# Radial solutions for Hénon type fully nonlinear equations in annuli and
exterior domains

Liliane<EMAIL_ADDRESS>Departamento de Matemática, Universidade de
Brasília, Brazil Gabrielle<EMAIL_ADDRESS>Instituto de
Ciências Matemáticas e de Computação, Universidade de São Paulo, Brazil

(August 27, 2024)

Abstract. In this note we study existence of positive radial solutions in
annuli and exterior domains for a class of nonlinear equations driven by Pucci
extremal operators subject to a Hénon type weight. Our approach is based on
the shooting method applied to the corresponding ODE problem, energy
arguments, and the associated flow of an autonomous quadratic dynamical
system.

Keywords. Fully nonlinear equations; radial positive solutions; dynamical
system; shooting method.

MSC2020. 35J15, 35J60, 35B09, 34A34.

##  1 Introduction

In this note we study existence of positive radial solutions of fully
nonlinear elliptic partial differential equations in the form

$\displaystyle\left\\{\begin{array}[]{rclcl}\mathcal{M}_{\lambda,\Lambda}^{\pm}(D^{2}u)+|x|^{a}\,u^{p}&=&0&\mbox{in}&\;\Omega\\\
u&>&0&\mbox{in}&\;\Omega\\\ u&=&0&\mbox{on}&\partial\Omega\end{array}\right.$
(1.4)

where $p>1$, $a>-1$, $0<\lambda\leq\Lambda$, and $\Omega$ is either an annulus
or an exterior domain in ${\mathbb{R}^{N}}$ for $N\geq 3$. Additionally, we
suppose $\tilde{N}_{+}>2$ in the case of $\mathcal{M}^{+}_{\lambda,\Lambda}$,
where $\tilde{N}_{\pm}$ are the dimension-like numbers

$\tilde{N}_{+}=\frac{\lambda}{\Lambda}(N-1)+1$,
$\tilde{N}_{-}=\frac{\Lambda}{\lambda}(N-1)+1$.

Here $\mathcal{M}_{\lambda,\Lambda}^{\pm}$ are the Pucci’s extremal operators,
which play an essential role in stochastic control theory and mean field
games. We deal with classical solutions of (1.4) that are $C^{2}$ in $\Omega$.

In [9] nonexistence results in exterior domains for weighted equations as in
(1.4) via a dynamical system approach were established. However, proving the
existence of such solutions, as well as solutions in annuli, is a difficult
task in terms of that approach, since the orbits in the flow there present
blow-up discontinuities. Our goal in this work is to complement the analysis
in [9], by showing existence of radial solutions in exterior domains and
annuli.

We mention that the analysis of the associated ODE problem for proving
existence of annular or exterior domain solutions has been performed in many
papers in semilinear cases [1, 8, 11].

In the case when $a=0$, results of this nature were obtained in [6, 5]. The
analysis in [5] was performed in light of the change of variables in [4]. In
[9] we have already noticed that employing quadratic dynamical systems is
effective to deal with these problems, in a simple and unified way. Here, our
shooting arguments to obtain existence of annular solutions are inspired by
those in [6]. On the other hand, in what concerns the existence in exterior
domains we introduce an alternative dynamical system, which is different from
those in [5] and [9], of quadratic type which do not present blow-up
discontinuities.

For solutions in annuli, our main result reads as follows.

######  Theorem 1.1.

For any $p>1$, and $0<\mathfrak{a}<\mathfrak{b}<+\infty$, problem (1.4) has a
positive radial solution in the annulus

$\Omega=\\{x\in{\mathbb{R}^{N}}\,:\,\mathfrak{a}<|x|<\mathfrak{b}\\}$, with
$0<\mathfrak{a}<\mathfrak{b}<+\infty$.

Note that solutions of (1.4) may not be radial in general, for instance the
Gidas-Ni-Nirenberg type symmetry result of [3] does not hold for annular
domains. Moreover, since $v=-u$ solves
$\mathcal{M}^{\mp}_{\lambda,\Lambda}(v)+|x|^{a}\,|u|^{p-1}u$ in $\Omega$, then
Theorem 1.1 also proves the existence of a negative solution for the
corresponding problem.

The proof of Theorem 1.1 relies on a careful study of the ODE problem,
shooting method and energy arguments. We mention that solutions in annuli can
be identified in correspondence with orbits in the dynamical system, but the
interior and exterior radii are not explicit in that approach neither can be
obtained for an arbitrary annulus via rescaling.

As far as exterior domain solutions are concerned, we obtain the following
result. We recall $p^{*}_{a\pm}$ are the critical exponents defined in [9] for
the operators $\mathcal{M}^{\pm}_{\lambda,\Lambda}$. They are the threshold
for the existence and nonexistence of radial positive regular solutions, that
is, solutions differentiably defined at $r=0$.

######  Theorem 1.2 (Exterior domain).

For any $p>p^{*}_{a\pm}$ and $R>0$, by setting
$\Omega={\mathbb{R}^{N}}\setminus B_{R}$, it holds:

  1. (i)

there exists a unique fast decaying solution of (1.4);

  2. (ii)

there exist infinitely many solutions of (1.4) with either slow or pseudo-slow
decay.

In addition, the ranges where pseudo-slow exterior domain solutions exist are
the same where pseudo-slow decay regular solutions exist in [9], see also
[12].

The choice of using quadratic systems to treat weighted equations is
categorical since the new dynamical system variables do not see the weight, as
in [9]. It is worth mentioning that earlier methods employed might be much
more involved, meanwhile a simple variable change which eliminates the weight
is not available for Pucci operators.

The text is organized as follows. In Section 2 we recall some preliminary
tools on radial solutions and shooting method. In Section 3 we write the
corresponding quadratic system and use it to prove Theorem 1.2, while Section
4 is devoted to the proof of Theorem 1.1.

##  2 Preliminaries

We start by recalling that Pucci’s extremal operators
$\mathcal{M}^{\pm}_{\lambda,\Lambda}$, for $0<\lambda\leq\Lambda$, are defined
as

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

where $A,X$ are $N\times N$ symmetric matrices, and $I$ is the identity
matrix. Equivalently, if we denote by $\\{e_{i}\\}_{1\leq i\leq N}$ the
eigenvalues of $X$, we can define the Pucci’s operators as

$\displaystyle\textstyle{\textrm{$\mathcal{M}_{\lambda,\Lambda}^{+}(X)=\Lambda\sum_{e_{i}\geq
0}e_{i}+\lambda\sum_{e_{i}<0}e_{i}$, \;\;\;
$\mathcal{M}_{\lambda,\Lambda}^{-}(X)=\lambda\sum_{e_{i}\geq
0}e_{i}+\Lambda\sum_{e_{i}<0}e_{i}$}.}$ (2.1)

From now on we will drop writing the parameters $\lambda,\Lambda$ in the
notations for the Pucci’s operators.

When $u$ is a radial function, for ease of notation we set $u(|x|)=u(r)$ for
$r=|x|$. If $u$ is $C^{2}$, the eigenvalues of the Hessian matrix $D^{2}u$ are
$\\{u^{\prime\prime},\frac{u^{\prime}(r)}{r},\ldots,\frac{u^{\prime}(r)}{r}\\}$
where $\frac{u^{\prime}(r)}{r}$ is repeated $N-1$ times.

The Lane-Emden system (1.4) for $\mathcal{M}^{+}$ is written in radial
coordinates as

$\displaystyle\begin{array}[]{l}u^{\prime\prime}\;=\;M_{+}(-r^{-1}(N-1)\,m_{+}(u^{\prime})-r^{a}u^{p}),\quad
u>0,\end{array}$ (2.3)

while for $\mathcal{M}^{-}$ one has

$\displaystyle\begin{array}[]{l}u^{\prime\prime}\;=\;M_{-}(-r^{-1}(N-1)\,m_{-}(u^{\prime})-r^{a}u^{p}),\quad
u>0,\end{array}$ (2.5)

where $M_{\pm}$ and $m_{\pm}$ are the Lipschitz functions

$\displaystyle m_{+}(s)=\begin{cases}\lambda s\;\textrm{ if }s\leq 0\\\
\Lambda s\;\textrm{ if }s>0\end{cases}\;\textrm{and}\quad
M_{+}(s)=\begin{cases}s/\lambda\;\textrm{ if }s\leq 0\\\ s/\Lambda\;\textrm{
if }s>0,\end{cases}$ (2.6) $\displaystyle m_{-}(s)=\begin{cases}\Lambda
s\;\textrm{ if }s\leq 0\\\ \lambda s\;\textrm{ if
}s>0\end{cases}\;\textrm{and}\quad M_{-}(s)=\begin{cases}s/\Lambda\;\textrm{
if }s\leq 0\\\ s/\lambda\;\textrm{ if }s>0.\end{cases}$ (2.7)

Equations (2.3) and (2.5) are understood in the maximal interval where $u$ is
positive.

######  Remark 2.1.

A positive function $u$ cannot be at the same time convex and increasing in an
interval, since $u^{\prime\prime}(r)<0$ as long as $u^{\prime}(r)\geq 0$. In
particular, any critical point of a positive solution $u$ is a local strict
maximum point for $u$.

By solution in annulus or exterior domain solution we mean a solution $u$ of
(2.3) or (2.5) defined in an interval $[\mathfrak{a},\rho)$, for
$\mathfrak{a}\in(0,+\infty)$ and $\rho\leq+\infty$, and verifying the
Dirichlet condition $u(\mathfrak{a})=\lim_{r\to\rho^{-}}u(r)=0$. We look at
the initial value problem

$\displaystyle\begin{cases}u^{\prime\prime}\;=\;M_{\pm}\left(-r^{-1}(N-1)\,m_{\pm}(u^{\prime})-r^{a}|u|^{p-1}u\,\right),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \quad
u(\mathfrak{a})=0,\;\;u^{\prime}(\mathfrak{a})=\delta,\qquad\delta>0,\end{cases}$
(2.8)

The equations (2.3), (2.5), together with (2.8) were studied in [5], [6].

For any $p>1$ and for each $\delta>0$, by ODE theory there exists a unique
solution $u=u_{\delta}$ defined in a maximal interval
$(\mathfrak{a},\rho_{\delta})$ where $u$ is positive,
$\mathfrak{a}<\rho_{\delta}\leq+\infty$.

If $\rho_{\delta}=+\infty$ we get a positive radial solution in the exterior
of the ball $B_{\mathfrak{a}}$. In the second case, a positive solution in the
annulus $(\mathfrak{a},\rho_{\delta})$ is produced. Note that equations (2.8)
are not invariant by rescaling.

######  Remark 2.2.

All results obtained for $\delta>0$ will be also true for $\delta<0$. Indeed,
negative shootings for an operator $F$ can be seen as positive shootings for
the operator $G$ defined as $G(x,X)=-F(x,-X)$, which is still elliptic and
satisfies all the properties we considered so far. In particular, the negative
solutions of $\mathcal{M}^{+}$ are positive solutions of $\mathcal{M}^{-}$ in
the same domain, and viceversa.

######  Remark 2.3.

Let $u$ be a positive radial solution of (2.8) with $u=u_{\delta}$, for some
positive constant $\delta$ in $[\mathfrak{a},\rho)$. Then the rescaled pair

${u}_{\gamma}=\gamma\,u(\gamma^{\frac{1}{\alpha}}r)$ , $\gamma>0$, (2.9)

for $\alpha$ as in (2.10), produces a positive solution pair of the same
equation in
$({\mathfrak{a}}{\gamma^{-\frac{1}{\alpha}}},{\rho}{\gamma^{-\frac{1}{\alpha}}})$
with initial values $u_{\gamma}({\mathfrak{a}}\gamma^{-\frac{1}{\alpha}})=0$,
as well as
$u_{\gamma}^{\prime}({\mathfrak{a}}\gamma^{-\frac{1}{\alpha}})=\gamma^{1+\frac{1}{\alpha}}\,\delta$.

Notation. Whenever $p>1$ and $\tilde{N}_{+}>2$, we set

$\displaystyle\textstyle{p^{p,a}_{\pm}=\frac{\tilde{N}_{\pm}+2a+2}{\tilde{N}_{\pm}-2},\quad
p^{s,a}_{\pm}=\frac{\tilde{N}_{\pm}+a}{\tilde{N}_{\pm}-2},\quad
p_{\Delta}^{a}=\frac{N+2+2a}{N-2},\quad\alpha=\frac{2+a}{p-1}.}$ (2.10)

##  3 The dynamical system and exterior domain solutions

Let $u,v$ be a positive solution pair of (2.3) or (2.5). Thus we can define
the new functions

$\displaystyle x(t)=-\frac{ru^{\prime}}{u},\qquad z(t)={r^{2+a}\,u^{p-1}},$
(3.1)

for $t=\mathrm{ln}(r)$, whenever $r>0$ is such that $u>0$. The phase space is
contained in $\mathbb{R}^{2}$. Throughout the text, we denote the first
quadrant as

$1Q=\\{\,(x,z)\in\mathbb{R}^{2}:\;x,z>0\,\\}$.

Since we are studying positive solutions, the points $(x(t),z(t))$ belong to
$1Q$ when $u^{\prime}<0$. Apart from $1Q$, we set

$2Q=\\{\,(x,z)\in\mathbb{R}^{2}:\;x<0,\;\;z>0\,\\}$,

that is, $2Q$ is the region in $\mathbb{R}^{2}$ such that the corresponding
$u$ satisfies $u^{\prime}>0$. In particular, $u^{\prime\prime}<0$ in $2Q$.

As a consequence of this monotonicity, the problems (2.3) and (2.5) become in
$1Q$ as:

$\displaystyle\textrm{for $\mathcal{M}^{+}$ in $1Q$}:\quad
u^{\prime\prime}\;=\;M_{+}(-\lambda r^{-1}(N-1)\,u^{\prime}-r^{a}u^{p}),\quad
u>0;$ (3.2) $\displaystyle\textrm{\;\;for $\mathcal{M}^{-}$ in $1Q$}:\quad
u^{\prime\prime}\;=\;M_{-}(-\Lambda r^{-1}(N-1)\,u^{\prime}-r^{a}v^{p}),\quad
u>0.$ (3.3)

In terms of the functions (3.1), we derive the following autonomous dynamical
system, corresponding to (3.2) for $\mathcal{M}^{+}$, where the dot $\dot{}$
stands for $\frac{\mathrm{d}}{\mathrm{d}t}$,

$\displaystyle\textrm{$\mathcal{M}^{+}$ in $1Q$ :
\quad}\left\\{\begin{array}[]{ccl}\dot{x}&=&\;x\,(x+1)-M_{+}(\lambda(N-1)x-z)\,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \dot{z}&=&\;z\,(x+2+a-px).\end{array}\right.$
(3.6)

Likewise one has for $\mathcal{M}^{-}$, associated to (3.3),

$\displaystyle\textrm{$\mathcal{M}^{-}$ in $1Q$ :
\quad}\left\\{\begin{array}[]{ccl}\dot{x}&=&\;x\,(x+1)-M_{-}(\Lambda(N-1)x-z)\,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \dot{z}&=&\;z\,(x+2+a-px).\end{array}\right.$
(3.9)

We stress that (3.6) and (3.9) correspond to positive, decreasing solutions of
(2.3) and (2.5).

On the other hand, given a trajectory $\tau=(x,z)$ of (3.6) or (3.9) in $1Q$,
we define

$\displaystyle\textstyle{u(r)=r^{-\alpha}\,z^{\frac{1}{p-1}}},\;\;\textrm{
where }\,r=e^{t},$ (3.10)

and then we deduce

$\displaystyle u^{\prime}(r)\textstyle=-\alpha
r^{-\alpha-1}\,z^{\frac{1}{p-1}}(t)+\frac{r^{-\alpha}}{p-1}\,z^{\frac{1}{p-1}-1}(t)\,\frac{\dot{z}}{r}=\frac{u}{r}\\{-\alpha+\frac{x+2+a-px}{p-1}\\}=-\frac{x(t)u(r)}{r}.$

Since $x\in C^{1}$, then $u\in C^{2}$. Moreover, $u$ satisfies either (2.3) or
(2.5) from the respective equations for $\dot{x},\dot{z}$ in the dynamical
system.

In other words, $(x,z)$ is a solution of equation (3.6) or (3.9) in $1Q$ if
and only if $u$ defined by (3.10) is a positive pair solution of (2.3) or
(2.5) with $u^{\prime}<0.$

An important role in the study of our problem is played by the following line
for $\mathcal{M}^{+}$,

$\displaystyle\ell_{+}=\\{\,(x,z):z=\lambda(N-1)x\,\\}\cap 1Q,$ (3.11)

which corresponds to the vanishing of $u^{\prime\prime}$, see also [9]. It
allows us to define the following regions

$\displaystyle R^{+}_{\lambda}=\\{(x,z)\in 1Q:z>\lambda(N-1)x\\},\quad
R^{-}_{\lambda}=\\{(x,z)\in 1Q:z<\lambda(N-1)x\\},$ (3.12)

which represent the sets where the function $u$ is concave or convex. More
precisely, $R^{+}_{\lambda}$ is the region of strictly concavity of $u$, while
$R^{-}_{\lambda}$ is the region of strictly convexity of $u$.

The respective notations for the operator $\mathcal{M}^{-}$ are

$\displaystyle\ell_{-}=\\{\,(x,z):z=\Lambda(N-1)x\,\\}\cap 1Q,$ (3.13)
$\displaystyle R^{+}_{\Lambda}=\\{(x,z)\in 1Q:z>\Lambda(N-1)x\\},\quad
R^{-}_{\Lambda}=\\{(x,z)\in 1Q:z<\Lambda(N-1)x\\}.$ (3.14)

At this stage it is worth observing that the systems (3.6) and (3.9) are
continuous on $\ell_{+}$ and $\ell_{-}$, respectively. More than that, the
right hand sides are locally Lipschitz functions of $x,z$, so the usual ODE
theory applies. Then one recovers existence, uniqueness, and continuity with
respect to initial data as well as continuity with respect to the parameter
$p$.

Since we are considering positive solutions of (1.4), and $u^{\prime}>0$
implies $u^{\prime\prime}<0$, one finds out the following ODEs:

$\displaystyle\textrm{\;\;for $\mathcal{M}^{+}$ in
$2Q$}:\quad\left\\{\begin{array}[]{l}\lambda u^{\prime\prime}\;=\;-\Lambda
r^{-1}(N-1)u^{\prime}-r^{a}u^{p},\quad u>0;\end{array}\right.$ (3.16)
$\displaystyle\textrm{\;\;for $\mathcal{M}^{-}$ in
$2Q$}:\quad\left\\{\begin{array}[]{l}\Lambda u^{\prime\prime}\;=\;-\lambda
r^{-1}(N-1)u^{\prime}-r^{a}u^{p},\quad u>0;\end{array}\right.$ (3.18)

Now, in terms of the corresponding dynamical system, we get

$\displaystyle\textrm{$\mathcal{M}^{+}$ in
$2Q$\,:\;\;}\left\\{\begin{array}[]{cc}\dot{x}=\;x\,(x-\tilde{N}_{-}+2)\,+\frac{z}{\lambda},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\dot{z}=\;z\,(\,x+2+a-px\,).\end{array}\right.$ (3.21)

On the other hand, for the operator $\mathcal{M}^{-}$ one has

$\displaystyle\textrm{$\mathcal{M}^{-}$ in
$2Q$\,:\;\;}\left\\{\begin{array}[]{cc}\dot{x}=\;x\,(x-\tilde{N}_{+}+2)\,+\frac{z}{\Lambda},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\dot{z}=\;z\,(\,x+2+a-px\,).\end{array}\right.$ (3.24)

######  Remark 3.1.

No stationary points exist in $2Q$. Indeed, since $u^{\prime}>0$ and $u>0$
yield $u^{\prime\prime}<0$ then $\dot{x}>0$ in $2Q$. In other words, we do not
have stationary points outside $1Q$ when we are considering positive solutions
$u$ of (1.4).

One may write the dynamical systems (3.6) and (3.9) in terms of the following
ODE first order autonomous equation

$\displaystyle(\dot{x},\dot{z})=(\,f(x,z),\,g(x,z)\,).$ (3.25)

For instance, in the case of the operator $\mathcal{M}^{+}$, then $f,g$ are
given by

$f(x,z)=\begin{cases}x\,(x-N+2)+\frac{z}{\lambda}\;\textrm{ in
}R^{+}_{\lambda}\\\ x\,(x-\tilde{N}_{+}+2)+\frac{z}{\Lambda}\;\textrm{ in
}R^{-}_{\lambda}\end{cases},\quad g(x,z)=z\,(x+2+a-px)$.

We first recall some standard definitions from the theory of dynamical
systems.

A stationary point $Q$ of (3.25) is a zero of the vector field $(f,g)$. If
$\sigma_{1}$ and $\sigma_{2}$ are the eigenvalues of the Jacobian matrix
$(Df(Q),Dg(Q))$, then $Q$ is hyperbolic if both $\sigma_{1},\sigma_{2}$ have
nonzero real parts. If this is the case, $Q$ is a source if
$\mathrm{Re}(\sigma_{1}),\mathrm{Re}(\sigma_{2})>0$, and a sink if
$\mathrm{Re}(\sigma_{1}),\mathrm{Re}(\sigma_{2})<0$; $Q$ is a saddle point if
$\mathrm{Re}(\sigma_{1})<0<\mathrm{Re}(\sigma_{2})$.

Next we recall a result on the local stable and unstable manifolds near saddle
points of the system (3.25); see [7, theorems 9.29, 9.35]. We will see that
the usual theory for autonomous planar systems applies since each stationary
point $Q$ possesses a neighborhood strictly contained in $R^{-}_{\lambda}$ in
the first quadrant where the function $f$ is $C^{1}$. In turn, $g$ is always a
$C^{1}$ function.

Sometimes we denote $\alpha(\tau)$, i.e. the $\alpha$-limit of the orbit
$\tau$, as the set of limit points of $\tau(t)$ as $t\to-\infty$. Similarly
one defines $\omega(\tau)$ i.e. the $\omega$-limit of $\tau$ at $+\infty$.

We observe that the $x$ axis is invariant by the flow in the sense that $z=0$
implies $\dot{z}=0$. Next, the set where $\dot{x}=0$ for the system (3.6),
with respect to the operator $\mathcal{M}^{+}$, is given by the parabola

$\displaystyle\pi_{1}^{+}=\\{(x,z):\,z=\Lambda(\tilde{N}_{+}-2)x-\Lambda
x^{2}\\}\cap 1Q.$ (3.26)

Note that a respective parabola $\\{\,(x,z):\,z=\lambda(N-2)x-\lambda
x^{2}\,\\}$ where $\dot{x}=0$ on the region $R_{\lambda}^{+}$ does not exist,
since it lies entirely below the concavity line $\ell_{+}$, namely

$z=\lambda(N-2)x-\lambda x^{2}<\lambda(N-2)x<\lambda(N-1)x$ for $x>0$.

Moreover , the parabola $\pi_{1}^{+}$ itself in (3.26) lies below the
concavity line $\ell_{+}$, so belonging to the region $R_{\lambda}^{-}$, that
is,

$z=\Lambda(\tilde{N}_{+}-2)x-\lambda
x^{2}<\Lambda(\tilde{N}_{+}-2)x<\lambda(N-1)x$ whenever $x>0$,

since $\tilde{N}_{+}-2<\tilde{N}_{+}-1=\frac{\lambda}{\Lambda}(N-1)$.
Analogously, for the operator $\mathcal{M}^{-}$, the parabola

$\displaystyle\pi_{1}^{-}=\\{\,(x,z):\,z=\lambda(\tilde{N}_{-}-2)x-\lambda
x^{2}\,\\}\cap 1Q$ (3.27)

represents the set where $\dot{x}=0$, which is contained in the region
$R_{\Lambda}^{-}$. Also, we define the line

$\displaystyle\pi_{2}=\pi_{2}^{\pm}=\\{\,(x,z):\,x=\alpha\,\\}\cap 1Q,$ (3.28)

which is the set where $\dot{z}=0$ and $z>0$ for both operators
$\mathcal{M}^{\pm}$.

######  Lemma 3.2.

The stationary points of the dynamical systems (3.6), (3.9), (3.21), and
(3.24) are:

for $\mathcal{M}^{+}$: $O=(0,0)$, $A_{0}=(\tilde{N}_{+}-2,0)$,
$M_{0}=(x_{0},z_{0})$,

where $x_{0}=\alpha$, and
$z_{0}=\alpha\Lambda(\tilde{N}_{+}-p\alpha+a)=\alpha\Lambda(\tilde{N}_{+}-2-\alpha)$;

for $\mathcal{M}^{-}$: $O=(0,0)$, $A_{0}=(\tilde{N}_{-}-2,0)$,
$M_{0}=(x_{0},z_{0})$,

where $x_{0}=\alpha$ and
$z_{0}=\alpha\lambda(\tilde{N}_{-}-p\alpha+a)=\alpha\lambda(\tilde{N}_{-}-2-\alpha)$.

###### Proof.

The stationary points are given by the intersection of the parabola
$\pi_{1}^{\pm}$ with the lines $\pi_{2}$ and
$\\{(x,z)\in\mathbb{R}^{2}:z=0\\}$. ∎

Next we analyze the directions of the vector field $F$ in (3.25) on the $x,z$
axes, on the concavity lines $\ell_{\pm}$, and on the sets $\pi_{1}^{\pm}$ and
$\pi_{2}$.

######  Proposition 3.3.

The systems (3.6) and (3.9) enjoy the following properties:

  1. (1)

Every trajectory of (3.6) in $1Q$ crosses the line $\ell_{+}$ transversely
except at the point $P=\frac{1+a}{p}(1,\lambda(N-1))$. It passes from
$R^{+}_{\lambda}$ to $R^{-}_{\lambda}$ if $x>\frac{1+a}{p}$, while it moves
from $R^{-}_{\lambda}$ to $R^{+}_{\lambda}$ if $x<\frac{1+a}{p}$. A similar
statement holds for (3.9), via $\ell_{-}$, $P=\frac{1+a}{p}(1,\Lambda(N-1))$,
$R^{+}_{\Lambda}$, $R^{-}_{\Lambda}$;

  2. (2)

The vector field at the point $P$ in item (1) is parallel to the line
$\ell_{+}$ (resp. $\ell_{-}$), and an orbit can only reach such a point from
$R^{-}_{\lambda}$ (resp. $R^{-}_{\Lambda}$);

  3. (3)

The flow induced by (3.25) on the $x$ axis points to the left for
$x\in(0,\tilde{N}_{\pm}-2)$, and to the right when $x>\tilde{N}_{\pm}-2$. On
the $z$ axis it always moves up and to the right;

  4. (4)

The vector field $(f,g)$ on the parabola $\pi_{1}^{\pm}$ is parallel to the
$z$ axis whenever $x\neq\alpha$. It points up if $x<\alpha$, and down if
$x>\alpha$.

  5. (5)

On the line $\pi_{2}$ the vector field $(f,g)$ is parallel to the $x$ axis for
$z\neq z_{0}$, where $z_{0}$ is the $z$-coordinate of $M_{0}$ in Lemma 3.2. It
moves to the left if $z<z_{0}$, and to the right if $z>z_{0}$.

###### Proof.

Let us consider the operator $\mathcal{M}^{+}$ since for $\mathcal{M}^{-}$ it
will be analogous.

(1)–(2) We have $\dot{x}>0$ on $\ell_{+}$ since $\pi_{1}^{+}$ is below the
line $\ell_{+}$. By the inverse function theorem, $t$ is a function of $x$ on
$\ell_{+}$, and so is $z$. In order to detect the transversality, one looks at
$\frac{\mathrm{d}z}{\mathrm{d}x}$ on $\ell_{+}$ and compare it with the slope
of $\ell_{+}$. Note that

$\displaystyle\textstyle\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{\dot{z}}{\dot{x}}=\frac{\lambda(N-1)(p-1)x(\alpha-x)}{x(x+1)}\quad\textrm{on\,\;
$\ell_{+}$}.$ (3.29)

Then $\frac{\mathrm{d}z}{\mathrm{d}x}>\lambda(N-1)$ for $x<\frac{1+a}{p}$,
while $\frac{\mathrm{d}z}{\mathrm{d}x}<\lambda(N-1)$ for $x>\frac{1+a}{p}$.

Now we infer that a change of concavity does not happen at $x=\frac{1+a}{p}$.
Indeed, if $\tau=(x,z)$ is a trajectory and $t_{0}$ is such that
$\tau(t_{0})=P$, then
$\frac{\mathrm{d}z}{\mathrm{d}x}(\frac{1+a}{p})=\lambda(N-1)$ i.e.
$\frac{\mathrm{d}\tilde{z}}{\mathrm{d}x}(\frac{1+a}{p})=0$, where
$\tilde{z}(x)=z(x)-\lambda(N-1)x$. Then $\tilde{z}$ has a maximum point at
$x=\frac{1+a}{p}$, and so $\tau$ stays in $R^{-}_{\lambda}$ in a neighborhood
of the point $P$.

(3) Since the $x$ axis is contained in $R^{-}_{\lambda}$ then
$\dot{x}=x(x-(\tilde{N}_{+}-2))$ which is positive for $x>\tilde{N}_{+}-2$ and
negative when $x<\tilde{N}_{+}-2.$ On the other hand, the $z$ axis is
contained in $R^{+}_{\lambda}$, so $\dot{x}=\frac{z}{\lambda}>0$ and
$\dot{z}=z(2+a)>0$ on $x=0$ for $a>-2$.

(4) $\dot{z}=(p-1)z(\alpha-x)$ on $\pi_{1}^{+}$ is positive for $x<\alpha$,
and negative when $x>\alpha$.

(5) On $\pi_{2}\cap R_{\lambda}^{-}$,
$\dot{x}=\alpha(\alpha-(\tilde{N}_{+}-2))+\frac{z}{\Lambda}$ is positive for
$z>z_{0}$ and negative when $z<z_{0}$, for $z_{0}$ given in Lemma 3.2. On
$\pi_{2}\cap(R_{\lambda}^{+}\cup\ell_{+})$ we have
$\dot{x}$$\geq\alpha(\alpha+1)>0$. ∎

Figure 1: The phase plane $(x,z)$ when $p>p^{s,a}_{+}$.

The next proposition gives us a study on a local analysis of the stationary
points. It follows by the correspondence with the variables $(X,Z)$ in [9],
and Propositions 2.2, 2.7, and 2.8, in addition to the appendix in there.

We recall that $X:=x$ and $Z(t):=-{r^{1+a}u^{p}(r)}/{u^{\prime}(r)}$,
$r=e^{t}$ from [9]. Note that $z=XZ$.

######  Proposition 3.4 ($\mathcal{M}^{\pm}$).

The following properties are verified for the systems (3.6) and (3.9).

  1. 1.

For every $p>1$ the origin $O$ is a saddle point, whose unstable direction is
given by

$z=\Lambda(\tilde{N}_{+}+a)x$ if the operator is $\mathcal{M}^{+}$,
$z=\lambda(\tilde{N}_{-}+a)x$ for $\mathcal{M}^{-}$.

Moreover, there is a unique trajectory coming out from $O$ at $-\infty$ with
slope as above, which we denote by $\Gamma_{p}$ , that is, for all $p>1$,
$\Gamma_{p}$ is such that $\alpha(\Gamma_{p})=O$.

  2. 2.

For $p>p^{s,a}_{\pm}$ the point $A_{0}$ is a saddle point whose linear stable
direction is

$\displaystyle z=-A_{\pm}\,x+A_{\pm}(\tilde{N}_{\pm}-2),$

where $A_{+}=\Lambda[(\tilde{N}_{+}-2)p-(2+a)]$ for $\mathcal{M}^{+}$, while
$A_{-}=\lambda[(\tilde{N}_{-}-2)p-(2+a)]$ for $\mathcal{M}^{-}$. Further,
there exists a unique trajectory arriving at $A_{0}$ at $+\infty$ with slope
as above, which we denote by $\Upsilon_{p}$ i.e. for all $p>p^{s,a}_{\pm}$,
$\Upsilon_{p}$ is such that $\omega(\Upsilon_{p})=A_{0}$.

  3. 3.

At $p=p^{s,a}_{\pm}$ the point $A_{0}$ coincides with $M_{0}$ and belongs to
the $x$ axis, while for $p<p_{\pm}^{s,a}$ the point $A_{0}$ is a source and
$M_{0}$ belongs to the fourth quadrant. Also, $M_{0}\in
1Q\,\Leftrightarrow\,p>p^{s,a}_{\pm}$ in which case: $M_{0}$ is a source if
$p_{\pm}^{s,a}<p<p_{\pm}^{p,a}$; $M_{0}$ is a sink for $p>p_{\pm}^{p,a}$;
$M_{0}$ is a center at $p=p^{p,a}_{\pm}$.

The trajectories $\Gamma_{p}$ and $\Upsilon_{p}$ uniquely determine the global
unstable and stable manifolds of the stationary points $O$ and $A_{0}$,
respectively. They are graphs of functions in a neighborhood of the stationary
points in their respective ranges of $p$. The tangent direction at $O$ is
always above the parabola $\pi_{1}^{\pm}$ with $\dot{z}>0$, while the tangent
direction at $A_{0}$ is above $\pi_{1}^{\pm}$ with $\dot{z}<0$ for all
$p>p^{s,a}_{\pm}$.

Next we translate the results obtained in [9] into the new variables in what
concerns periodic orbits, a priori bounds and blow-ups. We use the one to one
correspondence between the orbits in the system $(X,Z)$ in [9] and our
$(x,z)$.

Since $\dot{x}>0$ on $2Q$ then periodic orbits may only exist in $1Q$. From
this, we automatically recover the following proposition from [9]. Recall that
$p^{p,a}_{-}<p_{\Delta}^{a}<p^{p,a}_{+}$ from (2.10).

######  Proposition 3.5 (Dulac’s criterion).

Let $\lambda<\Lambda$. In the case of the operator $\mathcal{M}^{+}$ there are
no periodic orbits of (3.6) when $1<p\leq p_{\Delta}^{a}$ or $p>p^{p,a}_{+}$.
In the case of $\mathcal{M}^{-}$ no periodic orbits of (3.9) exist if
$1<p<p^{p,a}_{-}$ or $p\geq p_{\Delta}^{a}$. In addition, for
$\mathcal{M}^{+}$ ,

  1. (i)

there are no periodic orbits strictly contained in the region
$R^{+}_{\lambda}\cup\ell_{+}$ $($resp. $R^{+}_{\Lambda}\cup\ell_{-}$ for
$\mathcal{M}^{-})$, for any $p>1$;

  2. (ii)

periodic orbits contained in $R^{-}_{\lambda}\cup\ell_{+}$ $($resp.
$R^{-}_{\Lambda}\cup\ell_{-})$ are admissible only at $p=p^{p,a}_{\pm}$. Also,
no periodic orbits at $p^{p,a}_{\pm}$ can cross the concavity line
$\ell_{\pm}$ twice;

  3. (iii)

other limit cycles $\theta$ are admissible by the dynamical system as far as
they cross $\ell_{\pm}$ twice.

By Poincaré-Bendixson theorem, if a trajectory of (3.6) or (3.9) does not
converge to a stationary point neither to a periodic orbit, either forward or
backward in time, then it necessarily blows up. In the next propositions we
prove that a blow up may only occur in finite time. The admissible blow-ups
for $x$ in forward time are again in correspondence with $X$ in [9]. However,
blow-ups in $Z$ in $1Q$ from [9] do not occur in our system for $z$, since a
blow-up there occurs at a finite time $T$ where $u^{\prime}(T)=0$, $R=e^{T}$,
and for this we have $z(T)=R^{2+a}u^{p-1}(R)\in(0,+\infty)$.

It remains to characterize the types of blow-up for $x$ in $2Q$ backward in
time.

######  Lemma 3.6.

Any trajectory $\tau$ of (3.6)–(3.21) or (3.9)–(3.24) which passes through
$2Q$, with $\tau(t)=(x(t),z(t))$, is such that $x(t)\to-\infty$ and $z(t)\to
0$ as $t\to t_{1}$ for some $t_{1}\in\mathbb{R}$. Also, the vector field in
$2Q$ always points to the right and upwards, with $\dot{x}>0$ and $\dot{z}>0$.

###### Proof.

Let us consider the operator $\mathcal{M}^{+}$. In $2Q$ one uses the systems
(3.21) to write

$\dot{x}=x(x+2-\tilde{N}_{-})+\frac{z}{\lambda}>0$,
$\dot{z}=z(p-1)(\alpha-x)>0$,

since $x<0$ and $z>0$. Moreover, if $x(t_{0})<0$ for some
$t_{0}\in\mathbb{R}$, we write $\frac{\dot{x}}{x(x+2-\tilde{N}_{-})}\geq 1$
and so

$\displaystyle\textstyle{\frac{\mathrm{d}}{\mathrm{d}t}\,\mathrm{ln}\left(\frac{x(t)+2-\tilde{N}_{-}}{x(t)}\right)=\frac{\dot{x}}{x-(\tilde{N}_{-}-2)}-\frac{\dot{x}}{x}=\frac{(\tilde{N}_{-}-2)\dot{x}}{x(x+2-\tilde{N}_{-})}\geq\tilde{N}_{-}-2\;\;\textrm{
for }t\leq t_{0}.}$

By integrating in the interval $[t,t_{0}]$ we get

$\displaystyle\textstyle{c_{0}\,\frac{x(t)}{x(t)-(\tilde{N}_{-}-2)}\geq
e^{(\tilde{N}_{-}-2)(t_{0}-t)}}\;\Rightarrow\;\;\textstyle{x(t)\leq-\frac{\tilde{N}_{-}-2}{c_{0}e^{(\tilde{N}_{-}-2)(t-t_{0})}-1}}\,,\;\textrm{
where }\textstyle{c_{0}=1-\frac{\tilde{N}_{-}-2}{x(t_{0})}>1},$

and in particular $x$ blows up at the finite time
$t_{1}=t_{0}+\frac{\mathrm{ln}(1/c_{0})}{\tilde{N}_{-}-2}<t_{0}$. ∎

Regular or singular positive solutions of (2.3) and (2.5) enjoy the
monotonicity $u^{\prime}<0$ since they belong to $\overline{1Q}.$ Now we
obtain a priori bounds for trajectories of (3.6) or (3.9) defined for all $t$
in intervals of type $(\hat{t},+\infty)$ or $(-\infty,\hat{t})$.

######  Proposition 3.7.

Let $\tau$ be a trajectory of (3.6) or (3.9) in $1Q$, with
$\tau(t)=(x(t),z(t))$ defined for all $t\in(\hat{t},+\infty)$, for some
$\hat{t}\in\mathbb{R}$. Then $x(t)\in(0,\tilde{N}_{\pm}-2)$ for all
$t\geq\hat{t}$. If instead, $\tau$ is defined for all $t\in(-\infty,\hat{t})$,
for some $\hat{t}\in\mathbb{R}$, then

$z(t)<\lambda\alpha(N+a)$ in the case of $\mathcal{M}^{+}$,
$z(t)<\Lambda\alpha(N+a)$ for $\mathcal{M}^{-}$, for all $t\leq\hat{t}$.
(3.30)

In particular, if a global trajectory is defined for all $t\in\mathbb{R}$ in
$1Q$ then it remains inside the box
$(0,\tilde{N}_{+}-2)\times(0,\lambda\alpha(N+a))$ in the case of
$\mathcal{M}^{+}$; it stays in
$(0,\tilde{N}_{-}-2)\times(0,\Lambda\alpha(N+a))$ for $\mathcal{M}^{-}$.

###### Proof.

Since $z>0$, $\dot{x}\geq x(x+2-N)\textrm{ in }R_{\lambda}^{+}$ and
$\dot{x}\geq x(x+2-\tilde{N}_{+})\textrm{ in }R_{\lambda}^{-}$, the bound for
$x$ for a trajectory defined for all forward time is accomplished as in the
proof of [9, Proposition 2.11].

Meanwhile, with respect to the bound for $z$, we first claim that if a
trajectory $\tau$ intersects the line $z=\lambda(N+a)x$ then the trajectory
$\tau$ must cross the $z$ axis. Indeed, in this case $Z$ would attain the
value $Z=\lambda(N+a)$, and so a blow up at a backward time
$t_{0}\in\mathbb{R}$ in $Z$ would occur by the proof of (2.26) in [9,
Proposition 2.11]. Thus, $X(t)\to 0$ as $t\to t_{0}^{+}$, from which
$u^{\prime}(t_{0})=0$ with $u(t_{0})>0$. Thus, $x(t_{0})=0$. So the claim is
true. Next, we observe that a trajectory defined for all forward time attains
a maximum value for $z$ at the line $\pi_{2}$, therefore the a priori bound
(3.30) for $z$ is verified. ∎

Figure 2: The flow at $p=p^{*}_{a+}$. Figure 3: The flow for
$p\in(p^{*}_{a+},p^{p,a}_{+})$. Figure 4: The flow for $p>p^{p,a}_{+}$.

###### Proof of Theorem 1.2.

The existence follows by the existence of trajectories produced by the
dynamical system in $(x,z)$, which in turn comes from the dynamical system
analysis for $(X,Z)$ in [9] properly glued via the flow in $2Q$ originated by
$x,z$. More precisely, by [9, Lemma 4.11], for any $p>p^{*}_{a\pm}$, the orbit
$\Upsilon_{p}$ defined in Proposition 3.4(2) has a blow-up in $Z$ backwards at
finite time $T$, which corresponds to a trajectory in $(x,z)$ which crosses
the vertical $z$ axis at $T$. Then, by Lemma 3.6, $\Upsilon_{p}$ has a blow-up
in $x$ at $t_{0}<T$ such that $x(t)\to-\infty$ as $t\to t_{0}^{+}$. The
trajectory $\Upsilon_{p}$ corresponds to a fast decaying exterior domain
solution of (1.4) in ${\mathbb{R}^{N}}\setminus B_{r_{0}}$, with
$r_{0}=e^{t_{0}}$.

The remaining slow decaying and pseudo-slow decaying exterior domain solutions
come from the trajectories displayed in [9, Figures 5(b) and 6], which are
again glued through the $z$ axis by our dynamical system $(x,z)$, analogously.
∎

Even though solutions in annuli can be identified by the trajectories blowing
up both in backward ($x\to-\infty$) and forward ($x\to+\infty$) times in
Figures 2–4, the existence of an annular solution for an arbitrary annulus
$(\mathfrak{a},\mathfrak{b})$ is not ensured. Recall that the scaling in
Remark 2.3 does not work in this case since it changes both extrema of the
annulus. This will be accomplished in the next section, by using the shooting
method and energy functions.

##  4 Energy analysis and solutions in annuli

In this section we introduce some energy functions and use them to establish
existence of solutions in the annulus.

######  Proposition 4.1.

For each $\delta>0$, and $u_{\delta}$ solution of (2.8), we set

$\displaystyle\textstyle\mathcal{E}_{\sigma}(r)=\frac{1}{2r^{a}}(u^{\prime})^{2}+\frac{1}{\sigma(p+1)}|u|^{p+1}\
\;\;\;\textrm{ for\, }\sigma>0.$ (4.1)

Then the energy function

$\displaystyle\mathcal{E}(r)=\begin{cases}\;\mathcal{E}_{\Lambda}(r)\;\;\textrm{
if \;}uu^{\prime}>0\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\;\mathcal{E}_{\lambda}(r)\;\;\textrm{ if \;}uu^{\prime}<0\end{cases}$

is piecewisely monotone decreasing in $\\{u^{\prime}\neq 0\\}$ whenever
$\tilde{N}_{+}\geq 3/2$.

###### Proof.

To fix the ideas let $\delta>0$ and the operator $\mathcal{M}^{+}$. For
simplicity, we write

$u^{\prime\prime}+\frac{r^{a}}{m_{u^{\prime\prime}}}|u|^{p-1}u=-\frac{m_{u^{\prime}}}{m_{u^{\prime\prime}}}\frac{N-1}{r}\,u^{\prime}$

where $m_{s}$ is the step function defined through $m_{s}s=m_{+}(s)$, for
$s=u^{\prime}(r)$ or $s=u^{\prime\prime}(r)$, whenever $u^{\prime\prime}\neq
0$. Here $m_{+}(s)$ is the Lipschitz function given in (2.6).

Set $\hat{N}-1:=\frac{m_{u^{\prime}}}{m_{u^{\prime\prime}}}(N-1)$ which is
either $N-1$, $\tilde{N}_{+}-1$ or $\tilde{N}_{-}-1$, whenever
$u^{\prime\prime}\neq 0$. We have $\sigma=\Lambda\geq m_{u^{\prime\prime}}$
when $uu^{\prime}>0$; while $\sigma=\lambda\leq m_{u^{\prime\prime}}$ when
$uu^{\prime}<0$. Anyways it yields
$\frac{uu^{\prime}}{\sigma}\leq\frac{uu^{\prime}}{m_{u^{\prime\prime}}}$, then

$\displaystyle\textstyle\mathcal{E}_{\sigma}^{\prime}(r)$
$\displaystyle=-\textstyle\frac{a}{2}r^{-a-1}(u^{\prime})^{2}+r^{-a}u^{\prime}u^{\prime\prime}+\frac{1}{\sigma}\,|u|^{p-1}uu^{\prime}$
$\displaystyle\leq\textstyle-\frac{a}{2}r^{-a-1}(u^{\prime})^{2}+r^{-a}u^{\prime}\,\\{u^{\prime\prime}+\frac{r^{a}}{m_{u^{\prime\prime}}}\,|u|^{p-1}u\\}$
$\displaystyle=\textstyle-r^{-a-1}(u^{\prime})^{2}(\frac{a}{2}+\hat{N}-1)<0$

whenever $u^{\prime\prime}\neq 0$ and $u^{\prime}\neq 0$ and
$2(\hat{N}-1)+a>0$. The latter is ensured for instance when $a>-1$ and
$\tilde{N}_{+}\geq 3/2$. Note that at a point $r_{0}$ where
$u^{\prime\prime}(r_{0})=0$ we have $u^{\prime}(r_{0})$ and $u(r_{0})$ with
opposite signs since $u^{\prime}$ and $u$ cannot be both equal to zero by ODE
existence and uniqueness of the initial value problem for Lipschitz
nonlinearities. ∎

From the dynamical system we obtain a complete characterization of
monotonicity for solutions $u_{\delta}$ of (2.8) as follows. Specially in this
section we keep the notation in [6] for $\tau=\tau_{\delta}$ as a radius (and
not for trajectories as in the rest of the text).

######  Lemma 4.2.

For any $\delta>0$ such that $u_{\delta}$ is a positive solution of (2.8) in
$[\mathfrak{a},\rho]$, with $\rho=\rho_{\delta}\leq+\infty$, there exists a
unique number $\tau=\tau(\delta)$ with $\tau\in(\mathfrak{a},\rho)$, such that

$u^{\prime}(r)>0$ for $r\in[\mathfrak{a},\tau)$ , $u^{\prime}(\tau)=0$ ,
$u^{\prime}(r)<0$ for $r\in(\tau,\rho]\,.$

###### Proof.

Let us observe that a first critical point exists for $u$. To see this we look
at the dynamical system driven by $X,Z$. In this case, the behavior at the
third quadrant $X,Z<0$ is given by $\dot{X}>0$ and $\dot{Z}<0$, with a blow up
at finite time $T$ such that $u^{\prime}(e^{T})=0$ by [9, Remark 3.3], and so
$\tau=e^{T}$. At such time we have $u(\tau)>0$, and so $X(T)=x(T)=0$ and
$z(T)\in(0,+\infty)$. The uniqueness of $\tau$ follows by Remark 2.1. ∎

If $\rho_{\delta}=+\infty$ then $\lim_{r\to\infty}u_{\delta}(r)=0$. This comes
from the a priori bounds in Proposition 3.7. Thus, for any $\delta>0$ either
$\rho_{\delta}=+\infty$ and $\lim_{r\to\infty}u_{\delta}(r)=0$, or there
exists some $\rho_{\delta}<+\infty$ such that $u(\rho_{\delta})=0$. Moreover,
by continuous dependence on the initial data, the function
$\delta\mapsto\rho_{\delta}$ is continuous in a neighborhood of any $\delta>0$
where $\rho_{\delta}<+\infty$ whenever $p>1$.

We shall omit the dependence on the parameter $\delta>0$ whenever it is clear
from the context.

######  Proposition 4.3.

For any pair $\delta>0$, and $u$ of (2.8), the energy functions

$\displaystyle\textstyle E_{\lambda}(r)$
$\displaystyle=\,r^{2(\tilde{N}_{-}-1)+a}\;\mathcal{E}_{\lambda}(r)\;\;\textrm{
in }\;[\mathfrak{a},\tau]\vskip 6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle\textstyle E_{\Lambda}(r)$
$\displaystyle=\,r^{2(\tilde{N}_{-}-1)+a}\;\mathcal{E}_{\Lambda}(r)\;\;\textrm{
in }\;[\tau,\rho]$

are monotone increasing, where $\mathcal{E}_{\sigma}$ is given in (4.1) for
$\sigma\in\\{\lambda,\Lambda\\}$.

###### Proof.

Let us consider the operator $\mathcal{M}^{+}$. We recall that in the interval
$[\mathfrak{a},\tau]$ we have $u^{\prime}\geq 0$, $u^{\prime\prime}\leq 0$,
and so

$u^{\prime\prime}u^{\prime}+\frac{r^{a}}{\lambda}u^{p}u^{\prime}=-\frac{(\tilde{N}_{-}-1)}{r}(u^{\prime})^{2}.$

On the other hand, in $[\tau,\rho]$ we have $u^{\prime}\leq 0$, and

$u^{\prime\prime}u^{\prime}+\frac{r^{a}}{\Lambda}u^{p}u^{\prime}\geq
u^{\prime\prime}u^{\prime}+\frac{r^{a}}{m_{u^{\prime\prime}}}u^{p}u^{\prime}=-\frac{(\hat{N}-1)}{r}(u^{\prime})^{2}\geq-\frac{(\tilde{N}_{-}-1)}{r}(u^{\prime})^{2}$,

where $(m_{u^{\prime\prime}},\hat{N})$ is either $(\lambda,N)$ or
$(\Lambda,\tilde{N}_{+})$.

Set $\sigma=\lambda$ if $r\in[\mathfrak{a},\tau]$ and $\sigma=\Lambda$ if
$r\in[\tau,\rho]$. In any case, for $A=2(\tilde{N}_{-}-1)$ we obtain

$\displaystyle\textstyle E_{\sigma}^{\prime}(r)$ $\displaystyle=\textstyle
Ar^{A-1}\left\\{\frac{1}{2}(u^{\prime})^{2}+\frac{r^{a}}{\sigma(p+1)}u^{p+1}\right\\}+r^{A}\left\\{u^{\prime\prime}u^{\prime}+\frac{r^{a}}{\sigma}u^{p}u^{\prime}+\frac{ar^{a-1}}{\sigma(p+1)}u^{p+1}\right\\}\vskip
3.0pt plus 1.0pt minus 1.0pt$ $\displaystyle\textstyle\geq
r^{A-1}(u^{\prime})^{2}\,\\{\frac{A}{2}-(\tilde{N}_{-}-1)\\}=0$

where the inequality comes from $2(\tilde{N}_{-}-1)+a\geq 0$, which holds for
$a>-1$ and $N\geq 2$. ∎

In the remaining of the section we prove Theorem 1.1. This is reduced to show
that, for any given $+\infty>\mathfrak{b}>\mathfrak{a}>0$, there exists a
parameter $\delta>0$ such that $\rho_{\delta}=\mathfrak{b}$ in addition to
$u(\mathfrak{b})=0$.

We start analyzing the behavior of the solutions $u_{\delta}$ as $\delta$
approaches to $0$ and $+\infty$.

######  Lemma 4.4.

If $\delta\to 0$ then we have $u(\tau_{\delta})\to 0$ and
$\rho_{\delta}\to+\infty\,.$

###### Proof.

By Proposition 4.1 we have
$\mathcal{E}_{\Lambda}(r)\leq\mathcal{E}_{\Lambda}(\mathfrak{a})$ for all
$r\leq\tau$, that is,

$\textstyle\frac{{\mathfrak{a}}^{a}}{p+1}u^{p+1}(r)\leq\frac{\Lambda}{2}\,\delta^{2}\quad\textrm{
for all }\;r\in[\mathfrak{a},\tau],$ (4.2)

since $uu^{\prime}\geq 0$ in $[\mathfrak{a},\tau]$. In particular, at
$r=\tau=\tau_{\delta}$ ,

$\textstyle
u^{p+1}(\tau_{\delta})\leq\frac{\Lambda(p+1)}{2{\mathfrak{a}}^{a}}\,\delta^{2}\to
0\;\;\textrm{ when }\delta\to 0.$

Next we write the equation for $u$ in $[\mathfrak{a},\tau]$ as
$(u^{\prime}r^{\tilde{N}_{-}-1})^{\prime}=-\frac{r^{a}}{\lambda}u^{p}\,r^{\tilde{N}_{-}-1}$,
and so integrating from $\mathfrak{a}$ to $\tau$ produces

$\displaystyle
0=u^{\prime}(\tau)\,\tau^{\tilde{N}_{-}-1}=\delta\,{\mathfrak{a}}^{\tilde{N}_{-}-1}-\frac{1}{\lambda}\int_{\mathfrak{a}}^{\tau}s^{\tilde{N}_{-}-1+a}\,{u^{p}}\,.$
(4.3)

By combining the estimate for $u$ in (4.2) and equality (4.3) we obtain

$\displaystyle\delta=\frac{1}{\lambda{\mathfrak{a}}^{\tilde{N}_{-}-1}}\int_{\mathfrak{a}}^{\tau}s^{\tilde{N}_{-}-1+a}\,u^{p}\,\leq\,{C_{0}}\,\delta^{\frac{2p}{p+1}}\;\tau^{\tilde{N}_{-}+a}$

where $C_{0}$ depends only on $\mathfrak{a},p,a,N,\lambda,\Lambda$, and so

$\displaystyle\tau_{\delta}^{\tilde{N}_{-}+a}\,\geq\,\frac{1}{C_{0}\,\delta^{\frac{p-1}{p+1}}}\to+\infty\;\;\textrm{
as }\delta\to 0\,.$

In particular, $\rho_{\delta}\geq\tau_{\delta}\to+\infty$ as $\delta\to 0$. ∎

######  Lemma 4.5.

If $\delta\to+\infty$ then $\rho_{\delta}\to\mathfrak{a}$ and
$u(\tau_{\delta})\to+\infty$. Moreover, for every $C_{0}>0$ there exists a
positive constant $c_{0}$ depending only on
$C_{0},\mathfrak{a},N,p,\lambda,\Lambda$ such that

$\delta\leq C_{0}$ implies $\rho_{\delta}\geq\mathfrak{a}+c_{0}$.

###### Proof.

We denote
$A_{\mathfrak{r},\mathfrak{s}}=B_{\mathfrak{r}}\setminus\overline{B}_{\mathfrak{s}}$
for any $\mathfrak{r}>\mathfrak{s}$ and fix the operator $\mathcal{M}^{+}$,
for $\mathcal{M}^{-}$ it will be analogous.

Step 1) $u(\tau_{\delta})\to+\infty$ when $\delta\to\infty$.

Assume by contradiction that there exists a sequence $\delta_{k}\to\infty$
with respective solutions $u_{k}=u_{\delta_{k}}$ of (2.8), with
$\tau_{k}=\tau_{\delta_{k}}$ , $\rho_{k}=\rho_{\delta_{k}}$ , and $u_{k}\leq
M$ for all $k$.

Since $E_{\lambda}(r)\geq E_{\lambda}(\mathfrak{a})$ for all
$r\in[\mathfrak{a},\tau_{k}]$ by Proposition 4.3, then

$\displaystyle\textstyle\tau_{k}^{2(\tilde{N}_{-}-1)+a}\;u_{k}^{p+1}(\tau_{k})\geq\frac{\lambda(p+1)}{2}{\mathfrak{a}}^{2(\tilde{N}_{-}-1)}\;\delta_{k}^{2}\,\to+\infty.$
(4.4)

Since $u_{k}\leq M$, then $\tau_{k}\to\infty$ as $k\to\infty$. In particular,
$\tau_{k}\geq\mathfrak{a}+1$ for large $k$. Take $\varepsilon\in(0,1)$ with
$\varepsilon\leq\frac{\mathfrak{a}}{\tilde{N}_{-}-1}$ and
$r\in[\mathfrak{a},\mathfrak{a}+\varepsilon]\subset[\mathfrak{a},\tau_{k}]$.
Then we use Taylor expansion of $u_{k}$ at the point $\mathfrak{a}$ to write

$\displaystyle\textstyle
u_{k}(r)=u_{k}(\mathfrak{a})+u^{\prime}_{k}(\mathfrak{a})(r-\mathfrak{a})+\frac{1}{2}\,u^{\prime\prime}_{k}(c_{k})(r-\mathfrak{a})^{2},\;\;\textrm{
for some $c_{k}\in(\mathfrak{a},r)$.}$ (4.5)

Now we notice that

$\frac{\tilde{N}_{-}-1}{c_{k}}u^{\prime}_{k}(c_{k})\leq\frac{\tilde{N}_{-}-1}{\mathfrak{a}}\delta_{k}$
since $u^{\prime}_{k}(c_{k})\in(0,\delta_{k})$.

Moreover, since $u_{k}^{\prime}$ is decreasing in $(\mathfrak{a},r)$, we have
$u^{\prime}_{k}(c_{k})\leq\delta_{k}$ and so, by the second order PDE in (2.3)
and the fact that $u_{k}$ is increasing in $(\mathfrak{a},r)$, we deduce

$\displaystyle\textstyle
u^{\prime\prime}_{k}(c_{k})=-\frac{\tilde{N}_{-}-1}{c_{k}}u^{\prime}_{k}(c_{k})-\frac{c_{k}^{a}}{\lambda}u_{k}^{p}(c_{k})\geq-\frac{\tilde{N}_{-}-1}{\mathfrak{a}}\delta_{k}-\frac{(\mathfrak{a}+1)^{a}}{\lambda}u_{k}^{p}(r).$

Putting this estimate into (4.5) one finds

$\displaystyle\textstyle
u_{k}(r)\geq\delta_{k}(r-\mathfrak{a})-\frac{\tilde{N}_{-}-1}{2\mathfrak{a}}\delta_{k}(r-\mathfrak{a})^{2}-\frac{(\mathfrak{a}+1)^{a}}{2\lambda}u_{k}^{p}(r)\,(r-\mathfrak{a})^{2}.$

Finally, by evaluating it at $r=\mathfrak{a}+\varepsilon$ it yields

$\displaystyle\textstyle
u_{k}(\mathfrak{a}+\varepsilon)+\frac{(\mathfrak{a}+1)^{a}\varepsilon^{2}}{2\lambda}\,u_{k}^{p}(\mathfrak{a}+\varepsilon)\geq\,\delta_{k}\varepsilon\left\\{1-\frac{\tilde{N}_{-}-1}{2\mathfrak{a}}\varepsilon\right\\}\geq\frac{1}{2}\delta_{k}\varepsilon\;\;\textrm{
for sufficiently large }k,$

since $\frac{\tilde{N}_{-}-1}{2\mathfrak{a}}\varepsilon\leq\frac{1}{2}$. But
this is impossible since $\delta_{k}\to+\infty$ and $u_{k}$ is bounded. This
shows Step 1.

Step 2) $\rho_{\delta}\to\mathfrak{a}$ as $\delta\to+\infty$.

We first show that $\tau_{\delta}\to\mathfrak{a}$ as $\delta\to+\infty$. This
will be a consequence of Step 1 and the estimate

$\displaystyle\textstyle
u_{\delta}^{\frac{p-1}{2}}(\tau_{\delta})\leq{C_{0}}\,(\tau_{\delta}-\mathfrak{a})^{-1},$
(4.6)

where $C_{0}$ depends on $\mathfrak{a},a,p,\lambda$. In order to prove (4.6),
we write for $r\in[\mathfrak{a},\tau]$ where $\tau=\tau_{\delta}$,

$-\frac{1}{2}(\,u^{\prime}(r)^{2})^{\prime}\,\geq-u^{\prime\prime}u^{\prime}-\frac{\tilde{N}_{-}-1}{r}(u^{\prime})^{2}\geq\frac{r^{a}}{\lambda}u^{p}u^{\prime}\geq\frac{{\mathfrak{a}}^{a}}{\lambda}u^{p}u^{\prime}$,

and by integrating it in $[r,\tau]$, for $r\in[\mathfrak{a},\tau)$, one gets

$u^{\prime}(r)\,\geq\,C\,\\{u^{p+1}(\tau)-u^{p+1}(r)\\}^{1/2}$,

where $C$ depends on $\mathfrak{a},a,p,\lambda$. Another integration in
$[\mathfrak{a},\tau]$ yields

$\displaystyle\textstyle\int_{\mathfrak{a}}^{\tau}\frac{u^{\prime}\,\mathrm{d}r}{\sqrt{u^{p+1}(\tau)-u^{p+1}(r)}}\,\geq\,C\int_{\mathfrak{a}}^{\tau}\mathrm{d}r\,=\,C(\tau-\mathfrak{a}).$

By using $s=u(r)$ and $u^{\prime}\mathrm{d}r=\mathrm{d}s$ we get

$\displaystyle\textstyle
C(\tau-\mathfrak{a})\leq\int_{0}^{u(\tau)}\frac{\mathrm{d}s}{\sqrt{u^{p+1}(\tau)-s^{p+1}}}=\frac{1}{u^{\frac{p+1}{2}}(\tau)}\;\int_{0}^{1}\frac{u(\tau)\,\mathrm{d}\sigma}{\sqrt{1-\sigma^{p+1}}}=\frac{1}{u^{\frac{p-1}{2}}(\tau)}\int_{0}^{1}\frac{\mathrm{d}\sigma}{\sqrt{1-\sigma^{p+1}}}$

by taking $\sigma=\frac{s}{u(\tau)}$ and
$\mathrm{d}\sigma=\frac{\mathrm{d}s}{u(\tau)}$, from which we deduce (4.6).

Now it is enough to prove that

$\displaystyle\lim_{\delta\to\infty}\frac{\rho_{\delta}}{\tau_{\delta}}=1$.

If not, then there exists $\epsilon>0$ and a sequence $\delta_{k}\to\infty$,
with $\rho_{k}=\rho_{\delta_{k}}\leq+\infty$ and $\tau_{k}=\tau_{\delta_{k}}$
such that $\rho_{k}>(1+\epsilon)\tau_{k}$ for the solutions
$u_{k}=u_{\delta_{k}}$ of (2.8). In particular, $u_{k}$ is positive and
decreasing in the interval $[\tau_{k},(1+\epsilon)\tau_{k}]$.

For $r\in(\tau_{k},(1+\epsilon)\tau_{k}]$ we consider the annulus
$A_{k}=A_{\tau_{k},r}$ where $u_{k}$ solves

$-\mathcal{M}^{\pm}(D^{2}u_{k})\geq\,t_{k}|x|^{a}u_{k}$ in $A_{k}$ , $u_{k}>0$
in $A_{k}$,

where

$t_{k}=\min_{A_{k}}u_{k}^{p-1}=u_{k}^{p-1}(r)$.

Now, by the definition of first eigenvalue
$\lambda_{1}^{+}({D})=\lambda_{1}^{+}(\mathcal{M}^{+},{D})$ for the fully
nonlinear Lane-Emden equation driven by $\mathcal{M}^{+}$ in the domain ${D}$
with respect to the weight $|x|^{a}$ (see [2, 10, 13]), we have

$\displaystyle\textstyle u_{k}^{p-1}(r)\leq\lambda_{1}^{+}(A_{k}),\;\;\textrm{
for all }r\in(\tau_{k}\,,(1+\epsilon)\tau_{k}).$ (4.7)

Note that the following scaling holds

$\lambda_{1}^{+}(A_{\mathfrak{s},\,\mathfrak{s}(1+\epsilon)}\,)=\frac{1}{{\mathfrak{s}}^{2+a}}\,\lambda_{1}^{+}(A_{1,1+\epsilon})$,
for all $\mathfrak{s}>0$. (4.8)

In fact, if $\lambda_{1}^{+},\phi_{1}^{+}$ are a positive eigenvalue and
eigenfunction for the operator $\mathcal{M}^{+}$ with weight $|x|^{a}$ in
$A_{1,1+\epsilon}$ i.e.

$\mathcal{M}^{+}(D^{2}\phi_{1}^{+})+\lambda_{1}^{+}|x|^{a}\phi_{1}^{+}=0$,
$\phi_{1}^{+}>0$ in $A_{1,1+\epsilon}$, $\phi_{1}^{+}=0$ on $\partial
A_{1,1+\epsilon}$

then $\mu_{1}^{+},\psi_{1}^{+}$, where
$\mu_{1}^{+}={\lambda_{1}^{+}}{\mathfrak{s}^{-2-\mathfrak{a}}}$ and
$\psi_{1}^{+}(x)=\phi_{1}^{+}(\frac{x}{\mathfrak{s}})$ are a positive
eigenvalue and eigenfunction in $A_{\mathfrak{s},\mathfrak{s}(1+\epsilon)}$
for $\mathcal{M}^{+}$ with weight $|x|^{a}$.

Then, by combining (4.7) and (4.8) one finds

$\displaystyle\textstyle
u_{k}^{p-1}(r)\leq\,\frac{1}{{\mathfrak{a}}^{2+a}}\,\lambda_{1}^{+}(A_{1,1+\frac{\epsilon}{2}}),\;\;\textrm{
for all }r\in[(1+\frac{\epsilon}{2})\tau_{k},(1+\epsilon)\tau_{k}].$ (4.9)

Using $E_{\Lambda}(\tau_{k})\leq E_{\Lambda}(r)$ for
$r\in[\tau_{k},\rho_{k})$, it comes

$\displaystyle\textstyle
r^{2(\tilde{N}_{-}-1)}\left\\{\frac{r^{a}}{\Lambda(p+1)}u_{k}^{p+1}(r)+\frac{1}{2}(u_{k}^{\prime})^{2}(r)\right\\}$
$\displaystyle\geq\,\textstyle\frac{\tau_{k}^{{2(\tilde{N}_{-}-1)+a}}}{\Lambda(p+1)}\,u_{k}^{p+1}(\tau_{k})$
$\displaystyle\geq\,\textstyle\frac{{\mathfrak{a}}^{2(\tilde{N}_{-}-1)+a}}{\Lambda(p+1)}\,u_{k}^{p+1}(\tau_{k}).$
(4.10)

Since $\tau_{k}\to\mathfrak{a}$ as $k\to+\infty$ then

$r\leq(1+\epsilon)\tau_{k}\leq(1+\epsilon)(\mathfrak{a}+1)$ for large $k$.

Now, by putting the latter and (4.9) into (4) we derive

$(u_{k}^{\prime})^{2}(r)\geq J_{k}$ ,

where

$J_{k}:=C_{\epsilon,p,N,\Lambda}\\{(\mathfrak{a}+1)^{-2(\tilde{N}_{-}-1)}{\mathfrak{a}}^{2(\tilde{N}_{-}-1)+a}\,u_{k}^{p+1}(\tau_{k})-(\mathfrak{a}+1)^{a}{\mathfrak{a}}^{\frac{-(2+a)(p+1)}{p-1}}\\}$

and $J_{k}\to+\infty$ as $k\to\infty$ by Step 1. Hence

$-u_{k}^{\prime}(r)\geq J_{k}^{1/2}\to+\infty$ as $k\to\infty$, for all
$r\in[(1+\frac{\epsilon}{2})\tau_{k}\,,(1+\epsilon)\tau_{k}]$.

Via integration we get

$\displaystyle u_{k}((1+{\textstyle{\frac{\epsilon}{2}}})\tau_{k})$
$\displaystyle\geq\,u_{k}((1+{\textstyle{\frac{\epsilon}{2}}})\tau_{k})-u_{k}((1+{\epsilon})\tau_{k})$
$\displaystyle=\,-\int_{(1+\epsilon/2)\tau_{k}}^{(1+\epsilon)\tau_{k}}u_{k}^{\prime}(r)\mathrm{d}r\geq\textstyle{\frac{\epsilon\tau_{k}}{2}}J_{k}^{1/2}\to+\infty$

which contradicts (4.9).

Step 3) $\delta\leq C_{0}$ implies $\rho_{\delta}\geq\mathfrak{a}+c_{0}$.

Let us prove the contrapositive, that is, if $\rho_{\delta}\to\mathfrak{a}$
then $\delta\to+\infty$.

As in Step 2, if $s=\max_{A_{\mathfrak{a},\rho}}u^{p-1}=u^{p-1}(\tau)$ then
$u$ solves

$-\mathcal{M}^{\pm}(D^{2}u)\leq|x|^{a}u^{p}\leq s\,|x|^{a}u$ in
$A_{\mathfrak{a},\rho}$ , $u=0$ on $\partial A_{\mathfrak{a},\rho}$ .

Now, by the maximum principle for the fully nonlinear equation through the
characterization of the first eigenvalue in [2, 13] (see also [10] for the
weighted version) it follows

$\displaystyle
u^{p-1}(\tau)\geq\lambda_{1}^{+}(\mathcal{M}^{+},A_{\mathfrak{a},\rho}\,).$
(4.11)

In fact, if we had
$s<\lambda_{1}^{+}(\mathcal{M}^{+},A_{\mathfrak{a},\rho}\,)$ then by the
mentioned maximum principle we would obtain $u\leq 0$ in
$A_{\mathfrak{a},\rho}$ which is impossible.

Using the scaling for the eigenvalue in (4.8), (4.2), and (4.11), we derive

$\lambda_{1}^{+}(\mathcal{M}^{+},A_{1,\,{\rho}/{\mathfrak{a}}}\,)={\mathfrak{a}}^{2+a}\,\lambda_{1}^{+}(\mathcal{M}^{+},A_{\mathfrak{a},\rho}\,)\leq{\mathfrak{a}}^{2+a}\,(\,\frac{\Lambda(p+1)}{2\mathfrak{a}^{a}}\,\delta^{2}\,)^{\frac{p-1}{p+1}}.$

Again by the scaling as in Step 2,
$\lambda_{1}^{+}(\mathcal{M}^{+},D)\to+\infty$ as $|D|\to 0$. Then
$\rho_{\delta}\to\mathfrak{a}$ implies $\delta\to+\infty$. As a consequence,
the ratio ${\rho_{\delta}}/{\mathfrak{a}}$ remains bounded away from 1
whenever $\delta$ is bounded from above. ∎

###### Proof of Theorem 1.1..

We fix the annulus $A_{\mathfrak{a},\mathfrak{b}}$ for some
$0<\mathfrak{a}<\mathfrak{b}$. For every $\delta>0$, recall that $u_{\delta}$
is the unique radial solution of the initial value problem (2.8), with a
maximal radius of positivity given by
$\rho_{\delta}\in(\mathfrak{a},+\infty]$. Here, $u(\rho_{\delta})=0$ if
$\rho_{\delta}<+\infty$, while $u(r)\to 0$ as $r\to+\infty$ is
$\rho_{\delta}=+\infty$.

The mapping $\delta\to\rho_{\delta}$ is continuous by ODE continuous
dependence on initial data. In particular, the set

$\displaystyle\mathcal{D}=\mathcal{D}\,(p):=\\{\,\delta\in(0,+\infty):\;\rho_{\delta}<+\infty\,\\}$
(4.12)

is open. By Lemma 4.5, $\mathcal{D}$ is nonempty and contains an open
neighborhood of $+\infty$.

Let $\delta^{*}=\delta^{*}(p)$ be the infimum of the unbounded connected
component of $\mathcal{D}$. Since $\mathcal{D}$ is open, if $\delta^{*}>0$
then $\rho_{\delta^{*}}=+\infty$. If $\delta^{*}=0$ then $\lim_{\delta\to
0}\rho_{\delta}\geq\lim_{\delta\to 0}\tau_{\delta}=+\infty$ by Lemma 4.4.

The function $\delta\mapsto\rho_{\delta}$ is well defined and leads the
interval $(\delta^{*},+\infty)$ onto $(\mathfrak{a},+\infty)$ by the second
part of Lemma 4.5. Then there exists $\delta>0$ such that
$\rho_{\delta}=\mathfrak{b}$. The existence of negative solutions follows by
Remark 2.2. ∎

######  Remark 4.6.

For the non weighted case $a=0$, in [5] it was shown that
$\delta^{*}=\inf\mathcal{D}$ for all $p$, that is
$\mathcal{D}=(\delta^{*},+\infty)$ is an open interval. Moreover, they prove
there that at $\delta^{*}$ only a fast decaying solution is admissible.

Acknowledgments. The authors would like to thank Carmem Maia Gilardoni for
kindly making the figures, and for the precise comments of the anonymous
referee.

L. Maia was supported by FAPDF, CAPES, and CNPq grant 309866/2020-0. G.
Nornberg was supported by FAPESP grant 2018/04000-9, São Paulo Research
Foundation.

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