p}_{0}(\Omega)_{+}$. By the discrete Picone’s
inequality [5, Proposition 4.2], for all $x,y\in{\mathbb{R}}^{N}$ we have

$\displaystyle|\hat{u}_{1,m}(x)-\hat{u}_{1,m}(y)|^{p}$
$\displaystyle\geqslant|u_{n}(x)-u_{n}(y)|^{p-2}(u_{n}(x)-u_{n}(y))\Big{(}\frac{\hat{u}_{1,m}^{p}(x)}{u_{n}^{p-1}(x)}-\frac{\hat{u}_{1,m}^{p}(y)}{u_{n}^{p-1}(y)}\Big{)}$
$\displaystyle=|u(x)-u(y)|^{p-2}(u(x)-u(v))(v_{n}(x)-v_{n}(y)).$

Using the inequality above and testing (3.4) with $v_{n}\in
W^{s,p}_{0}(\Omega)_{+}$ we have

(3.5) $\displaystyle\|\hat{u}_{1,m}\|^{p}$
$\displaystyle\geqslant\iint_{{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(v))(v_{n}(x)-v_{n}(y))}{|x-y|^{N+ps}}\,dx\,dy$
$\displaystyle=\lambda\int_{\Omega}m(x)u^{p-1}v_{n}\,dx+\int_{\Omega}\beta(x)v_{n}\,dx.$

From (3.5) we have

$\int_{\Omega}\beta(x)v_{n}\,dx\leqslant\|\hat{u}_{1,m}\|^{p}<\infty,$

hence $(\beta v_{n})$ is a bounded sequence in $L^{1}(\Omega)$, tending to
$\frac{\beta\hat{u}_{1,m}^{p}}{u^{p-1}}$ in $\Omega$. By Lebesgue’s dominated
convergence theorem we have $\frac{\beta\hat{u}_{1,m}^{p}}{u^{p-1}}\in
L^{1}(\Omega)$. Besides, clearly $mu^{p-1}v_{n}\to m\hat{u}_{1,m}^{p}$ in
$L^{1}(\Omega)$. Passing to the limit in (3.5) as $n\to\infty$, we have

$\displaystyle\|\hat{u}_{1,m}\|^{p}$
$\displaystyle\geqslant\lambda\int_{\Omega}m(x)\hat{u}_{1,m}^{p}\,dx+\int_{\Omega}\beta(x)\frac{\hat{u}_{1,m}^{p}}{u^{p-1}}\,dx$
$\displaystyle>\lambda_{1}(m)\int_{\Omega}m(x)\hat{u}_{1,m}^{p}\,dx,$

against Proposition 3.5. Thus, we conclude that $u^{-}\not\equiv 0$. ∎

######  Remark 3.10.

Most results in this section also hold in the singular case $p\in(1,2)$. The
assumption $p\geqslant 2$ is only required to have regularity as in
Proposition 3.2 and the consequent Proposition 3.3 (see [23, 24]).

##  4\. Two nontrivial solutions for jumping reactions

In this section we study problem (1.1) under the following hypotheses, which
imply a symmetric ’jump’ over the principal eigenvalue between $0$ and
$\pm\infty$:

  * ${\bf H}_{1}$

$f:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a Carathéodory mapping
satisfying

    1. $(i)$

for all $M>0$ there exists $a_{M}\in L^{\infty}(\Omega)_{+}$ s.t. for a.e.
$x\in\Omega$ and all $|t|\leq M$

$|f(x,t)|\leq a_{M}(x);$

    2. $(ii)$

there exist $\theta_{1},\theta_{2}\in L^{\infty}(\Omega)_{+}$ s.t.
$\theta_{1}\leq\theta_{2}\leq\lambda_{1}$ in $\Omega$,
$\theta_{2}\not\equiv\lambda_{1}$, and uniformly for a.e. $x\in\Omega$

$\theta_{1}(x)\leq\liminf_{|t|\to\infty}\frac{f(x,t)}{|t|^{p-2}t}\leq\limsup_{|t|\to\infty}\frac{f(x,t)}{|t|^{p-2}t}\leq\theta_{2}(x);$

    3. $(iii)$

there exist $\eta_{1},\eta_{2}\in L^{\infty}(\Omega)$ s.t.
$\lambda_{1}\leq\eta_{1}\leq\eta_{2}<\lambda_{2}$ in $\Omega$,
$\eta_{1}\not\equiv\lambda_{1}$, and uniformly for a.e. $x\in\Omega$

$\eta_{1}(x)\leq\liminf_{t\to 0}\frac{f(x,t)}{|t|^{p-2}t}\leq\limsup_{t\to
0}\frac{f(x,t)}{|t|^{p-2}t}\leq\eta_{2}(x).$

Note that we assume non-resonance both at $0$ and $\pm\infty$, with a relevant
difference: non-resonance with $\lambda_{1}$ is only required on a subset of
$\Omega$ with positive measure, while non-resonance with $\lambda_{2}$ must
hold on the whole $\Omega$. Clearly, ${\bf H}_{1}$ implies ${\bf H}_{0}$ (with
$q=p$). So, all the results of Section 3 apply here. Besides, from ${\bf
H}_{1}$ $(iii)$ we immediately see that $f(\cdot,0)=0$ in $\Omega$, hence
problem (1.1) admits the trivial solution $u=0$. We aim at proving the
existence of nontrivial solutions, so we may assume, without loss of
generality, that (1.1) has finitely many solutions.

######  Example 4.1.

The autonomous mapping $f\in C({\mathbb{R}})$ defined by

$f(t)=\theta|t|^{p-2}t+(\eta-\theta)|t|^{p-2}t\frac{\ln(1+|t|)}{|t|},$

with $\theta<\lambda_{1}<\eta<\lambda_{2}$, satisfies ${\bf H}_{1}$.

In the following lemmas we study the behavior of the operator $A-N_{f}$. We
begin with an existence result:

######  Lemma 4.2.

If ${\bf H}_{1}$ holds, then (1.1) has a solution $u_{0}\in
C_{s}^{\alpha}(\overline{\Omega})\setminus\\{0\\}$. Moreover, there exists
$\rho_{0}>0$ s.t. for all $\rho\in(0,\rho_{0}]$

$\deg_{(S)_{+}}(A-N_{f},B_{\rho}(u_{0}),0)=1.$

###### Proof.

By ${\bf H}_{1}$ $(ii)$ and Proposition 3.8, there exists $\sigma>0$ s.t. for
all $u\in W^{s,p}_{0}(\Omega)$

(4.1)
$\|u\|^{p}-\int_{\Omega}\theta_{2}(x)|u|^{p}\,dx\geqslant\sigma\|u\|^{p}.$

Fix $\varepsilon\in(0,\sigma\lambda_{1})$. By ${\bf H}_{1}$ $(ii)$ we can find
$M>0$ s.t. for a.e. $x\in\Omega$ and all $|t|>M$

$\frac{f(x,t)}{|t|^{p-2}t}\leqslant\theta_{2}(x)+\varepsilon.$

By ${\bf H}_{1}$ $(i)$, for a.e. $x\in\Omega$ and all $|t|\leqslant M$ we have

$|f(x,t)|\leqslant a_{M}(x).$

So, for a.e. $x\in\Omega$ and all $t>M$ we get

$\displaystyle F(x,t)$
$\displaystyle\leqslant\int_{0}^{M}|f(x,\tau)|\,d\tau+\int_{M}^{t}(\theta_{2}(x)+\varepsilon)\tau^{p-1}\,d\tau$
$\displaystyle\leqslant
Ma_{M}(x)+\frac{\theta_{2}(x)+\varepsilon}{p}(t^{p}-M^{p})$
$\displaystyle\leqslant\frac{\theta_{2}(x)+\varepsilon}{p}t^{p}+C.$

Similar estimates hold for $t\leqslant M$, so for a.e. $x\in\Omega$ and all
$t\in{\mathbb{R}}$ we have

(4.2) $F(x,t)\leqslant\frac{\theta_{2}(x)+\varepsilon}{p}|t|^{p}+C.$

Define $\Phi\in C^{1}(W^{s,p}_{0}(\Omega))$ as in Section 3. By (4.1), (4.2),
and Proposition 3.5 we have for all $u\in W^{s,p}_{0}(\Omega)$

$\displaystyle\Phi(u)$
$\displaystyle\geqslant\frac{\|u\|^{p}}{p}-\int_{\Omega}\Big{(}\frac{\theta_{2}(x)+\varepsilon}{p}|u|^{p}+C\Big{)}\,dx$
$\displaystyle\geqslant\frac{\sigma}{p}\|u\|^{p}-\frac{\varepsilon}{p}\|u\|_{p}^{p}-C$
$\displaystyle\geqslant\Big{(}\sigma-\frac{\varepsilon}{\lambda_{1}}\Big{)}\frac{\|u\|^{p}}{p}-C,$

and the latter tends to $\infty$ as $\|u\|\to\infty$. So, $\Phi$ is coercive
in $W^{s,p}_{0}(\Omega)$. Plus, it is sequentially weakly l.s.c. Thus, there
exists $u_{0}\in W^{s,p}_{0}(\Omega)$ s.t.

(4.3) $\Phi(u_{0})=\inf_{u\in W^{s,p}_{0}(\Omega)}\Phi(u)=:\mu_{0}.$

Let $\hat{u}_{1}\in{\rm int}(C^{0}_{s}(\overline{\Omega})_{+})$ be as in
Proposition 3.5 (with $m\equiv 1$). By ${\bf H}_{1}$ $(iii)$ we have

$\int_{\Omega}\eta_{1}(x)\hat{u}_{1}^{p}\,dx>\lambda_{1}.$

Fix now $\varepsilon>0$ s.t.

$\varepsilon<\int_{\Omega}\eta_{1}(x)\hat{u}_{1}^{p}\,dx-\lambda_{1}.$

By ${\bf H}_{1}$ $(iii)$ there exists $\delta>0$ s.t. for a.e. $x\in\Omega$
and all $t\in(0,\delta]$

$\frac{f(x,t)}{t^{p-1}}\geqslant\eta_{1}(x)-\varepsilon,$

hence

$F(x,t)\geqslant\frac{\eta_{1}(x)-\varepsilon}{p}t^{p}.$

For all $\tau>0$ small enough we have $0<\tau\hat{u}_{1}\leqslant\delta$ in
$\Omega$, so, recalling Proposition 3.5, we have

$\displaystyle\Phi(\tau\hat{u}_{1})$
$\displaystyle\leqslant\frac{\tau^{p}}{p}\|\hat{u}_{1}\|^{p}-\int_{\Omega}\frac{\eta_{1}(x)-\varepsilon}{p}(\tau\hat{u}_{1})^{p}\,dx$
$\displaystyle=\frac{\tau^{p}}{p}\Big{(}\lambda_{1}-\int_{\Omega}\eta_{1}(x)\hat{u}_{1}^{p}\,dx+\varepsilon\Big{)}<0.$

Then we have $\mu_{0}<0$ in (4.3), in particular $u_{0}\neq 0$. From (4.3) we
have $\Phi^{\prime}(u_{0})=0$ in $W^{-s,p^{\prime}}(\Omega)$, so $u_{0}$
solves (1.1). By Proposition 3.2 we have $u_{0}\in
C^{\alpha}_{s}(\overline{\Omega})\setminus\\{0\\}$. Finally, recalling that
$\Phi^{\prime}=A-N_{f}$ is a demicontinuous $(S)_{+}$-map and $u_{0}$ is a
local minimizer of $\Phi$ and an isolated critical point (by the assumption
that $\Phi$ has only finitely many such points), by Proposition 2.2 there
exists $\rho_{0}>0$ s.t. for all $\rho\in(0,\rho_{0}]$

$\deg_{(S)_{+}}(A-N_{f},B_{\rho}(u_{0}),0)=1,$

which concludes the proof. ∎

The next lemma deals with the asymptotic behavior of $A-N_{f}$:

######  Lemma 4.3.

If ${\bf H}_{1}$ holds, then there exists $R_{0}>0$ s.t. for all $R\geqslant
R_{0}$

$\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)=1.$

###### Proof.

Fix $m_{\infty}\in L^{\infty}(\Omega)_{+}$ s.t. $\theta_{1}\leqslant
m_{\infty}\leqslant\theta_{2}$ in $\Omega$, and define
$K_{m_{\infty}}:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ as in
Section 3, hence $A-K_{m_{\infty}}$ is a demicontinuous $(S)_{+}$-map. Now set
for all $(t,u)\in[0,1]\times W^{s,p}_{0}(\Omega)$

$h_{\infty}(t,u)=A(u)-(1-t)N_{f}(u)-tK_{m_{\infty}}(u).$

As seen in Section 2, $h_{\infty}:[0,1]\times W^{s,p}_{0}(\Omega)\to
W^{-s,p^{\prime}}(\Omega)$ is a $(S)_{+}$-homotopy. We claim that there exists
$R_{0}>0$ s.t.

(4.4) $h_{\infty}(t,u)\neq 0\ \text{for all $t\in[0,1]$, $\|u\|\geqslant
R_{0}$.}$

Arguing by contradiction, assume that there exist sequences $(t_{n})$ in
$[0,1]$, $(u_{n})$ in $W^{s,p}_{0}(\Omega)$ s.t. $\|u_{n}\|\to\infty$ and for
all $n\in{\mathbb{N}}$ we have $h_{\infty}(t_{n},u_{n})=0$ in
$W^{-s,p^{\prime}}(\Omega)$, i.e.,

$A(u_{n})=(1-t_{n})N_{f}(u_{n})+t_{n}K_{m_{\infty}}(u_{n}).$

Passing to a subsequence if necessary, we have $t_{n}\to t$ and $\|u_{n}\|>0$.
Set for all $n\in{\mathbb{N}}$

$v_{n}=\frac{u_{n}}{\|u_{n}\|}.$

The sequence $(v_{n})$ is obviously bounded in $W^{s,p}_{0}(\Omega)$, so
passing to a further subsequence we have $v_{n}\rightharpoonup v$ in
$W^{s,p}_{0}(\Omega)$, $v_{n}\to v$ in $L^{p}(\Omega)$, and $v_{n}(x)\to v(x)$
for a.e. $x\in\Omega$. Dividing the equality above by $\|u_{n}\|^{p-1}$ we get
for all $n\in{\mathbb{N}}$

(4.5) $A(v_{n})=(1-t_{n})g_{n}+t_{n}K_{m_{\infty}}(v_{n}),$

where we have set for all $x\in\Omega$

$g_{n}(x)=\frac{f(x,u_{n}(x))}{\|u_{n}\|^{p-1}}.$

Reasoning as in Lemma 4.2 we see that there exists $C>0$ s.t. for a.e.
$x\in\Omega$ and all $t\in{\mathbb{R}}$

(4.6) $|f(x,t)|\leqslant C(1+|t|^{p-1}).$

We focus on the first term on the right-hand side of (4.5). By (4.6), we have
for all $n\in{\mathbb{N}}$

$\displaystyle\int_{\Omega}|g_{n}(x)|^{p^{\prime}}\,dx$
$\displaystyle\leqslant
C\int_{\Omega}\frac{(1+|u_{n}|^{p-1})^{p^{\prime}}}{\|u_{n}\|^{p}}\,dx$
$\displaystyle\leqslant C\frac{1+\|u_{n}\|_{p}^{p}}{\|u_{n}\|^{p}},$

and the latter is bounded by the continuous embedding
$W^{s,p}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)$. So, $(g_{n})$ is a bounded
sequence in $L^{p^{\prime}}(\Omega)$. Passing to a subsequence, we have
$g_{n}\rightharpoonup g_{\infty}$ in $L^{p^{\prime}}(\Omega)$. We claim that
there exists $\hat{g}_{\infty}\in L^{\infty}(\Omega)$ s.t. in $\Omega$

(4.7) $g_{\infty}=\hat{g}_{\infty}|v|^{p-2}v,\
\theta_{1}\leqslant\hat{g}_{\infty}\leqslant\theta_{2}.$

Indeed, set

$\Omega^{+}=\big{\\{}x\in\Omega:\,v(x)>0\big{\\}},\
\Omega^{-}=\big{\\{}x\in\Omega:\,v(x)<0\big{\\}},\
\Omega^{0}=\big{\\{}x\in\Omega:\,v(x)=0\big{\\}}.$

Then fix $\varepsilon>0$ and set for all $n\in{\mathbb{N}}$

$\Omega^{+}_{\varepsilon,n}=\Big{\\{}x\in\Omega:\,u_{n}(x)>0,\,\theta_{1}(x)-\varepsilon\leqslant\frac{f(x,u_{n}(x))}{u_{n}^{p-1}(x)}\leqslant\theta_{2}(x)+\varepsilon\Big{\\}}.$

For a.e. $x\in\Omega^{+}$ we have $v_{n}(x)\to v(x)>0$ as $n\to\infty$, hence
$u_{n}(x)\to\infty$. Recalling ${\bf H}_{1}$ $(ii)$, for all
$n\in{\mathbb{N}}$ big enough we have $u_{n}(x)>0$ and

$\theta_{1}(x)-\varepsilon\leqslant\frac{f(x,u_{n}(x))}{u_{n}^{p-1}(x)}\leqslant\theta_{2}(x)+\varepsilon,$

i.e., $x\in\Omega^{+}_{\varepsilon,n}$. In other words,
$\chi_{\Omega^{+}_{\varepsilon,n}}\to 1$ a.e. in $\Omega^{+}$, with bounded
convergence, which implies
$\chi_{\Omega^{+}_{\varepsilon,n}}g_{n}\rightharpoonup g_{\infty}$ in
$L^{p^{\prime}}(\Omega^{+})$. By definition of $v_{n}$, for all
$n\in{\mathbb{N}}$ big enough we have in $\Omega^{+}$

$\chi_{\Omega^{+}_{\varepsilon,n}}(\theta_{1}-\varepsilon)v_{n}^{p-1}\leqslant\chi_{\Omega^{+}_{\varepsilon,n}}g_{n}\leqslant\chi_{\Omega^{+}_{\varepsilon,n}}(\theta_{2}+\varepsilon)v_{n}^{p-1}.$

Passing to the limit as $n\to\infty$ we get in $\Omega^{+}$

$(\theta_{1}-\varepsilon)v^{p-1}\leqslant
g_{\infty}\leqslant(\theta_{2}+\varepsilon)v^{p-1}.$

Further, letting $\varepsilon\to 0^{+}$, we get in $\Omega^{+}$

$\theta_{1}v^{p-1}\leqslant g_{\infty}\leqslant\theta_{2}v^{p-1},$

which proves the claim in $\Omega^{+}$. Similarly, considering the set

$\Omega^{-}_{\varepsilon,n}=\Big{\\{}x\in\Omega:\,u_{n}(x)<0,\,\theta_{1}(x)-\varepsilon\leqslant\frac{f(x,u_{n}(x))}{|u_{n}(x)|^{p-2}u_{n}(x)}\leqslant\theta_{2}(x)+\varepsilon\Big{\\}},$

we get in $\Omega^{-}$

$\theta_{2}|v|^{p-2}v\leqslant g_{\infty}\leqslant\theta_{1}|v|^{p-2}v.$

Finally, for a.e. $x\in\Omega^{0}$ we have $v_{n}(x)\to 0$, which with
$\|u_{n}\|\to\infty$ and (4.6) implies $g_{n}(x)\to 0$. So we have
$g_{\infty}=0$ in $\Omega^{0}$, which completes the argument for (4.7). Now go
back to (4.5), which we test with $v_{n}-v\in W^{s,p}_{0}(\Omega)$ getting

$\langle
A(v_{n}),v_{n}-v\rangle=(1-t_{n})\int_{\Omega}g_{n}(x)(v_{n}-v)\,dx+t_{n}\int_{\Omega}m_{\infty}(x)|v_{n}|^{p-2}v_{n}(v_{n}-v)\,dx,$

and the latter tends to $0$ as $n\to\infty$, so we have

$\limsup_{n}\langle A(v_{n}),v_{n}-v\rangle\leqslant 0.$

By the $(S)_{+}$-property of $A$, we deduce that $v_{n}\to v$ in
$W^{s,p}_{0}(\Omega)$, in particular $\|v\|=1$. Besides, passing to the limit
in (4.5) as $n\to\infty$ and applying (4.7), we have in
$W^{-s,p^{\prime}}(\Omega)$

$A(v)=\tilde{g}_{\infty}|v|^{p-2}v,$

where we have set for all $x\in\Omega$

$\tilde{g}_{\infty}(x)=(1-t)\hat{g}_{\infty}(x)+tm_{\infty}(x).$

In other words, $v$ solves the weighted eigenvalue problem

(4.8)
$\begin{cases}(-\Delta)_{p}^{s}\,v=\tilde{g}_{\infty}(x)|v|^{p-2}v&\text{in
$\Omega$}\\\ v=0&\text{in $\Omega^{c}$.}\end{cases}$

Clearly $\tilde{g}_{\infty}\in L^{\infty}(\Omega)$, and due to (4.7) and the
choice of $m_{\infty}$ it satisfies
$\theta_{1}\leqslant\tilde{g}_{\infty}\leqslant\theta_{2}$ in $\Omega$. By
${\bf H}_{1}$ $(ii)$, then, we have $\tilde{g}_{\infty}\leqslant\lambda_{1}$
in $\Omega$ and $\tilde{g}_{\infty}\not\equiv\lambda_{1}$, hence by
Proposition 3.5

$\lambda_{1}(\tilde{g}_{\infty})>\lambda_{1}(\lambda_{1})=1.$

Thus, by (4.8), $v\neq 0$ is an eigenfunction with weight
$\tilde{g}_{\infty}$, associated to the eigenvalue $1$, against Proposition
3.5. This proves (4.4). Now we can apply Proposition 2.1 $(iv)$ (homotopy
invariance), which gives for all $R\geqslant R_{0}$

(4.9)
$\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)=\deg_{(S)_{+}}(A-K_{m_{\infty}},B_{R}(0),0).$

To conclude, we compute the degree of $A-K_{m_{\infty}}$. By ${\bf H}_{1}$
$(ii)$ we have $m_{\infty}\leqslant\lambda_{1}$ in $\Omega$ and
$m_{\infty}\not\equiv\lambda_{1}$, hence by Proposition 3.5 we have

$\lambda_{1}(m_{\infty})>\lambda_{1}(\lambda_{1})=1.$

Therefore, by Proposition 3.7 $(i)$ we have for all $R>0$

$\deg_{(S)_{+}}(A-K_{m_{\infty}},B_{R}(0),0)=1,$

which along with (4.9) gives for all $R\geqslant R_{0}$

$\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)=1,$

thus concluding the proof. ∎

The last lemma deals with the behavior of $A-N_{f}$ near $0$:

######  Lemma 4.4.

If ${\bf H}_{1}$ holds, then there exists $r_{0}>0$ s.t. for all
$r\in(0,r_{0}]$

$\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)=-1.$

###### Proof.

Fix $m_{0}\in L^{\infty}(\Omega)$ s.t. $\eta_{1}\leqslant
m_{0}\leqslant\eta_{2}$ in $\Omega$, and define
$K_{m_{0}}:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ as in Section 3,
hence $A-K_{m_{0}}$ is a demicontinuous $(S)_{+}$-map. As in Lemma 4.3, we
define a $(S)_{+}$-homotopy $h_{0}:[0,1]\times W^{s,p}_{0}(\Omega)\to
W^{-s,p^{\prime}}(\Omega)$ by setting for all $(t,u)\in[0,1]\times
W^{s,p}_{0}(\Omega)$

$h_{0}(t,u)=A(u)-(1-t)N_{f}(u)-tK_{m_{0}}(u).$

We claim that there exists $r_{0}>0$ s.t.

(4.10) $h_{0}(t,u)\neq 0\ \text{for all $t\in[0,1]$, $0<\|u\|\leqslant
r_{0}$.}$

Arguing as above by contradiction, assume that there exist sequences $(t_{n})$
in $[0,1]$, $(u_{n})$ in $W^{s,p}_{0}(\Omega)\setminus\\{0\\}$ s.t. $u_{n}\to
0$ in $W^{s,p}_{0}(\Omega)$ and for all $n\in{\mathbb{N}}$ we have
$h_{0}(t_{n},u_{n})=0$ in $W^{-s,p^{\prime}}(\Omega)$. Set for all
$n\in{\mathbb{N}}$

$v_{n}=\frac{u_{n}}{\|u_{n}\|}.$

Passing to a subsequence, we have $t_{n}\to t$, as well as
$v_{n}\rightharpoonup v$ in $W^{s,p}_{0}(\Omega)$, $v_{n}\to v$ in
$L^{p}(\Omega)$, and $v_{n}(x)\to v(x)$ for a.e. $x\in\Omega$. Plus, for all
$n\in{\mathbb{N}}$ we have

(4.11) $A(v_{n})=(1-t_{n})g_{n}+t_{n}K_{m_{0}}(v_{n}),$

where we have set for all $x\in\Omega$

$g_{n}(x)=\frac{f(x,u_{n}(x))}{\|u_{n}\|^{p-1}}.$

By ${\bf H}_{1}$ $(ii)$ $(iii)$, we can find $C>0$, $\delta\in(0,1)$ s.t. for
a.e. $x\in\Omega$ and all $t\in{\mathbb{R}}$ with either $|t|<\delta$ or
$|t|>\delta^{-1}$

$|f(x,t)|\leqslant C|t|^{p-1}.$

Besides, by ${\bf H}_{1}$ $(i)$ with $M=\delta^{-1}>0$, for a.e. $x\in\Omega$
and all $\delta\leqslant|t|\leqslant\delta^{-1}$ we have

$|f(x,t)|\leqslant
a_{M}(x)\leqslant\frac{\|a_{M}\|_{\infty}}{\delta^{p-1}}|t|^{p-1}.$

All in all, taking $C>0$ even bigger if necessary, for a.e. $x\in\Omega$ and
all $t\in{\mathbb{R}}$ we have

$|f(x,t)|\leqslant C|t|^{p-1}.$

Now for all $n\in{\mathbb{N}}$ we have

$\displaystyle\int_{\Omega}|g_{n}(x)|^{p^{\prime}}\,dx$
$\displaystyle\leqslant\int_{\Omega}\Big{(}\frac{C|u_{n}|^{p-1}}{\|u_{n}\|^{p-1}}\Big{)}^{p^{\prime}}\,dx$
$\displaystyle\leqslant
C\frac{\|u_{n}\|_{p}^{p}}{\|u_{n}\|^{p}}=C\|v_{n}\|_{p}^{p},$

and the latter is bounded by the continuous embedding
$W^{s,p}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)$. So we see that $(g_{n})$
is a bounded sequence in $L^{p^{\prime}}(\Omega)$, hence up to a further
subsequence $g_{n}\rightharpoonup g_{0}$ in $L^{p^{\prime}}(\Omega)$. Arguing
as in Lemma 4.3 and defining this time the sets

$\Omega^{+}_{\varepsilon,n}=\Big{\\{}x\in\Omega:\,u_{n}(x)>0,\,\eta_{1}(x)-\varepsilon\leqslant\frac{f(x,u_{n}(x))}{u_{n}^{p-1}(x)}\leqslant\eta_{2}(x)+\varepsilon\Big{\\}},$
$\Omega^{-}_{\varepsilon,n}=\Big{\\{}x\in\Omega:\,u_{n}(x)<0,\,\eta_{1}(x)-\varepsilon\leqslant\frac{f(x,u_{n}(x))}{|u_{n}(x)|^{p-2}u_{n}(x)}\leqslant\eta_{2}(x)+\varepsilon\Big{\\}}$

for all $\varepsilon>0$, $n\in{\mathbb{N}}$, we find $\hat{g}_{0}\in
L^{\infty}(\Omega)$ s.t. in $\Omega$

(4.12) $g_{0}=\hat{g}_{0}|v|^{p-2}v,\
\eta_{1}\leqslant\hat{g}_{0}\leqslant\eta_{2}.$

Testing (4.11) with $v_{n}-v\in W^{s,p}_{0}(\Omega)$ and using the
$(S)_{+}$-property of $A$, we see that $v_{n}\to v$ in $W^{s,p}_{0}(\Omega)$,
hence $\|v\|=1$. Passing to the limit in (4.11) as $n\to\infty$, we see that
$v\neq 0$ solves

(4.13) $\begin{cases}(-\Delta)_{p}^{s}\,v=\tilde{g}_{0}(x)|v|^{p-2}v&\text{in
$\Omega$}\\\ v=0&\text{in $\Omega^{c}$,}\end{cases}$

where we have set for all $x\in\Omega$

$\tilde{g}_{0}(x)=(1-t)\hat{g}_{0}(x)+tm_{0}(x).$

Clearly, $\tilde{g}_{0}\in L^{\infty}(\Omega)$, and due to (4.12) and the
choice of $m_{0}$ it satisfies
$\eta_{1}\leqslant\tilde{g}_{0}\leqslant\eta_{2}$ in $\Omega$. By ${\bf
H}_{1}$ $(iii)$, then, we have $\lambda_{1}\leqslant\tilde{g}_{0}<\lambda_{2}$
in $\Omega$ and $\tilde{g}_{0}\not\equiv\lambda_{1}$, so from Proposition 3.5
we have

$\lambda_{1}(\tilde{g}_{0})<\lambda_{1}(\lambda_{1})=1,$

while by Proposition 3.6 we have

$\lambda_{2}(\tilde{g}_{0})>\lambda_{2}(\lambda_{2})=1.$

This is against Proposition 3.6, as problem (4.13) has no eigenvalue in the
interval $(\lambda_{1}(\tilde{g}_{0}),\lambda_{2}(\tilde{g}_{0}))$. So (4.10)
is proved. Now we can apply Proposition 2.1 $(iv)$ (homotopy invariance),
which gives for all $r\in(0,r_{0}]$

(4.14)
$\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)=\deg_{(S)_{+}}(A-K_{m_{0}},B_{r}(0),0).$

To conclude, we compute the degree of $A-K_{m_{0}}$. By ${\bf H}_{1}$ $(iii)$
we have $\lambda_{1}\leqslant m_{0}<\lambda_{2}$ in $\Omega$ and
$m_{0}\not\equiv\lambda_{1}$, so by Propositions 3.5, 3.6 we have

$\lambda_{1}(m_{0})<1<\lambda_{2}(m_{0}).$

Hence, by Proposition 3.7 $(ii)$ we have for all $r>0$

$\deg_{(S)_{+}}(A-K_{m_{0}},B_{r}(0),0)=-1,$

which along with (4.14) gives for all $r\in(0,r_{0}]$

$\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)=-1,$

thus concluding the proof. ∎

Using the Lemmas above we can prove our first multiplicity result (an
analogous result for the Neumann $p$-Laplacian with set-valued reactions is
[2, Theorem 3]):

######  Theorem 4.5.

If ${\bf H}_{1}$ holds, then problem (1.1) has at least two nontrivial
solutions $u_{0},u_{1}\in C^{\alpha}_{s}(\overline{\Omega})\setminus\\{0\\}$.

###### Proof.

First, from ${\bf H}_{1}$ we know that $0$ solves (1.1). From Lemma 4.2 we
know that there exists a solution $u_{0}\in
C^{\alpha}_{s}(\overline{\Omega})\setminus\\{0\\}$ s.t. for all $\rho>0$ small
enough

$\deg_{(S)_{+}}(A-N_{f},B_{\rho}(u_{0}),0)=1.$

Besides, from Lemma 4.3 we know that for all $R>0$ big enough

$\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)=1,$

and from Lemma 4.4 that for all $r>0$ small enough

$\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)=-1.$

Choosing $\rho,r>0$ even smaller and $R>0$ bigger if necessary, we can ensure

$\overline{B}_{\rho}(u_{0})\cup\overline{B}_{r}(0)\subset B_{R}(0),\
\overline{B}_{\rho}(u_{0})\cap\overline{B}_{r}(0)=\emptyset.$

Besides, by our standing assumption that $A-N_{f}$ vanishes at finitely many
points, we can find $\rho,r>0$ s.t. $A(u)-N_{f}(u)\neq 0$ for all
$u\in\partial B_{\rho}(u_{0})\cup\partial B_{r}(0)$. So, by Proposition 2.1
$(ii)$ (domain additivity) we have

$\displaystyle\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)$
$\displaystyle=\deg_{(S)_{+}}(A-N_{f},B_{\rho}(u_{0}),0)+\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)$
$\displaystyle+\deg_{(S)_{+}}(A-N_{f},B_{R}(0)\setminus\overline{(B_{\rho}(u_{0})\cup
B_{r}(0))},0),$

which amounts to

$\deg_{(S)_{+}}(A-N_{f},B_{R}(0)\setminus\overline{(B_{\rho}(u_{0})\cup
B_{r}(0))},0)=1.$

By Proposition 2.1 $(v)$ (solution property), there exists $u_{1}\in
B_{R}(0)\setminus\overline{(B_{\rho}(u_{0})\cup B_{r}(0))}$ s.t. in
$W^{-s,p^{\prime}}(\Omega)$

$A(u_{1})-N_{f}(u_{1})=0.$

By Proposition 3.2, finally, we conclude that $u_{1}\in
C^{\alpha}_{s}(\overline{\Omega})\setminus\\{0,u_{0}\\}$ is a second solution
of (1.1). ∎

######  Remark 4.6.

The proof of Theorem 4.5 can be performed using the properties of the degree
in different ways, for instance exploiting Proposition 2.1 $(iii)$ (excision
property).

##  5\. Two nontrivial solutions for doubly asymmetric reactions

In this section we study problem (1.1) under different hypotheses, implying an
asymmetric ’jump’, from above the principal eigenvalue to below between $0$
and $\infty$ and vice versa between $0$ and $-\infty$:

  * ${\bf H}_{2}$

$f:\Omega\times{\mathbb{R}}\to{\mathbb{R}}$ is a Carathéodory mapping
satisfying

    1. $(i)$

for all $M>0$ there exists $a_{M}\in L^{\infty}(\Omega)_{+}$ s.t. for a.e.
$x\in\Omega$ and $|t|\leq M$

$|f(x,t)|\leq a_{M}(x);$

    2. $(ii)$

there exist $\theta_{1},\theta_{2}\in L^{\infty}(\Omega)$ s.t.
$\theta_{1}\leq\theta_{2}\leq\lambda_{1}$ in $\Omega$,
$\theta_{2}\not\equiv\lambda_{1}$, and uniformly for a.e. $x\in\Omega$

$\theta_{1}(x)\leq\liminf_{t\to\infty}\frac{f(x,t)}{t^{p-1}}\leq\limsup_{t\to\infty}\frac{f(x,t)}{t^{p-1}}\leq\theta_{2}(x);$

    3. $(iii)$

there exist $\eta_{1},\eta_{2}\in L^{\infty}(\Omega)$ s.t.
$\lambda_{1}\leq\eta_{1}\leq\eta_{2}$ in $\Omega$,
$\eta_{1}\not\equiv\lambda_{1}$, and uniformly for a.e. $x\in\Omega$

$\eta_{1}(x)\leq\liminf_{t\to 0^{+}}\frac{f(x,t)}{t^{p-1}}\leq\limsup_{t\to
0^{+}}\frac{f(x,t)}{t^{p-1}}\leq\eta_{2}(x);$

    4. $(iv)$

there exist $\xi_{1},\xi_{2}\in L^{\infty}(\Omega)$ s.t.
$\lambda_{1}\leq\xi_{1}\leq\xi_{2}$ in $\Omega$,
$\xi_{1}\not\equiv\lambda_{1}$, and uniformly for a.e. $x\in\Omega$

$\xi_{1}(x)\leq\liminf_{t\to-\infty}\frac{f(x,t)}{|t|^{p-2}t}\leq\limsup_{t\to-\infty}\frac{f(x,t)}{|t|^{p-2}t}\leq\xi_{2}(x);$

    5. $(v)$

uniformly for a.e. $x\in\Omega$

$\lim_{t\to 0^{-}}\frac{f(x,t)}{|t|^{p-2}t}=0.$

Hypotheses ${\bf H}_{2}$ conjure a doubly asymmetric behavior of $f(x,\cdot)$,
which is bounded below $\lambda_{1}$ at $\infty$, bounded above $\lambda_{1}$
both at $0^{+}$ and at $-\infty$ (without resonance on a positive measure
subset of $\Omega$), while we assume that it is $(p-1)$-superlinear at
$0^{-}$. Clearly ${\bf H}_{2}$ implies ${\bf H}_{0}$ (with $q=p$), so all the
results of Section 3 apply. As in Section 4, ${\bf H}_{2}$ $(iii)$ $(v)$ imply
that (1.1) admits the trivial solution $u=0$. Without loss of generality, we
may assume that (1.1) has only finitely many solutions.

######  Example 5.1.

The autonomous mapping $f\in C({\mathbb{R}})$ defined by

$f(t)=\begin{cases}\eta\,t^{p-1}+\frac{2}{\pi}(\theta-\eta)t^{p-1}\arctan(t)&\text{if
$t>0$}\\\ \xi\,\frac{2}{\pi}|t|^{p-2}t\arctan(-t)&\text{if $t\leq
0$,}\end{cases}$

with $\theta<\lambda_{1}$ and $\xi,\eta>\lambda_{1}$, satisfies ${\bf H}_{2}$.

Dealing with this case, we need to introduce truncated reactions, along with
the corresponding operators and functionals. So we set for all
$(x,t)\in\Omega\times{\mathbb{R}}$

$f_{\pm}(x,t)=f(x,\pm t^{\pm}),\
F_{\pm}(x,t)=\int_{0}^{t}f_{\pm}(x,\tau)\,d\tau.$

Further, define the completely continuous maps
$N^{\pm}_{f}:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ by setting for
all $u,v\in W^{s,p}_{0}(\Omega)$

$\langle N^{\pm}_{f}(u),v\rangle=\int_{\Omega}f_{\pm}(x,u)v\,dx,$

and the functionals $\Phi_{\pm}\in C^{1}(W^{s,p}_{0}(\Omega))$ by setting for
all $u\in W^{s,p}_{0}(\Omega)$

$\Phi_{\pm}(u)=\frac{\|u\|^{p}}{p}-\int_{\Omega}F_{\pm}(x,u)\,dx,$

satisfying $\Phi^{\prime}_{\pm}=A-N^{\pm}_{f}$. Finally, for any $m\in
L^{\infty}(\Omega)_{+}\setminus\\{0\\}$ we define two completely continuous
maps $K^{\pm}_{m}:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ by setting
for all $u,v\in W^{s,p}_{0}(\Omega)$

$\langle K^{\pm}_{m}(u),v\rangle=\pm\int_{\Omega}m(x)(u^{\pm})^{p-1}v\,dx.$

Our first existence results bears a sign information this time:

######  Lemma 5.2.

If ${\bf H}_{2}$ holds, then (1.1) has a solution $u_{0}\in
C_{s}^{\alpha}(\overline{\Omega})\cap{\rm
int}(C^{0}_{s}(\overline{\Omega})_{+})$. Moreover, there exists $\rho_{0}>0$
s.t. for all $\rho\in(0,\rho_{0}]$

$\deg_{(S)_{+}}(A-N_{f},B_{\rho}(u_{0}),0)=1.$

###### Proof.

We closely follow the argument of Lemma 4.2. Using ${\bf H}_{2}$ $(i)$ $(ii)$
and Proposition 3.8 we prove that $\Phi_{+}$ is coercive in
$W^{s,p}_{0}(\Omega)$. Besides, it is sequentially weakly l.s.c. Thus, there
exists $u_{0}\in W^{s,p}_{0}(\Omega)$ s.t.

(5.1) $\Phi_{+}(u_{0})=\inf_{u\in
W^{s,p}_{0}(\Omega)}\Phi_{+}(u_{0})=:\mu^{+}_{0}.$

Then, using ${\bf H}_{2}$ $(iii)$, we see that $\mu^{+}_{0}<0$, hence
$u_{0}\neq 0$. By (5.1), we have $\Phi^{\prime}_{+}(u_{0})=0$ in
$W^{-s,p^{\prime}}(\Omega)$, i.e., for all $v\in W^{s,p}_{0}(\Omega)$

(5.2) $\langle A(u_{0}),v\rangle=\int_{\Omega}f_{+}(x,u_{0})v\,dx.$

Testing (5.2) with $-u_{0}^{-}\in W^{s,p}_{0}(\Omega)$, by (3.1) we have

$\displaystyle\|u_{0}^{-}\|^{p}$ $\displaystyle\leqslant\langle
A(u_{0}),-u^{-}_{0}\rangle$
$\displaystyle=\int_{\Omega}f_{+}(x,u_{0})(-u_{0}^{-})\,dx=0.$

so $u_{0}\geqslant 0$ in $\Omega$. Therefore, (5.2) rephrases as (1.1). Since
$u_{0}\in W^{s,p}_{0}(\Omega)_{+}\setminus\\{0\\}$ solves (1.1), by
Propositions 3.2, 3.4 we have $u_{0}\in
C_{s}^{\alpha}(\overline{\Omega})\cap{\rm
int}(C^{0}_{s}(\overline{\Omega})_{+})$. Since $\Phi=\Phi_{+}$ in
$W^{s,p}_{0}(\Omega)_{+}$, from (5.1) we see that $u_{0}\in{\rm
int}(C^{0}_{s}(\overline{\Omega})_{+})$ is a local minimizer of $\Phi$ in
$C^{0}_{s}(\overline{\Omega})$. So, by Proposition 3.3, it is as well a local
minimizer of $\Phi$ in $W^{s,p}_{0}(\Omega)$. By our standing assumption that
$\Phi$ has only finitely many critical points, $u_{0}$ is an isolated critical
point of $\Phi$. So, by Proposition 2.2, there exists $\rho_{0}>0$ s.t. for
all $\rho\in(0,\rho_{0}]$

$\deg_{(S)_{+}}(A-N_{f},B_{\rho}(u_{0}),0)=1,$

which concludes the proof. ∎

Again we study the asymptotic behavior of $A-N_{f}$, which mainly relies on
the growth of $f(x,\cdot)$ at $-\infty$:

######  Lemma 5.3.

If ${\bf H}_{2}$ holds, then there exists $R_{0}>0$ s.t. for all $R\geqslant
R_{0}$

$\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)=0.$

###### Proof.

Fix $m_{\infty}\in L^{\infty}(\Omega)_{+}$ s.t. $\xi_{1}\leqslant
m_{\infty}\leqslant\xi_{2}$ in $\Omega$, and define
$K_{m_{\infty}}^{-}:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ as
above. The first part of the proof follows that of Lemma 4.3. We define a
$(S)_{+}$-homotopy $h^{-}_{\infty}:[0,1]\times W^{s,p}_{0}(\Omega)\to
W^{-s,p^{\prime}}(\Omega)$ by setting for all $(t,u)\in[0,1]\times
W^{s,p}_{0}(\Omega)$

$h^{-}_{\infty}(t,u)=A(u)-(1-t)N_{f}(u)-tK_{m_{\infty}}^{-}(u).$

We claim that there exists $R_{0}>0$ s.t.

(5.3) $h^{-}_{\infty}(t,u)\neq 0\ \text{for all $t\in[0,1]$, $\|u\|\geqslant
R_{0}$.}$

Arguing by contradiction, assume that there exist sequences $(t_{n})$ in
$[0,1]$, $(u_{n})$ in $W^{s,p}_{0}(\Omega)$ s.t. $\|u_{n}\|\to\infty$ and for
all $n\in{\mathbb{N}}$ we have $h^{-}_{\infty}(t_{n},u_{n})=0$ in
$W^{-s,p^{\prime}}(\Omega)$, i.e.,

(5.4) $A(u_{n})=(1-t_{n})N_{f}(u_{n})+t_{n}K^{-}_{m_{\infty}}(u_{n}).$

By ${\bf H}_{2}$ $(ii)$ and Proposition 3.8, there exists $\sigma>0$ s.t. for
all $u\in W^{s,p}_{0}(\Omega)$

$\|u\|^{p}-\int_{\Omega}\theta_{2}(x)|u|^{p}\,dx\geqslant\sigma\|u\|^{p}.$

Fix $\varepsilon\in(0,\sigma\lambda_{1})$. Then, by ${\bf H}_{2}$ $(ii)$ we
can find $M>0$ s.t. for a.e. $x\in\Omega$ and all $t\geqslant M$

$f(x,t)\leqslant(\theta_{2}(x)+\varepsilon)t^{p-1}.$

Also, for a.e. $x\in\Omega$ and all $t\in[0,M]$ we have by ${\bf H}_{2}$ $(i)$

$f(x,t)\leqslant a_{M}(x).$

All in all, we can find $C>0$ s.t. for a.e. $x\in\Omega$ and all $t\geqslant
0$

$f(x,t)\leqslant(\theta_{2}(x)+\varepsilon)t^{p-1}+C.$

By applying (3.1) and testing (5.4) with $u^{+}_{n}\in
W^{s,p}_{0}(\Omega)_{+}$, we get for all $n\in{\mathbb{N}}$

$\displaystyle\|u^{+}_{n}\|^{p}$ $\displaystyle\leqslant\langle
A(u_{n}),u^{+}_{n}\rangle$
$\displaystyle=(1-t_{n})\int_{\Omega}f(x,u_{n})u^{+}_{n}\,dx-
t_{n}\int_{\Omega}m_{\infty}(x)(u^{-}_{n})^{p-1}u^{+}_{n}\,dx$
$\displaystyle\leqslant\int_{\Omega}(\theta_{2}(x)+\varepsilon)(u^{+}_{n})^{p}\,dx+C\|u^{+}_{n}\|_{1},$

which, along with the continuous embedding $W^{s,p}_{0}(\Omega)\hookrightarrow
L^{1}(\Omega)$, implies

$\Big{(}\sigma-\frac{\varepsilon}{\lambda_{1}}\Big{)}\|u^{+}_{n}\|^{p}\leqslant
C\|u^{+}_{n}\|.$

Hence $(u^{+}_{n})$ is bounded in $W^{s,p}_{0}(\Omega)$. Besides, by the
triangle inequality we have for all $n\in{\mathbb{N}}$

$\|u^{-}_{n}\|\geqslant\|u_{n}\|-\|u^{+}_{n}\|,$

and the latter tends to $\infty$ as $n\to\infty$. So we have
$\|u^{-}_{n}\|\to\infty$. Passing if necessary to a subsequence, we have
$t_{n}\to t$ and $\|u_{n}\|>0$. Set for all $n\in{\mathbb{N}}$

$v_{n}=\frac{u_{n}}{\|u_{n}\|}.$

Since $(v_{n})$ is bounded in $W^{s,p}_{0}(\Omega)$, passing to a further
subsequence we have $v_{n}\rightharpoonup v$ in $W^{s,p}_{0}(\Omega)$,
$v_{n}\to v$ in $L^{p}(\Omega)$, and $v_{n}(x)\to v(x)$ for a.e. $x\in\Omega$.
Note that

$\|v^{+}_{n}\|=\frac{\|u^{+}_{n}\|}{\|u_{n}\|}\to 0$

as $n\to\infty$. So, for a.e. $x\in\Omega$ we have

$v^{-}_{n}(x)=v^{+}_{n}(x)-v_{n}(x)\to-v(x),$

which implies $v(x)\leqslant 0$ in $\Omega$. Dividing (5.4) by
$\|u_{n}\|^{p-1}$ we have for all $n\in{\mathbb{N}}$

(5.5) $A(v_{n})=(1-t_{n})g_{n}+t_{n}K^{-}_{m_{\infty}}(v_{n}),$

where we have set for all $x\in\Omega$

$g_{n}(x)=\frac{f(x,u_{n}(x))}{\|u_{n}\|^{p-1}}.$

Reasoning as in Lemma 4.3 we see that $(g_{n})$ is bounded in
$L^{p^{\prime}}(\Omega)$, hence, passing to a subsequence,
$g_{n}\rightharpoonup g_{\infty}$ in $L^{p^{\prime}}(\Omega)$. We will now
prove that there exists $\hat{g}_{\infty}\in L^{\infty}(\Omega)$ s.t. in
$\Omega$

(5.6) $g_{\infty}=\hat{g}_{\infty}|v|^{p-2}v,\
\xi_{1}\leqslant\hat{g}_{\infty}\leqslant\xi_{2}.$

Indeed, recall that $v\leqslant 0$ in $\Omega$. Set

$\Omega^{-}=\big{\\{}x\in\Omega:\,v(x)<0\big{\\}},\
\Omega^{0}=\big{\\{}x\in\Omega:\,v(x)=0\big{\\}}.$

Then, for all $\varepsilon>0$, $n\in{\mathbb{N}}$ set

$\Omega^{-}_{\varepsilon,n}=\Big{\\{}x\in\Omega:\,u_{n}(x)<0,\,\xi_{1}(x)-\varepsilon\leqslant\frac{f(x,u_{n}(x))}{|u_{n}(x)|^{p-2}u_{n}(x)}\leqslant\xi_{2}(x)+\varepsilon\Big{\\}}.$

By ${\bf H}_{2}$ $(iv)$ we have $\chi_{\Omega^{-}_{\varepsilon,n}}\to 1$ in
$\Omega^{-}$ with bounded convergence, hence
$\chi_{\Omega^{-}_{\varepsilon,n}}g_{n}\rightharpoonup g_{\infty}$ in
$L^{p^{\prime}}(\Omega^{-})$. Besides, by definition of $v_{n}$, for all
$\varepsilon>0$ and all $n\in{\mathbb{N}}$ big enough, in $\Omega^{-}$ we have
$v_{n}<0$ and

$\chi_{\Omega^{-}_{\varepsilon,n}}(\xi_{2}+\varepsilon)|v_{n}|^{p-2}v_{n}\leqslant\chi_{\Omega^{-}_{\varepsilon,n}}g_{n}\leqslant\chi_{\Omega^{-}_{\varepsilon,n}}(\xi_{1}-\varepsilon)|v_{n}|^{p-2}v_{n}.$

Passing to the limit as $n\to\infty$ and $\varepsilon\to 0^{+}$, we get in
$\Omega^{-}$

$\xi_{2}|v|^{p-2}v\leqslant g_{\infty}\leqslant\xi_{1}|v|^{p-2}v.$

Similarly, we get $g_{\infty}=0$ in $\Omega^{0}$, which completes the argument
for (5.6). Now we test (5.5) with $v_{n}-v\in W^{s,p}_{0}(\Omega)$, so we get
for all $n\in{\mathbb{N}}$

$\langle A(v_{n}),v_{n}-v\rangle=(1-t_{n})\int_{\Omega}g_{n}(x)(v_{n}-v)\,dx-
t_{n}\int_{\Omega}m_{\infty}(x)(v_{n}^{-})^{p-1}(v_{n}-v)\,dx.$

The latter tends to $0$ as $n\to\infty$, so by the $(S)_{+}$-property of $A$
we have $v_{n}\to v$ in $W^{s,p}_{0}(\Omega)$, hence in particular $\|v\|=1$.
Passing to the limit in (5.5) as $n\to\infty$ and using (5.6), we see that $v$
solves the weighted eigenvalue problem

(5.7)
$\begin{cases}(-\Delta)_{p}^{s}\,v=\tilde{g}_{\infty}(x)|v|^{p-2}v&\text{in
$\Omega$}\\\ v=0&\text{in $\Omega^{c}$,}\end{cases}$

where we have set for all $x\in\Omega$

$\tilde{g}_{\infty}(x)=(1-t)\hat{g}_{\infty}(x)+tm_{\infty}(x).$

Clearly $\tilde{g}_{\infty}\in L^{\infty}(\Omega)$, in addition by (5.6) and
the choice of $m_{\infty}$ we have
$\xi_{1}\leqslant\tilde{g}_{\infty}\leqslant\xi_{2}$ in $\Omega$, hence by
${\bf H}_{2}$ $(iv)$ $\tilde{g}_{\infty}\geqslant\lambda_{1}$ in $\Omega$ with
$\tilde{g}_{\infty}\not\equiv\lambda_{1}$. By Proposition 3.5 we have

$\lambda_{1}(\tilde{g}_{\infty})<\lambda_{1}(\lambda_{1})=1.$

So, $v$ is a non-principal eigenfunction of (5.7), hence nodal (again by
Proposition 3.5), against $v\leqslant 0$ in $\Omega$. The contradiction proves
(5.3). We can now apply Proposition 2.1 $(iv)$ (homotopy invariance) and get
for all $R\geqslant R_{0}$

(5.8)
$\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)=\deg_{(S)_{+}}(A-K^{-}_{m_{\infty}},B_{R}(0),0).$

So we are led to computing the degree of $A-K^{-}_{m_{\infty}}$, which this
time cannot be done directly. Fix $\beta_{\infty}\in
L^{\infty}(\Omega)_{+}\setminus\\{0\\}$, and for all $(t,u)\in[0,1]\times
W^{s,p}_{0}(\Omega)$ set

$\hat{h}_{\infty}(t,u)=A(u)-K^{-}_{m_{\infty}}(u)+t\beta_{\infty}$

(here we identify $\beta_{\infty}$ with an element of
$W^{-s,p^{\prime}}(\Omega)$). Clearly $\hat{h}_{\infty}:[0,1]\times
W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ is a $(S)_{+}$-homotopy. We
claim that for all $t\in[0,1]$ and all $u\in
W^{s,p}_{0}(\Omega)\setminus\\{0\\}$

(5.9) $\hat{h}_{\infty}(t,u)\neq 0.$

Arguing by contradiction, let $t\in[0,1]$, $u\in
W^{s,p}_{0}(\Omega)\setminus\\{0\\}$ be s.t. in $W^{-s,p^{\prime}}(\Omega)$

(5.10) $A(u)=K^{-}_{m_{\infty}}(u)-t\beta_{\infty}.$

We distinguish two cases:

  * $(a)$

If $t=0$, then (5.10) rephrases as

$A(u)=K^{-}_{m_{\infty}}(u).$

Testing with $u^{+}\in W^{s,p}_{0}(\Omega)$ and applying (3.1), we have

$\displaystyle\|u^{+}\|^{p}$ $\displaystyle\leqslant\langle A(u),u^{+}\rangle$
$\displaystyle=-\int_{\Omega}m_{\infty}(x)(u^{-})^{p-1}u^{+}\,dx=0,$

so $u\leqslant 0$ in $\Omega$. Then $u$ solves in fact

$\begin{cases}(-\Delta)_{p}^{s}\,u=m_{\infty}(x)|u|^{p-2}u&\text{in
$\Omega$}\\\ u=0&\text{in $\Omega^{c}$.}\end{cases}$

By ${\bf H}_{2}$ $(iv)$ and the choice of $m_{\infty}$ we have
$m_{\infty}\geqslant\lambda_{1}$ in $\Omega$ and
$m_{\infty}\not\equiv\lambda_{1}$, hence

$\lambda_{1}(m_{\infty})<\lambda_{1}(\lambda_{1})=1.$

So, $u\in-W^{s,p}_{0}(\Omega)_{+}\setminus\\{0\\}$ is a non-principal
eigenfunction with weight $m_{\infty}$, hence nodal by Proposition 3.5, a
contradiction.

  * $(b)$

If $t\in(0,1]$, then testing (5.10) with $u^{+}\in W^{s,p}_{0}(\Omega)_{+}$
and applying (3.1) we have

$\displaystyle\|u^{+}\|^{p}$ $\displaystyle\leqslant\langle A(u),u^{+}\rangle$
$\displaystyle=-\int_{\Omega}m_{\infty}(x)(u^{-})^{p-1}u^{+}\,dx-t\int_{\Omega}\beta(x)u^{+}\,dx\leqslant
0,$

so again $u\leqslant 0$ in $\Omega$. Then, $-u\in W^{s,p}_{0}(\Omega)_{+}$
satisfies

$\begin{cases}(-\Delta)_{p}^{s}\,(-u)=m_{\infty}(x)(-u)^{p-1}+t\beta_{\infty}(x)&\text{in
$\Omega$}\\\ -u=0&\text{in $\Omega^{c}$.}\end{cases}$

As above $\lambda_{1}(m_{\infty})<1$, so this violates Lemma 3.9.

In both cases we reach a contradiction, thus proving (5.9). Again by
Proposition 2.1 $(iv)$ (homotopy invariance) we have for all $R>0$

(5.11)
$\deg_{(S)_{+}}(A-K^{-}_{m_{\infty}},B_{R}(0),0)=\deg_{(S)_{+}}(A-K^{-}_{m_{\infty}}+\beta_{\infty},B_{R}(0),0).$

Finally, for all $R>0$ we have

(5.12) $\deg_{(S)_{+}}(A-K^{-}_{m_{\infty}}+\beta_{\infty},B_{R}(0),0)=0.$

Arguing by contradiction, assume that the degree above does not vanish for
some $R>0$. Then, by Proposition 2.1 $(v)$ (solution property) there exists
$u\in B_{R}(0)$ s.t. in $W^{-s,p^{\prime}}(\Omega)$

$A(u)=K^{-}_{m_{\infty}}(u)-\beta_{\infty},$

namely,

$\begin{cases}(-\Delta)_{p}^{s}\,u=-m_{\infty}(x)(u^{-})^{p-1}-\beta_{\infty}(x)&\text{in
$\Omega$}\\\ u=0&\text{in $\Omega^{c}$.}\end{cases}$

Arguing as in case $(b)$ above we see that $u\leqslant 0$ in $\Omega$ and
reach a contradiction to Lemma 3.9, thus proving (5.12). Now, concatenating
(5.8), (5.11), and (5.12), we have for all $R\geqslant R_{0}$

$\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)=0,$

thus concluding the proof. ∎

We complete the picture by studying the behavior of $A-N_{f}$ near $0$, which
is affected by the behavior of $f(x,\cdot)$ near $0^{+}$:

######  Lemma 5.4.

If ${\bf H}_{2}$ holds, then there exists $r_{0}>0$ s.t. for all
$r\in(0,r_{0}]$

$\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)=0.$

###### Proof.

Fix $m_{0}\in L^{\infty}(\Omega)$ s.t. $\eta_{1}\leqslant
m_{0}\leqslant\eta_{2}$ in $\Omega$, and define the map
$K^{+}_{m_{0}}:W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ as above. At
first we follow the proof of Lemma 4.4. We define a $(S)_{+}$-homotopy
$h^{+}_{0}:[0,1]\times W^{s,p}_{0}(\Omega)\to W^{-s,p^{\prime}}(\Omega)$ by
setting for all $(t,u)\in[0,1]\times W^{s,p}_{0}(\Omega)$

$h^{+}_{0}(t,u)=A(u)-(1-t)N_{f}(u)-tK^{+}_{m_{0}}(u).$

We claim that there exists $r_{0}>0$ s.t.

(5.13) $h^{+}_{0}(t,u)\neq 0\ \text{for all $t\in[0,1]$, $0<\|u\|\leqslant
r_{0}$.}$

Arguing by contradiction, assume that there exist sequences $(t_{n})$ in
$[0,1]$, $(u_{n})$ in $W^{s,p}_{0}(\Omega)\setminus\\{0\\}$ s.t. $u_{n}\to 0$
in $W^{s,p}_{0}(\Omega)$ and $h^{+}_{0}(t_{n},u_{n})=0$ in
$W^{-s,p^{\prime}}(\Omega)$ for all $n\in{\mathbb{N}}$. Set for all
$n\in{\mathbb{N}}$

$v_{n}=\frac{u_{n}}{\|u_{n}\|}.$

Passing if necessary to a subsequence, we have $t_{n}\to t$ as well as
$v_{n}\rightharpoonup v$ in $W^{s,p}_{0}(\Omega)$, $v_{n}\to v$ in
$L^{p}(\Omega)$, and $v_{n}(x)\to v(x)$ for a.e. $x\in\Omega$. Besides, for
all $n\in{\mathbb{N}}$ we have in $W^{-s,p^{\prime}}(\Omega)$

(5.14) $A(v_{n})=(1-t_{n})g_{n}+t_{n}K^{+}_{m_{0}}(v_{n}),$

where we have set for all $x\in\Omega$

$g_{n}(x)=\frac{f(x,u_{n}(x))}{\|u_{n}\|^{p-1}}.$

Reasoning as in Lemma 4.4 we see that $(g_{n})$ is a bounded sequence in
$L^{p^{\prime}}(\Omega)$, so passing to a further subsequence we have
$g_{n}\rightharpoonup g_{0}$ in $L^{p^{\prime}}(\Omega)$. We claim that there
exists $\hat{g}_{0}\in L^{\infty}(\Omega)$ s.t. in $\Omega$

(5.15) $g_{0}=\hat{g}_{0}(v^{+})^{p-1},\
\eta_{1}\leqslant\hat{g}_{0}\leqslant\eta_{2}.$

Indeed, define the set

$\Omega^{+}=\big{\\{}x\in\Omega:\,v(x)>0\big{\\}},$

and for all $\varepsilon>0$, $n\in{\mathbb{N}}$

$\Omega^{+}_{\varepsilon,n}=\Big{\\{}x\in\Omega:\,u_{n}(x)>0,\,\eta_{1}(x)-\varepsilon\leqslant\frac{f(x,u_{n}(x))}{u_{n}^{p-1}(x)}\leqslant\eta_{2}(x)+\varepsilon\Big{\\}}.$

By ${\bf H}_{2}$ $(iii)$ we have $\chi_{\Omega^{+}_{\varepsilon,n}}\to 1$ a.e.
in $\Omega^{+}$, with bounded convergence, hence
$\chi_{\Omega^{+}_{\varepsilon,n}}g_{n}\rightharpoonup g_{0}$ in
$L^{p^{\prime}}(\Omega^{+})$. Recalling the definition of $v_{n}$, for all
$\varepsilon>0$ and all $n\in{\mathbb{N}}$ big enough, in $\Omega^{+}$ we have
$v_{n}>0$ and

$\chi_{\Omega^{+}_{\varepsilon,n}}(\eta_{1}-\varepsilon)v_{n}^{p-1}\leqslant\chi_{\Omega^{+}_{\varepsilon,n}}g_{n}\leqslant\chi_{\Omega^{+}_{\varepsilon,n}}(\eta_{2}+\varepsilon)v_{n}^{p-1}.$

Passing to the limit as $n\to\infty$ and then as $\varepsilon\to 0^{+}$, we
get in $\Omega^{+}$

$\eta_{1}v^{p-1}\leqslant g_{0}\leqslant\eta_{2}v^{p-1}.$

Similarly, using ${\bf H}_{2}$ $(v)$ we get $g_{0}=0$ in
$\Omega\setminus\Omega^{+}$, which completes the argument for (5.15). Now we
go back to (5.14). Testing with $v_{n}-v\in W^{s,p}_{0}(\Omega)$, we have

$\langle
A(v_{n}),v_{n}-v\rangle=(1-t_{n})\int_{\Omega}g_{n}(x)(v_{n}-v)\,dx+t_{n}\int_{\Omega}m_{0}(x)(v_{n}^{+})^{p-1}(v_{n}-v)\,dx,$

and the latter tends to $0$ as $n\to\infty$. So, by the $(S)_{+}$-property of
$A$ we have $v_{n}\to v$ in $W^{s,p}_{0}(\Omega)$, hence $\|v\|=1$. Passing to
the limit in (5.14) as $n\to\infty$ and using (5.15), we get in
$W^{-s,p^{\prime}}(\Omega)$

(5.16) $A(v)=\tilde{g}_{0}(v^{+})^{p-1},$

where we have set for all $x\in\Omega$

$\tilde{g}_{0}(x)=(1-t)\hat{g}_{0}+tm_{0}(x).$

Clearly $\tilde{g}_{0}\in L^{\infty}(\Omega)$, and by (5.15) and the choice of
$m_{0}$ we have $\eta_{1}\leqslant\tilde{g}_{0}\leqslant\eta_{2}$ in $\Omega$.
Testing (5.16) with $-v^{-}\in W^{s,p}_{0}(\Omega)$ and applying (3.1), we
have

$\displaystyle\|v^{-}\|^{p}$ $\displaystyle\leqslant\langle
A(v),-v^{-}\rangle$
$\displaystyle=\int_{\Omega}\tilde{g}_{0}(x)(v^{+})^{p-1}(-v^{-})\,dx=0,$

hence $v\geqslant 0$ in $\Omega$. We can therefore rephrase (5.16) as the
weighted eigenvalue problem

(5.17) $\begin{cases}(-\Delta)_{p}^{s}\,v=\tilde{g}_{0}(x)v^{p-1}&\text{in
$\Omega$}\\\ v=0&\text{in $\Omega^{c}$.}\end{cases}$

By ${\bf H}_{2}$ $(iii)$ we have $\tilde{g}_{0}\geqslant\lambda_{1}$ in
$\Omega$ with $\tilde{g}_{0}\not\equiv\lambda_{1}$, so by Proposition 3.5

$\lambda_{1}(\tilde{g}_{0})<\lambda_{1}(\lambda_{1})=1.$

Thus, $v\neq 0$ is a non-principal eigenfunction of (5.17), hence nodal (again
by Proposition 3.5), against $v\geqslant 0$. This contradiction proves (5.13).
We apply Proposition 2.1 $(iv)$ (homotopy invariance) and get for all
$r\in(0,r_{0}]$

(5.18)
$\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)=\deg_{(S)_{+}}(A-K^{+}_{m_{0}},B_{r}(0),0).$

Now there remains to compute the right-hand side. We proceed as in Lemma 5.3,
fixing $\beta_{0}\in L^{\infty}(\Omega)_{+}\setminus\\{0\\}$ and setting for
all $(t,u)\in[0,1]\times W^{s,p}_{0}(\Omega)$

$\hat{h}^{+}_{0}(t,u)=A(u)-K^{+}_{m_{0}}(u)-t\beta_{0}.$

Clearly, $\hat{h}^{+}_{0}:[0,1]\times W^{s,p}_{0}(\Omega)\to
W^{-s,p^{\prime}}(\Omega)$ is a $(S)_{+}$-homotopy. Again we claim that for
all $t\in[0,1]$ and all $u\in W^{s,p}_{0}(\Omega)\setminus\\{0\\}$

(5.19) $\hat{h}^{+}_{0}(t,u)\neq 0.$

Arguing by contradiction, let $t\in[0,1]$, $u\in
W^{s,p}_{0}(\Omega)\setminus\\{0\\}$ be s.t. in $W^{-s,p^{\prime}}(\Omega)$

(5.20) $A(u)=K^{+}_{m_{0}}(u)+t\beta_{0}.$

We distinguish two cases:

  * $(a)$

If $t=0$, then (5.20) rephrases as

$A(u)=K^{+}_{m_{0}}(u).$

Testing with $-u^{-}\in W^{s,p}_{0}(\Omega)$ and applying (3.1) we have

$\displaystyle\|u^{-}\|^{p}$ $\displaystyle\leqslant\langle
A(u),-u^{-}\rangle$
$\displaystyle=\int_{\Omega}m_{0}(x)(u^{+})^{p-1}(-u^{-})\,dx=0,$

so $u\geqslant 0$ in $\Omega$. Then $u$ solves in fact

$\begin{cases}(-\Delta)_{p}^{s}\,u=m_{0}(x)u^{p-1}&\text{in $\Omega$}\\\
u=0&\text{in $\Omega^{c}$.}\end{cases}$

By ${\bf H}_{2}$ $(iii)$ we have $m_{0}\geqslant\lambda_{1}$ in $\Omega$ with
$m_{0}\not\equiv\lambda_{1}$, so by Proposition 3.5

$\lambda_{1}(m_{0})<\lambda_{1}(\lambda_{1})=1.$

So, $u$ is a non-principal eigenfunction with weight $m_{0}$, hence nodal,
against $u\geqslant 0$.

  * $(b)$

If $t\in(0,1]$, then testing (5.20) with $-u^{-}\in W^{s,p}_{0}(\Omega)$ and
applying (3.1) we have

$\displaystyle\|u^{-}\|^{p}$ $\displaystyle\leqslant\langle
A(u),-u^{-}\rangle$
$\displaystyle=\int_{\Omega}m_{0}(x)(u^{+})^{p-1}(-u^{-})\,dx+t\int_{\Omega}\beta_{0}(x)(-u^{-})\,dx\leqslant
0,$

so $u\geqslant 0$ in $\Omega$. Then (5.20) becomes

$\begin{cases}(-\Delta)_{p}^{s}\,u=m_{0}(x)u^{p-1}+t\beta_{0}(x)&\text{in
$\Omega$}\\\ u=0&\text{in $\Omega^{c}$,}\end{cases}$

with $1>\lambda_{1}(m_{0})$ as above and $u\geqslant 0$ in $\Omega$, against
Lemma 3.9.

In both cases we reach a contradiction, thus proving (5.19). This in turn
allows us to apply Proposition 2.1 $(iv)$ (homotopy invariance) and have for
all $r>0$

(5.21)
$\deg_{(S)_{+}}(A-K^{+}_{m_{0}},B_{r}(0),0)=\deg_{(S)_{+}}(A-K^{+}_{m_{0}}-\beta_{0},B_{r}(0),0).$

To conclude, we claim that for all $r>0$

(5.22) $\deg_{(S)_{+}}(A-K^{+}_{m_{0}}-\beta_{0},B_{r}(0),0)=0.$

Arguing by contradiction, assume that for some $r>0$

$\deg_{(S)_{+}}(A-K^{+}_{m_{0}}-\beta_{0},B_{r}(0),0)\neq 0.$

By Proposition 2.1 $(v)$ (solution property) there exists $u\in B_{r}(0)$ s.t.

$\begin{cases}(-\Delta)_{p}^{s}\,u=m_{0}(x)(u^{+})^{p-1}+\beta_{0}(x)&\text{in
$\Omega$}\\\ u=0&\text{in $\Omega^{c}$.}\end{cases}$

Arguing as in case $(b)$ we see that $u\geqslant 0$ in $\Omega$, violating
Lemma 3.9. So (5.22) is proved. Finally, concatenating (5.18), (5.21), and
(5.22) we get for all $r\in(0,r_{0}]$

$\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)=0,$

which concludes the proof. ∎

Our multiplicity result for this case is the following:

######  Theorem 5.5.

If ${\bf H}_{2}$ holds, then problem (1.1) has at least two nontrivial
solutions $u_{0}\in C_{s}^{\alpha}(\overline{\Omega})\cap{\rm
int}(C^{0}_{s}(\overline{\Omega})_{+})$, $u_{1}\in
C_{s}^{\alpha}(\overline{\Omega})\setminus\\{0\\}$.

###### Proof.

The proof is similar to that of Theorem 4.5. First, from ${\bf H}_{2}$ we know
that $0$ solves (1.1). By Lemma 5.2 there exists a solution $u_{0}\in
C_{s}^{\alpha}(\overline{\Omega})\cap{\rm
int}(C^{0}_{s}(\overline{\Omega})_{+})$ s.t. for all $\rho>0$ small enough

$\deg_{(S)_{+}}(A-N_{f},B_{\rho}(u_{0}),0)=1.$

Besides, by Lemma 5.3 we have for all $R>0$ big enough

$\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)=0,$

and by Lemma 5.4 we have for all $r>0$ small enough

$\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)=0.$

Choosing $\rho,r>0$ even smaller and $R>0$ bigger if necessary, we can ensure

$\overline{B}_{\rho}(u_{0})\cup\overline{B}_{r}(0)\subset B_{R}(0),\
\overline{B}_{\rho}(u_{0})\cap\overline{B}_{r}(0)=\emptyset.$

By our standing assumption that $A-N_{f}$ vanishes at finitely many points, we
can find $\rho,r>0$ s.t. $A(u)-N_{f}(u)\neq 0$ for all $u\in\partial
B_{\rho}(u_{0})\cup\partial B_{r}(0)$. So, by Proposition 2.1 $(ii)$ (domain
additivity) we have

$\displaystyle\deg_{(S)_{+}}(A-N_{f},B_{R}(0),0)$
$\displaystyle=\deg_{(S)_{+}}(A-N_{f},B_{\rho}(u_{0}),0)+\deg_{(S)_{+}}(A-N_{f},B_{r}(0),0)$
$\displaystyle+\deg_{(S)_{+}}(A-N_{f},B_{R}(0)\setminus\overline{(B_{\rho}(u_{0})\cup
B_{r}(0))},0),$

which amounts to

$\deg_{(S)_{+}}(A-N_{f},B_{R}(0)\setminus\overline{(B_{\rho}(u_{0})\cup
B_{r}(0))},0)=-1.$

By Proposition 2.1 $(v)$ (solution property), there exists $u_{1}\in
B_{R}(0)\setminus\overline{(B_{\rho}(u_{0})\cup B_{r}(0))}$ s.t. in
$W^{-s,p^{\prime}}(\Omega)$

$A(u_{1})-N_{f}(u_{1})=0.$

By Proposition 3.2, finally, we conclude that $u_{1}\in
C^{\alpha}_{s}(\overline{\Omega})\setminus\\{0,u_{0}\\}$ is a second solution
of (1.1). ∎

######  Remark 5.6.

Formally, Theorem 5.5 above is analogous to [1, Theorem 32]. Nevertheless, the
nonlocal nature of the operator $(-\Delta)_{p}^{s}\,$ bears significant
differences. The main issue is that, in general, for $u\in
W^{s,p}_{0}(\Omega)$ we have

$(-\Delta)_{p}^{s}\,u\neq(-\Delta)_{p}^{s}\,u^{+}-(-\Delta)_{p}^{s}\,u^{-},$

which demands a different technique in dealing with the positive and negative
parts, with respect to the case of the $p$-Laplacian (compare for instance [1,
Proposition 30] to our Lemma 5.3).

Acknowledgement. Both authors are members of GNAMPA (Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM
(Istituto Nazionale di Alta Matematica ’Francesco Severi’) and are supported
by the research project _Evolutive and Stationary Partial Differential
Equations with a Focus on Biomathematics_ , funded by Fondazione di Sardegna
(2019). A. Iannizzotto is also supported by the grant PRIN-2017AYM8XW:
_Nonlinear Differential Problems via Variational, Topological and Set-valued
Methods_. We warmly thank L. Brasco for precious suggestions about the
eigenvalue problem.

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