\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 151, pp. 1--26.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/151\hfil Stability of ground states]
{Stability of ground states for a nonlinear parabolic equation}

\author[L. Bisconti, M. Franca \hfil EJDE-2018/151\hfilneg]
{Luca Bisconti, Matteo Franca}

\address{Luca Bisconti \newline
Dipartimento di Matematica e Informatica ``U. Dini'',
Universit\`a degli Studi di Firenze,
Via S.\ Marta 3, I-50139 Firenze, Italy}
\email{luca.bisconti@unifi.it}

\address{Matteo Franca \newline
Dipartimento di Ingegneria Industriale e Scienze Matematiche,
Universit\`a Politecnica delle Marche,
Via Brecce Bianche, I-60131 Ancona, Italy}
\email{franca@dipmat.univpm.it}

\dedicatory{Communicated by Zhaosheng Feng}

\thanks{Submitted April 5, 2018. Published August 10, 2018.}
\subjclass[2010]{35k58, 35k91, 34e05, 35b08, 35b35}
\keywords{Weak asymptotic stability; supercritical parabolic  equations;
\hfill\break\indent  ground states; asymptotic expansion}

\begin{abstract}
 We consider the Cauchy-problem for the parabolic equation
 \[
 u_t = \Delta u+ f(u,|x|),
 \]
 where $x \in \mathbb R^n$, $n >2$, and $f(u,|x|)$ is either critical or
 supercritical with respect to the Joseph-Lundgren exponent.
 In particular, we improve and generalize some known results concerning
 stability and weak asymptotic stability of positive ground states.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\newcommand{\norm}[1]{{|\kern-0.25ex|\kern-0.25ex|#1|\kern-0.25ex|\kern-0.25ex|}}

\section{Introduction}

In this article we discuss the stability
properties of positive radial solutions of the  equation
\begin{equation}\label{laplace}
 \Delta u+ f(u,|x|)=0 ,
\end{equation}
where $x \in \mathbb{R}^n$, $n >2$, and $f=f(u, |x|)$ is a potential
(which is null for $u=0$) superlinear in $u$, and supercritical in a
sense that will be specified just below. Such solutions correspond to
the positive steady states of the following Cauchy problem
\begin{gather}
 u_t= \Delta u+ f(u,|x|), \label{parab} \\
 u(x,0) = \phi(x), \label{data}
\end{gather}
where $\phi$ is the initial value.

Let $u(x,t;\phi)$ be the solution of \eqref{parab}--\eqref{data}. The
analysis of the long time behavior of $u(x,t;\phi)$ is strongly based
on the separation properties of the radial solutions of
\eqref{laplace}. If $u(x)=U(|x|)$ is a radial solutions of \eqref{laplace},
 we find that $U=U(r)$ solves
\begin{equation}\label{radsta}
 U''+\frac{n-1}{r} U'+ f(U,r)=0 \, ,
\end{equation}
where $``\,'\, "$ denotes the derivative with respect to $r$. In the
whole paper we denote by $U(r,\alpha)$ the unique solution of
\eqref{radsta} with the initial condition $U(0,\alpha)=\alpha>0$.

In the previous decades the Cauchy problem \eqref{parab}--\eqref{data} has
raised a great interest in the mathematical community, starting from the model case
$f(u,|x|)=u^{q-1}$, and it has been analyzed by several authors (see,
e.g., \cite{BF,FWY,GNW1,GNW2, BH, PY1,PY2,PY3,QS, W}).

Since in the whole paper we are interested in positive solutions, there is no
ambiguity in using the notation $u^{q-1}$. It is well known that the
behavior of solutions of \eqref{radsta}, and consequently of
\eqref{parab}, changes drastically as $q$ passes through some critical
values. Here we focus on the case where
$q>2^*:=2n/(n-2)$, so that for any $\alpha>0$ the solution $U(r,\alpha)$
of \eqref{radsta} is positive and bounded for any $r>0$, i.e. it is a
Ground State (GS), and especially in the case $q \ge \sigma^*$, where
\begin{equation}\label{Josep}
 \sigma^*:=  \begin{cases}
\frac{(n-2)^2-4n+8 \sqrt{n-1}}{(n-2)(n-10)} & \text{if }n>10, \\
 +\infty &  \text{if } n\le 10,
 \end{cases}
\end{equation}
so that GSs gain some stability properties (see \cite{W}).
Let us recall that $2^*$ is the Sobolev critical exponent,
while $\sigma^*$ is the Joseph-Lundgren exponent (see \cite{JL}).

When $2^*<q<\sigma^*$ all the GSs intersect each other infinitely
many times, and this fact is used to construct suitable sub- and super-solutions for
\eqref{laplace}. Then, it is possible to show that, in this range of
parameters, GSs determine the threshold between solutions of
\eqref{parab} that blow up in finite time, and solutions that exist
for any time $t$ and fade away.

\begin{theorem}[\cite{W,GNW1}] \label{thmB}
 Assume $f(u,r)=u^{q-1}$, $2^*<q< \sigma^*$.
 \begin{itemize}
 \item[(1)] If there is $\alpha>0$ such that $\phi(x) \gneqq
 U(|x|,\alpha)$, then there is $T(\phi)$ such that $\lim_{t \to
 T(\phi)^-}\|u(t,x; \phi)\|_{\infty}=+\infty$.

 \item[(2)] If there is $\alpha> 0$ such that $\phi(x) \lneqq
 U(|x|,\alpha)$, then $\lim_{t \to +\infty}\|u(t,x;
 \phi)\|_{\infty}=0$.
 \end{itemize}
\end{theorem}

On the other hand, when $q \ge \sigma^*$, GSs are well ordered, and
gain some stability properties as we will see  below.
In fact, already in \cite{W}, the whole argument was generalized to
embrace the so called Henon-equation, i.e.\ when $f(u,r)=r^{\delta}
u^{q-1}$, and $\delta>-2$. In this case there is a shift in the
critical exponents, so we find convenient to introduce the following
parameters (see Section~\ref{steadyasympt} below, see also \cite{BF}
for more details) which will be widely used through this article:
\begin{equation} \label{eq:l0}
l_s:=2 \frac{q+\delta}{2+\delta}\quad
 \text{and}\quad m(l_s):=\frac{2}{l_s-2}= \frac{2+\delta}{q-2}.
\end{equation}
In this context, the previous discussion is still valid, but we have
stability whenever $l_s \ge \sigma^*$, and we lose it for
$2^*<l_s<\sigma^*$ (see \cite{W}). Notice that $l_s$ reduces to $q$
for $\delta=0$. In both cases the GSs, $U$, decay as $U(r) \sim
U(r,+\infty)=P_1 r^{-m(l_s)}$ for $r \to +\infty$, and $U(r,+\infty)$
is the unique singular solution of \eqref{radsta}.

To clarify the notion of stability we will use in the sequel, we
recall the definition of the following weighted norms
 (see, e.g., \cite{GNW1}), i.e.
 \begin{gather*}
 \|\psi\|_\lambda  := \sup_{x\in \mathbb{R}^n} |(1 +
 |x|^\lambda)\psi (x)|, \\
\norm{\psi}_{\lambda}  :=
 \sup_{x\in \mathbb{R}^n} \Big| \frac{(1 +
 |x|^{\lambda})}{[\ln(2+|x|)]}\psi (x)\Big|,
 \end{gather*}
where $\psi$ is continuous, $\lambda \in \mathbb R$, and $k \in \mathbb{N}$.

\begin{definition}\rm
 We say that a GS, $U(|x|)=U(|x|, \alpha)$, is \emph{stable}
 with respect to the norm $\|\cdot \|_\lambda$ if for every
 $\epsilon>0$ there exists $\delta >0$ such that, when $\| \varphi -
 U\|_\lambda< \delta$, we have $\|u(\cdot , t, \varphi) -
 U(|\cdot|)\|_\lambda < \epsilon$ for all $t> 0$.

 Further, we say that $U(|x|)$ is \emph{weakly asymptotically stable} with
 respect to $\|\cdot \|_\lambda$ when $U(|x|)$ is stable with respect
 to $\|\cdot \|_\lambda$, and there exists $\delta >0$ such that
 $\|u(\cdot , t, \varphi) - U(|\cdot|)\|_{\lambda'} \to 0$ as $t\to
 \infty$ for all $\lambda' <
 \lambda$, if $\|\varphi - U\|_\lambda< \delta$.
\end{definition}

Let us consider the  quadratic equation in $\lambda$,
\begin{equation}\label{defLa}
 \lambda^2+ \Big(n-2-\frac{4}{q-2}\Big) \lambda +  2\Big(n-2-\frac{2}{q-2}\Big)=0.
\end{equation}
This problem admits two real and negative solutions, say
$\lambda_2 \le \lambda_1 <0$ if and only if $q \ge \sigma^*$, and they coincide
if and only if $q=\sigma^*$.

Gui et al \cite{GNW1} proved the following theorem.

\begin{theorem}[\cite{W,GNW1}]\label{thmD}
 Assume $f(u,r)=u^{q-1}$, $q \ge \sigma^*$. Let $\lambda_2 \le \lambda_1$ be
 the roots of equation \eqref{defLa}.
 \begin{enumerate}
 \item If $q>\sigma^*$ any GS $U(r,\alpha)$ is stable with respect to
 the norm $\|\cdot \|_{m(q)+ |\lambda_1|}$ and weakly asymptotically
 stable with respect to the norm $\|\cdot \|_{m(q)+ |\lambda_2|}$. 

 \item If $q=\sigma^*$ any GS $U(r,\alpha)$ is stable with respect to
 $\norm{\cdot}_{m(q)+ |\lambda_1|}$ and weakly asymptotically stable
 with respect to the norm $\| \cdot \|_{m(q)+ |\lambda_1|}$.
 \end{enumerate}
\end{theorem}

There are several results aimed at extending the previous
analysis to more general potentials $f=f(u, |x|)$ (see, e.g.,
\cite{Bae1,DLL,Dnew,YZ,BF}). In particular, the instability result
given by Theorem~\ref{thmB}, and the stability result
Theorems~\ref{thmD}, have been generalized also to the following equation
\begin{equation} \label{eq:kmodel-1}
 u_t = \Delta u + k(r)r^\delta
 u^{q-1}, \quad \text{where $\delta>-2$ and $r=|x|$}
\end{equation}
assuming $k(r)$ decreasing, uniformly positive and bounded, in the
cases $l_s > \sigma^*$ (see \cite{DLL}), and $l_s=\sigma^*$ (see
\cite{Dnew}). In particular, these hypotheses imply that the
singular radial solution $U(r,+\infty)$ of \eqref{laplace} behaves
like $r^{-m(l_s)}$ both as $r \to 0$ and as $r \to +\infty$.

In such a case $q$ is replaced by $l_s$ and also the values of
$\lambda_1,\lambda_2$ change accordingly, i.e.\ they solve
\begin{equation}\label{defLaDelta}
 \lambda^2+ \Big(n-2-2\frac{2 + \delta}{q-2}\Big) \lambda+
 \frac{2+\delta}{q-2} \Big(n-2-\frac{2+\delta}{q-2}\Big)=0.
\end{equation}

In \cite{BF} we proposed a unifying approach which allows to extend
Theorem \ref{thmB} to a more general class of nonlinearities $f$,
including \eqref{eq:kmodel-1}, but also more involved dependence on
$u$.

The goal of this paper is to continue the analysis of \cite{BF},
extending the stability results proved in Theorem \ref{thmD} to a class of
potentials $f=f(u, |x|)$ larger than the one considered therein.
This purpose is achieved with an approach
obtained through the combination of the main ideas in \cite{W,GNW1,
 DLL}, techniques borrowed from the theory of non-autonomous
dynamical systems (see \cite{JPY,BF}), along with the use of some new
arguments.

As far as \eqref{eq:kmodel-1} is concerned we are able to drop the
assumption of boundedness on $k$, replacing it by the following:
\begin{equation} \label{asintotico}
 k(r) \sim r^{-\eta}, \quad \text{as $r\to 0$  with  } 0\leq \eta < 2 + \delta.
\end{equation}
Then, we can allow two different behaviors for singular and slow decay
solutions (see \cite{BF}), namely:
$U(r) \sim r^{-m(l_s)}$ as
$r \to +\infty$ and $U(r) \sim r^{-m(l_u)}$ as $r \to 0$, where
\begin{equation}\label{esempioM}
 l_u= 2\frac{q+\delta-\eta}{2+\delta-\eta}\quad \text{and}\quad
 m(l_u)= \frac{2+\delta-\eta}{q-2}.
\end{equation}
We prove the following result. \newpage

\begin{theorem} \label{main-1}
Let $f(u,r)$ be as in \eqref{eq:kmodel-1},
 where $k(r)\in C^1$ satisfies \eqref{asintotico}, is decreasing, and
 $\lim_{r\to +\infty} k(r)>0$. Then
 \begin{enumerate}
 \item If $l_s>\sigma^*$ any GS $U(r,\alpha)$ is stable with respect
 to the norm $\|\cdot \|_{m(l_s)+ |\lambda_1|}$ and weakly
 asymptotically stable with respect to the norm $\|\cdot
 \|_{m(l_s)+ |\lambda_2|}$.

 \item If $l_s=\sigma^*$ any GS $U(r,\alpha)$ is stable with respect
 to $\norm{\cdot }_{m(l_s)+ |\lambda_1|}$ and weakly asymptotically
 stable with respect to the norm $\|\cdot \|_{m(l_s)+ |\lambda_1|}$.
 \end{enumerate}
\end{theorem}

Our approach is flexible enough to consider $f=f(u, |x|)$ as a finite sum
of powers in $u$, i.e.
\begin{equation}
  f(u, |x|) = k_1(|x|)r^{\delta_1}|u|^{q_1-1} +
 k_2(|x|)r^{\delta_2}|u|^{q_2-1},
 \label{eq:potential-1}
\end{equation}
where $q_1<q_2$, $k_i=k_i(|x|)$, $i= 1,2$, are supposed to be $C^1$
(see Theorems \ref{stabile} and \ref{asint.stabile} below).

Equation~\eqref{eq:potential-1} has been already considered by Yang
and Zhang  \cite{YZ}, but just in the particular situation of
$k_1(r)=k_2(r) \equiv 1$. We emphasize that, even if it is not
explicitly stated, in \cite{YZ} it is required that
\begin{equation}\label{condizione}
 \frac{2+\delta_2}{q_2-2}(q_2-q_1)+ \delta_1 < 0,
\end{equation}
which excludes the relevant case $\delta_1=\delta_2=0$. With these
assumptions, Yang and Zhang were able to prove
Theorem~\ref{thmD}-(1), replacing $q$ by $l_s= 2
\frac{q_2+\delta_2}{2+\delta_2}$, and changing the values of $m(l_s)$
and of $\lambda_i$ accordingly. We stress that condition
\eqref{condizione} is in fact needed to use the approach of
\cite{YZ}, and also for the schema proposed in this paper;
See the discussion after  Lemma~\ref{ord1} for
more details on this point. However we believe that \eqref{condizione}
is just a technical
requirement and that it might be removed with the approach
used by Bae and Naito in \cite{BaeN}.

As a consequence of our main results we are able to generalize the
results in \cite{YZ} and to prove Theorem~\ref{thmD}, allowing $k_i$
to depend on $r$, and even to be unbounded, i.e.
\begin{equation} \label{asintotico1} k_1(r)\sim r^{-\eta_1}\,\,
 \text{ and }\,\, k_2(r)\sim r^{-\eta_2}, \,\, \text{ as }\,\,
 r\to 0,
\end{equation}
with $0\leq \eta_i < 2+ \delta_i$, $i=1, 2$. However we still need to
require \eqref{condizione}.

\begin{theorem} \label{main-2} Let $f(u,r)$ be as in
 \eqref{eq:potential-1}, and assume \eqref{condizione}, and
 \eqref{asintotico1}. Suppose that both $k_1(r)r^{
 \frac{2+\delta_2}{q_2-2}(q_2-q_1)+\delta_1}$ and $k_2(r)$ are
 decreasing, $k_1(r)$ is positive and $k_2(r)$ is uniformly positive.
 Then, setting $l_s = 2 \frac{q_2 + \delta_2}{2 + \delta_2}$, we obtain
 the same conclusions as in Theorem~\ref{main-1}.
\end{theorem}

Notice that we can deal with non-monotone functions $k_1(r)$. Under,
these assumptions we are able to prove Theorem \ref{thmD}--(2) which
is new even in the case $k_1(r)=k_2(r) \equiv 1$ considered in
\cite{YZ}.

The main ingredients to obtain our results on \eqref{parab} are the
separation and the asymptotic properties of GSs. The separation
properties are a result of independent interest, and generalize the
ones obtained in \cite[Theorems 1,2]{DLL2}, \cite[Theorem 2]{YZ2}. As
a consequence we also get Proposition \ref{ord3-}, which gives an
insight on the behavior of the singular solution of \eqref{radsta},
which seems to play a key role in determining the threshold between
blowing up and fading solutions (see the Introduction in \cite{W}).

To prove weak asymptotic stability, we need a suitable asymptotic
expansion for GSs, which refines and generalizes the ones of
\cite{DLL,YZ} (see Proposition \ref{sintetizzo}, below). In fact in
\cite{DLL,YZ} the highly nontrivial proof relies on an iterative
scheme developed by \cite{W} in a simpler (and still nontrivial)
context. Here, we followed a different idea: in fact we proved
an asymptotic results for nonlinear systems of ODEs, which seems to be
new to the best of our knowledge, and that, in our opinion, is of
intrinsic mathematical interest (even for systems of ODEs). In this
more general framework the statement assumes a more comprehensible
aspect, and the proof is simplified, even if it is still quite
cumbersome. So to keep the technical analytic machinery to the
minimum, we rely on the Appendix of reference \cite{BF-arxiv} for a
detailed proof of this result.

Now, we briefly review some results which have been proved just in the
setting of Theorems \ref{thmB}, \ref{thmD}. First, using some sub-
and super-solutions constructed on the self-similar solutions,
\cite{GNW2,Na} proved that $U(|x|,\alpha)$ is weakly asymptotically
stable in the norm $\| \cdot \|_\lambda$ for any
$m(q)+\lambda_1<\lambda<m(q)+\lambda_2+2$. Further Naito in \cite{Na} showed that
this result is optimal, i.e. in this range asymptotic stability does
not hold. Moreover Gui et al. in \cite{GNW2} proved that GSs are not
even stable if we use too coarse, but surprisingly also too fine
norms, namely for $\lambda<m(q)+\lambda_1$ and for $\lambda \ge n$. Notice that we
have stability for $\lambda=m(q)+\lambda_1$, but still there is a small gap for
$m(q)+\lambda_2+2 < n$. Similarly the null solution is weakly
asymptotically stable if $m(q) \le \lambda < n$ and unstable otherwise,
\cite{GNW2}.

Moreover, in a series of papers (see \cite{FWY2,HoYa, Na}) the authors showed
that the speed of convergence of solutions $u(t,x; \phi)$ depends
linearly on the weight used to measure the distance with respect to
the GS. Namely if $\|\phi(x)-U(|x|,\alpha)\|_\lambda$ is small enough then
$t^{\nu} \|u(t,x;\phi)-U(|x|,\alpha)\|_{\lambda'}$ is bounded for any $t>0$,
where $\nu=\frac{1}{2} \max\{\lambda-\lambda', \lambda-m(q)-\lambda_1 \}$, whenever
$m(q)+\lambda_1<\lambda<m(q)+\lambda_2+2$ and $0<\lambda'<\lambda$.

If either the assumptions of
Theorem~\ref{thmB} or of Theorem~\ref{thmD} are satisfied, following
\cite{BF} we can construct a family of sub-solutions $\phi$ for
\eqref{radsta} with arbitrarily small $L^{\infty}$-norm and decaying
like $r^{2-n}$ for $r$ large, and such that the solution $u(t,x,\phi)$
blows up in finite time. This type of behavior contradicts the idea
that the decay of the singular solution, i.e.\ $r^{-m(q)}$, is the
critical one to determine the threshold between fading and blowing up
solutions: The situation is indeed more intricate. This result in fact
extends to more general non-linearities $f=f(u, |x|)$ (see \cite{BF}).

To conclude, we recall that when the non-linearity $f(u,r)$
becomes unbounded as $r \to 0$, in general it is not possible to find
classical solutions of \eqref{parab}--\eqref{data}. However it is
still possible to obtain mild solutions assuming that $f(u,r)r^{\ell}$ is
bounded for a certain $0<\ell<2$, and in fact the solutions $u$ are classical for $x
\ne 0$ and $t>0$, and they are $C^{\alpha,\alpha/2}$ also for $x=0$
and $t=0$ for any $\alpha \in (0, 2-\ell)$. For an exhaustive exposition
about such a topic we refer to \cite{W} (see also \cite{BF}).

 This article is organized as follows: In
Section~\ref{steadyasympt} we collect all the preliminary results
concerning the solutions of \eqref{radsta}. We prove ordering
properties and asymptotic estimates for positive solutions of such a
problem. Section~\ref{sec:stability} is devoted to the proof of the
main results of the paper (from which Theorems~\ref{main-1} and
\ref{main-2} follow directly).

\section{Ordering results and asymptotic estimates for the
stationary problem}\label{steadyasympt}

In this section we give some preliminary results which are crucial for our analysis.
These results are obtained by applying the Fowler transformation to
\eqref{radsta}. To this end we introduce the following quantities that
will appear frequently in the whole paper, i.e.
\begin{equation}\label{constants}
 m(l)=\frac{2}{l-2} \,,\quad  A(l)=n-2-2m(l) \,,\quad  B(l)=m(l)[n-2-m(l)],
\end{equation}
where $l>2$ is a parameter (which is related to $l_s$ and $l_u$, in
\eqref{eq:l0} and in \eqref{esempioM}, respectively) whose role will
be explained few lines below. Set
\begin{equation}\label{transf1}
 \begin{gathered}
 r=\mathrm{e}^s\,, \quad y_1(s,l)=U(\mathrm{e}^s)\mathrm{e}^{m(l) s}\,,\quad
 y_2(s,l)=\dot{y}_1(s,l), \\
 g(y_1,s;l)=f(y_1 \mathrm{e}^{-m(l) s},  \mathrm{e}^s) \mathrm{e}^{(m(l)+2)s}\,.
 \end{gathered}
\end{equation}
In what follows with ``$\,\dot{ }\, $'' we will denote the
differentiation with respect to $s$ (recall that ``$\,'\,$" indicates
differentiation with respect to $r$). Using these transformations we
pass from \eqref{radsta} to the  system
\begin{equation} \label{si.na}
\begin{pmatrix} \dot{y}_1 \\
 \dot{y}_2 \end{pmatrix}
 =  \begin{pmatrix} 0 & 1 \\
 B(l) & -A(l)
 \end{pmatrix}
\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}
 - \begin{pmatrix} 0 \\ g(y_1,s;l)\end{pmatrix}.
\end{equation}
Here and in the sequel, we write
\begin{equation} \label{traj-nuovo}
\mathbf{y}(s,\tau;
\mathbf{Q};\bar{l})=\big(y_1(s,\tau; \mathbf{Q};\bar{l}), y_2(s,\tau;
\mathbf{Q};\bar{l}) \big)
\end{equation}
to denote a trajectory of \eqref{si.na}, where $l=\bar{l}$, evaluated
at $s$ and starting from $\mathbf{Q} \in \mathbb R^2$ for $s=\tau$.

Assume first $f(u,r)= r^{\delta} u^{q-1}$ and set $ l
= 2\frac{q+\delta}{2+\delta}$, so that
\eqref{si.na} reduces to the  autonomous system
\begin{equation} \label{si.a}
 \begin{pmatrix} \dot{y}_1 \\
 \dot{y}_2 \end{pmatrix}
=  \begin{pmatrix} 0 & 1 \\
 B(l) & -A(l)
 \end{pmatrix}
 \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}
-\begin{pmatrix} 0 \\ (y_1)^{q-1} \end{pmatrix}.
\end{equation}
In this case we passed from a singular non-autonomous ODE to an
autonomous system from which the singularity has been removed. Also
note that when $\delta=0$ we can simply take $l=q$. The sign of the
constants $A(l)$, $B(l)$ defined in \eqref{constants} determines
 whether the system is sub- or supercritical, if there are slow
decay solutions ($B(l) \ge 0$) or if they do not exist ($B(l) < 0$).

\begin{remark} \label{slow-fast} \rm
Under the previous assumptions,  as $r \to 0$, positive solutions $U(r)$
of \eqref{radsta} have  two possible behaviors: \emph{Regular},
 i.e.\ $\lim_{r\to 0} U(r) = \alpha>0$, or \emph{Singular}, i.e.\ $\lim_{r\to 0} U(r) =
 +\infty$.

 Similarly, as $r \to +\infty$, we either have $\lim_{r\to +\infty} U(r)r^{n-2}=
 \beta>0$ and we say that $U(r)$ has \emph{fast decay}, or $\lim_{r\to +\infty}
 U(r)r^{n-2}= +\infty$ and we say that $U(r)$ has \emph{slow decay}.

 In fact, the behavior of singular and slow decay solutions can be
 specified better, see Proposition \ref{asympt} below), and
 Proposition \ref{sintetizzo}.
\end{remark}

In this article we restrict the whole discussion to the case $l>2^*$.
Therefore $A(l)>0$ and $B(l)>0$. System \eqref{si.a} admits three
critical points for $l>2^*$: The origin $O=(0,0)$, $\mathbf{P}=(P_1,0)$
and $-\mathbf{P}$, where $P_1= [B(l)]^{1/(q-2)}>0$. The origin is a
saddle point and admits a one-dimensional $C^1$ stable manifold
$\overline{M}^s$ and a one-dimensional $C^1$ unstable manifold
$\overline{M}^u$, see Figure \ref{livelli}. The origin splits
$\overline{M}^u$ in two relatively open components: We denote by $M^u$
the component which leaves the origin and enters into the semi-plane
$y_1\ge 0$. Since we are just interested in positive solutions, with a
slight abuse of notation, we will refer to $M^u$ as the unstable
manifold.

\begin{remark}\label{criticalP} \rm
 The critical point $\mathbf{P}$ of \eqref{si.a} is a stable focus if
 $2^*<l<\sigma^*$ and a stable node if $l \ge \sigma^*$.
\end{remark}

As a consequence of some asymptotic estimates we deduce the following
useful fact (see, e.g. \cite{Fjdde, Fcamq}).

\begin{remark}\label{corrispondenze2} \rm
 Let $u(r)$ be a solution of \eqref{radsta} and let $\mathbf{Y}(s;l)$ be
 the corresponding trajectory for the system \eqref{si.a}, with
 $l>2^*$. Then $u(r)$ is regular (respectively has fast decay) if and
 only if $\mathbf{Y}(s;l)$ converges to the origin as $s \to -\infty$
 (resp. as $s \to +\infty$), $u(r)$ is singular (respectively has
 slow decay) if and only if $\mathbf{Y}(s;l)$ converges to $\mathbf{P}$ as $s
 \to -\infty$ (resp. as $s \to +\infty$).
\end{remark}

Using the Pohozaev identity introduced in \cite{Po}, and adapted to
this context in \cite{FArch}, we can draw a picture of the phase
portrait of \eqref{si.a} (see Figure \ref{livelli} below) and deduce
information on positive solutions of \eqref{radsta}. Then it is not
hard to classify positive solutions: In the supercritical case
($l>2^*$) all the regular solutions are GSs with slow decay, and there is
a unique Singular Ground State (SGS) with slow decay.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig1}
\end{center}
\caption{Sketches of the phase portrait of \eqref{si.na}, for $q>2$  fixed.}
\label{livelli}
\end{figure}

We stress that all the previous arguments concerning the autonomous
Equation \eqref{si.na} still hold  for any autonomous super-linear
system \eqref{si.na}. More precisely, whenever $g(y_1,s;l)\equiv
g(y_1;l)$ and $g(y_1;l)$ has the following property, denoted by
(A0) (see \cite{FAnn} for a proof in the general $p$-Laplace
context, see also \cite{BF}).
\begin{itemize}
\item[(A0)] There is $l>2^*$ such that $g(0;l)=0=\partial_{y_1}
 g(0,l)$ and $\partial_{y_1} g(y_1,l)$ is a positive strictly
 increasing function for $y_1>0$ and $\lim_{y_1 \to +\infty}
 \partial_{y_1} g(y_1,l)=+\infty$.
\end{itemize}
When (A0) holds, we denote by $P_1$ the unique positive
solution in $y_1$ of $ g(y_1; l)=B(l) y_1$. Hence $(P_1,0)$ is again a
critical point for \eqref{si.a}.
Further, we let $\sigma_*<\sigma^*$ be the real solutions of the
equation in $l$ given by
\begin{equation}\label{defsigma}
 A(l)^2-4\big[\partial_y g(P_1,l)-B(l)\big]=0 ,
\end{equation}
which reduces to $A(l)^2-4(q-2)B(l)=0$ for $g(y_1)=(y_1)^{q-1}$. We
emphasize that when $f(u,r)=u^{q-1}$ the value of $\sigma^*$ coincides
with the one given in \eqref{Josep}. Notice that Remarks
\ref{criticalP}, \ref{corrispondenze2} continue to hold in this
slightly more general context (see \cite{Fcamq, Fjdde}).


\subsection{Main assumptions and preliminaries} \label{ssec:main-assumptions}

We collect  below the assumptions used in our main results.
\begin{itemize}
\item[(A1)] There is $l_u \ge \sigma^*$ such that for any
 $y_1>0$ the function $ g(y_1,s;l_u)$ converges to a $s$-independent
 $C^1$ function $g(y_1,-\infty; l_u)\not\equiv 0$ as $s \to -\infty$,
 uniformly on compact intervals. The function $g(y_1,s; l_u)$
 satisfies (A0) for any $s \in \mathbb R$. Further, there is
 $\varpi>0$ such that $\lim_{s \to -\infty} \mathrm{e}^{-\varpi s}\partial_s
 g(y_1,s; l_u)=0$. 

\item[(A2)] There is $l_s \ge \sigma^*$ such that for any
 $y_1>0$ the function $ g(y_1,s; l_s)$ converges to a $s$-independent
 $C^1$ function $g(y_1,+\infty, l_s)\not\equiv 0$ as $s \to +\infty$,
 uniformly on compact intervals. The function $g(y_1, s; l_s)$
 satisfies (A0) for any $s \in \mathbb R$. Further, there is
 $\varpi>0$ such that $\lim_{s \to
 +\infty} \mathrm{e}^{+\varpi s}\partial_s g(y_1,s;l_s)=0$. 

\item[(A3)] Condition (A2) holds and $g(y_1,s;l_s)$ and
 $\partial_{y_1} g(y_1,s;l_s)$ are
 decreasing in $s$ for any $y_1>0$. 

\item[(A4)] Condition (A2) is verified with $\varpi=\gamma$
 satisfying
 \[
 g(P_1^+,s;l_s)=g(P_1^+, {+\infty}; l_s)+ c \mathrm{e}^{-\gamma s}
 +o(\mathrm{e}^{-\gamma s})
\]
for a certain $c \ne 0$.

\item[(A5)] Either $f$ is as in \eqref{eq:kmodel-1} or $f$ is as
 in \eqref{eq:potential-1} and satisfies \eqref{condizione}.
\end{itemize}
%
Assumptions (A1), (A2) are used to ensure that the phase
portrait of \eqref{si.na} converges to an autonomous system of the
form \eqref{si.a} (with $l \ge \sigma^*$), respectively as $s \to \pm
\infty$.

 Instead, (A3) is needed to prove the ordering properties of positive
solutions and generalizes the condition required in \cite{DLL}.

Assumption (A4) is used to derive asymptotic estimates on slow decay
solutions of \eqref{radsta}, and it gives back the standard
requirement when $f(u,r)=k(r) u^{q-1}$, i.e. $k(r)=k(\infty)+c
r^{-\gamma}+o(r^{-\gamma})$ (see \cite{DLL}).
Condition (A4) is assumed for definiteness and may be weakened, at the
price of some additional cumbersome technicalities.

Finally, condition (A5) is just a technical requirement we are not able to
avoid, which is in fact implicitly assumed also in \cite{YZ}. It
implies that there is $c>0$ such that
\begin{equation}\label{Kimplica}
 B(l_s)= \frac{g(P_1^+, {+\infty}; l_s)}{P_1^+}= c |P_1^+|^{\overline q-2} =
 \frac{\partial_{y_1}g(P_1^+, {+\infty}; l_s)}{ \overline q-1}
\end{equation}
with $\overline q = q$ in the case of \eqref{eq:kmodel-1}, and
$\overline q= q_2$ for the potential \eqref{eq:potential-1}.

\begin{remark}\label{spiegoluls} \rm
 Observe that (A1) and (A2) are satisfied, e.g., in the
 following cases:
 \begin{itemize}
 \item For equation \eqref{eq:kmodel-1} with $k$ satisfying
 \eqref{asintotico}: $l_s$ and $l_u$ are as in \eqref{eq:l0}
 and \eqref{esempioM}, respectively.

 \item When $f$ is as in \eqref{eq:potential-1} and
 \eqref{asintotico1} holds: In this case $l_s$ is as in Theorem
 \ref{main-2}, i.e. $l_s= \min
 \big\{2\frac{q_i+\delta_i}{2+\delta_i} : i=1,2 \big\}$, while
 $l_u= \max \big\{2\frac{q_i+\delta_i-\eta_i}{2+\delta_i-\eta_i}
 : i=1,2 \big\}$. We also emphasize that, if we consider
 \eqref{eq:potential-1}, then \eqref{condizione} amounts to ask for
 $2\frac{q_2+\delta_2}{2+\delta_2} \le
 2\frac{q_1+\delta_1}{2+\delta_1}$; so (A5) is not satisfied if
 $\delta_i=\eta_i=0$, since we find $l_s=q_1 < q_2=l_u$.
 \end{itemize}
\end{remark}

\begin{lemma}\label{decresce}
 Assume {\rm (A2)} and {\rm (A3)}. Then we have the following condition
 \begin{itemize}
 \item[(A6)] The function $G(y_1,s;2^*):=\int_0^{y_1}g(a,s;
 2^\ast) da$ is decreasing in $s$ for any $y_1 >0$ strictly for
 some $s$.
 \end{itemize}
\end{lemma}

\begin{proof}
 Set $G(z,s,l_s)= \int_0^z g(a,s,l_s) da$, $H(z,s)=G(z,s,l_s)/z$.
 Then
 \[%\label{gcala}
 G(z,s,l_s)=\int_0^z \frac{g(a,s,l_s)}{a}a da
\le \frac{g(z,s,l_s)}{z} \int_{0}^{z}a da= \frac{z g(z,s,l_s)}{2}
 \]
Therefore $zg-G \ge zg-2G \ge 0$. Since $\partial_z H= (zg-G)/z^2$,
 then $H(z,s)$ is increasing in $z$ and decreasing in $s$ for
 (A3). Hence
 \[
 G(y_1,s,2^*)=G(y_1 \mathrm{e}^{-\delta s},s,l_s) \mathrm{e}^{\delta s}=H(y_1
 \mathrm{e}^{-\delta s},s)y_1 \, ,
 \]
 so we conclude that $G(y_1,s,2^*)$ is decreasing in $s$.
\end{proof}

Observe that (A6) means that the system is supercritical with
respect to $2^*$, and this ensures the existence of GSs for
\eqref{radsta} (see e.g. \cite[Proposition 2.12]{BF}). In the sequel,
in some cases, it will be convenient to use (A6) along with (A2), in place of the
combination of (A2) and (A3).
In fact, the first couple of requirements is slightly weaker than the second.

\subsection{Stationary problem: spatial dependent case}
\label{subsec-stationary-prob}

Now we consider \eqref{si.na} in the $s$-dependent case. The
first step is to extend invariant manifold theory to the
non-autonomous setting.

Assume (A1). We introduce the following $3$-dimensional
autonomous system, obtained from \eqref{si.na} by adding the extra
variable $z=\text{e}^{\varpi t}$, i.e.,
\begin{equation}\label{si.naa}
 \begin{pmatrix}
 \dot{y}_{1} \\ \dot{y}_{2} \\ \dot{z}
 \end{pmatrix}
 = \begin{pmatrix} 0 & 1 &0 \\ B(l_u) & -A(l_u) & 0 \\ 0 & 0 & \varpi
 \end{pmatrix}
 \begin{pmatrix}
 y_1 \\ y_2 \\ z
 \end{pmatrix} -
 \begin{pmatrix}
 0 \\ g(y_1,\frac{\ln(z)}{\varpi};l_u)\\ 0
 \end{pmatrix}.
\end{equation}
Similarly if (A2) is satisfied we set $l=l_s$ and
$\zeta(t)=\mathrm{e}^{-\varpi t}$ and we consider
\begin{equation}\label{si.naas}
 \begin{pmatrix}
 \dot{y}_{1} \\ \dot{y}_{2} \\ \dot{\zeta}
 \end{pmatrix} =
 \begin{pmatrix}
 0 & 1 &0 \\ B(l_s) & -A(l_s) & 0 \\ 0 & 0 & \varpi
 \end{pmatrix}
 \begin{pmatrix}
 y_1 \\ y_2 \\ \zeta
 \end{pmatrix}
 - \begin{pmatrix} 0 \\ g(y_1,-\frac{\ln(\zeta)}{\varpi};l_s)\\ 0
 \end{pmatrix}.
\end{equation}
The technical assumptions at the end of (A1), (A2) are
needed in order to ensure that the systems are smooth respectively for
$z=0$ and $\zeta=0$.

We recall that if a trajectory of \eqref{si.na} does not cross the
coordinate axes indefinitely then it is continuable for any $s \in
\mathbb R$ (see e.g.\ \cite[Lemma 3.9]{Fcamq}, \cite{fDmF}). Consider
\eqref{si.naa} {(respectively \eqref{si.naas}) each trajectory
 corresponding to a definitively positive solution $u(r)$ of
 \eqref{radsta} is such that its $\alpha$-limit set is contained in
 the $z=0$ plane (respectively its $\omega$-limit set is contained in
 the $\zeta=0$ plane). Moreover such a plane is invariant and the
 dynamics reduced to $z=0$ (respectively, $\zeta=0$) coincides with
 the one of the autonomous system \eqref{si.na} where
$g(y_1,s;l_u)\equiv g(y_1,-\infty; l_u)$ (respectively,
$g(y_1,s;l_s)\equiv g(y_1,+\infty; l_s)$).

Note that the origin of \eqref{si.naa} admits a $2$-dimensional
 unstable manifold $\mathbf{W^u}(l_u)$ which is transversal to $z=0$,
 and a $1$-dimensional stable manifold $M^s$ contained in $z=0$.

 Following \cite{Fdie} (see also \cite{JPY}), for any $\tau \in \mathbb R$
 we have that
 \[
 W^u(\tau;l_u)=\mathbf{W^u}(l_u) \cap \{ z= \mathrm{e}^{\varpi \tau} \}\quad
 \text{and}\quad  W^u(-\infty;l_u)=\mathbf{W}^u(l_u) \cap \{ z= 0 \}
 \]
 are 1-dimensional immersed manifolds, i.e.\ the graph of $C^1$
 regular curves. Moreover, they inherit the same smoothness as
 \eqref{si.naa} and \eqref{si.naas}, that is: Let $K$ be a segment
 which intersects $W^u(\tau_0;l_u)$ transversally in a point
 $\mathbf{Q}(\tau_0)$ for $\tau_0 \in[-\infty,+\infty)$, then there is a
 neighborhood $I$ of $\tau_0$ such that $W^u(\tau;l_u)$ intersects
 $K$ in a point $\mathbf{Q}(\tau)$ for any $\tau \in I$, and $\mathbf{Q}(\tau)$ is as
 smooth as \eqref{si.naa}.

 Since we need to compare $W^u(\tau;l_u)$ and $W^s(\tau;l_s)$, we
 introduce the following manifolds:
 \begin{equation}\label{cambioL}
 W^u(\tau;l_s):= \big\{\mathbf{R}=\mathbf{Q}
 \exp \big(\big(m(l_s)-m(l_u)\big)\tau \big) \in \mathbb R^2
 : \mathbf{Q} \in W^u(\tau;l_u) \big\}.
 \end{equation}
 Note that $W^u(\tau;l_u)$ and $W^u(\tau;l_s)$ are homothetic, since
 they are obtained from each other simply multiplying by an
 exponential scalar. However, if $l_u >l_s$, $W^u(\tau;l_s)$ becomes
 unbounded as $\tau \to -\infty$.

 To deal with bounded  sets, we also define the following manifold which will
 be useful in
 Section~\ref{sec:stability}, i.e.
 \begin{align}\label{manistella}
 \begin{aligned} W^u(\tau;l_*):= & \left\{ \begin{array}{ll}
 W^u(\tau;l_u) & \text{if } \tau \le 0 \\ W^u(\tau;l_s) &
 \text{if } \tau \ge 0
 \end{array}
 \right. , \quad \xi(\tau) := \left\{ \begin{array}{ll} z(\tau)
 & \text{if } \tau \le 0 \\ 2-\zeta(\tau) & \text{if }
 \tau \ge 0
 \end{array} \right. \\
 \intertext{and} & \mathbf{W}^u(l_*):= \big\{ (\mathbf{Q}, \xi(\tau)) \mid \mathbf{Q} \in
 W^u(\tau;l_*) \big\}.
 \end{aligned} \hspace{-2 cm}
 \end{align}

 The sets $W^u(\tau;l_u)$ may be constructed also using the argument
 of \cite[\S 13]{CoLe}, simply requiring that \eqref{si.na} is $C^1$
 in $\mathbf{y}$ uniformly with respect to $t$ for $t \le \tau$ in a
 fixed neighborhood of the origin. With this second method we see that the
 tangent space to $W^u(\tau;l_u)$ is simply the unstable space of
 the linearization of \eqref{si.na} in the origin, and we obtain the
 following.

\begin{remark}\label{tangente} \rm
 Assume (A1). Then, in the origin $W^u(\tau;l_u)$ is tangent to
 the line $y_2=m(l_u)y_1$, for any $\tau \in \mathbb R$. Since
 $W^u(\tau;l_u)$, $W^u(\tau;l_s)$ and $W^u(\tau;l_*)$ are homothetic,
 they are all tangent to $y_2=m(l_u)y_1$ in the origin.
\end{remark}

As in the $s$-independent case, we see that the \emph{regular
 solutions} correspond to the trajectories in $W^u$
(see \cite{Fdie, Fcamq}). More precisely, from \cite[Lemma 3.5]{Fcamq}, we obtain the
following result.

\begin{lemma}\label{corrispondenze}
 Assume {\rm (A1), (A2)}. Consider the trajectory
 $\mathbf{y}(s,\tau,\mathbf{Q};l_u)$ of \eqref{si.na} with $l=l_u$, the
 corresponding trajectory $\mathbf{y}(t,\tau,\mathbf{R}; l_s)$ of \eqref{si.na}
 with $l=l_s$ and let $u(r)$ be the corresponding solution of
 \eqref{radsta}. Then $\mathbf{R}=\mathbf{Q} \exp [ (m(l_s)-m(l_u)) \tau]$.

 Further $u(r)$ is a regular solution if and only if $\mathbf{Q} \in
 W^u(\tau; l_u)$ or equivalently $\mathbf{R} \in W^u(\tau; l_s)$.
\end{lemma}

Now, we consider singular and slow decay solutions of
\eqref{radsta}. Let $P_1^-$, $P_1^+$ be the unique positive solutions
in $y_1$ respectively of $B(l_u)y_1=g(y_1,-\infty;l_u)$ and of
$B(l_s)y_1=g(y_1,+\infty;l_s)$, and set $\mathbf{P}^{\pm}=(P_1^{\pm},0)$.
Then, it follows that $(\mathbf{P}^-,0)$ and $(\mathbf{P}^+,0)$ are
respectively critical points of \eqref{si.naa} and \eqref{si.naas}.

If $l_u \ge 2^*$, then $(\mathbf{P}^{-},0)$ admits a $1$-dimensional
exponentially unstable manifold, transversal to $z=0$ (the graph of a
trajectory which will be denoted by $\mathbf{y}^*(s,*;l_u)$) for system
\eqref{si.naa}, while if $l_s > 2^*$ then $(\mathbf{P}^{+},0)$ is stable
for \eqref{si.naas}, so it admits a $3$-dimensional stable manifold
(an open set).

From \cite[Proposition 2.12]{BF} we find the following proposition.

\begin{proposition}[\cite{BF}]\label{super}
 Assume {\rm (A1), (A2), (A6)}. Then, all the regular
 solutions $U(r,\alpha)$ of \eqref{radsta} are GSs with slow decay,
 there is a unique singular solution, say $U(r,\infty)$, and it is a
 SGS with slow decay.
\end{proposition}

\begin{proposition}[\cite{BF}] \label{asympt}
 Assume {\rm (A1), (A2)}. Then if $u(r)$ and $v(r)$ are
 respectively a singular and a slow decay solution of \eqref{radsta}
 we have $u(r)r^{m(l_u)} \to P_1^-$ as $r \to 0$ and $u(r)r^{m(l_s)}
 \to P_1^+$ as $r \to +\infty$.
\end{proposition}

\subsection{Separation properties of stationary solutions}

In this subsection we adapt the argument of \cite{DLL} and of \cite{YZ}
to obtain separation properties of \eqref{radsta}. We begin by the
following Lemma which is rephrased from \cite[Theorem 4.1]{YZ2}, which
is a slight adaption of \cite[Lemma 2.11]{DLL}. We emphasize that
condition (A5) is needed to prove estimate \eqref{graph2} below,
and it is in fact implicitly required in \cite[Theorem 4.1]{YZ2}, even
if it is not explicitly stated.

\begin{lemma}\label{ord0}
 Assume {\rm (A1), (A2), (A3), (A5)}. Let
 $\bar{\mathbf{y}}(s)$ be the trajectory of \eqref{si.na} corresponding
 to the GS $U(r,\alpha)$ of \eqref{radsta}. Then, for any $s \in
 \mathbb R$ we have $\bar{y}_2(s)=\dot{\bar{y}}_1(s) \ge 0$,
 $0<\bar{y}_1(s) < P^+_1$ and
 \begin{equation}
 g(\bar{y}_1(s),s;l_s) < B(l_s) \bar{y}_1(s) \label{eq:g}
 \end{equation}
\end{lemma}

\begin{proof}
 Let us recall that all the regular solutions are GSs: This is a
 direct consequence of Proposition~\ref{super} and
 Lemma~\ref{corrispondenze}. Let $\bar{\mathbf{y}}(s;l_u)=
 \bar{\mathbf{y}}(s) \mathrm{e}^{(\alpha_{l_s}-\alpha_{l_u})s}$ be the
 corresponding trajectory of \eqref{si.na} where $l=l_u$, then, by
 standard facts in dynamical system theory, see \cite{CoLe}, we see
 that there are $c_i>0$ such that $\bar{y}_i(s;l_u)
 \mathrm{e}^{-\alpha_{l_u} s} \to c_i$ as $s \to -\infty$ for $i=1,2$. Hence
 $ \bar{y}_i (s) \sim c_i \mathrm{e}^{\alpha_{l_s}s} \to 0$ as $s \to
 -\infty$ for $i=1,2$: So \eqref{eq:g} is satisfied for $s \ll 0$.

 Let us set
 \begin{equation}\label{s0}
 s_0:= \sup \big\{ S \in \mathbb R \mid g(\bar{y}_1(s),s;l_s) <
 B(l_s)\bar{y}_1(s)  \text{ for any $s<S$ } \big\},
 \end{equation}
 so that \eqref{eq:g} holds for $s<s_0$.

 It follows that $\dot{\bar{y}}_2(s)+A(l_s)\bar{y}_2(s)>0$ for
 $s<s_0$, hence $w(s)= \bar{y}_2(s) \mathrm{e}^{A(l_s) s}$ is increasing for
 $s<s_0$. Since $w(s)\to 0$ as $s \to -\infty$ we find that
 $\bar{y}_2(s)>0$, for $s \le s_0$.

 Further, assume by contradiction that there is $\tilde{s}< s_0$ such
 that $\bar{y}_1(\tilde{s})=P^+_1$. Then, from (A3) we have
 \[
 g(\bar{y}_1(\tilde{s}),+\infty;l_s) \le
 g(\bar{y}_1(\tilde{s}),\tilde{s};l_s) <
 B(l_s)\bar{y}_1(\tilde{s})=g(P^+_1,+\infty;l_s).
 \]
 Since $g(\cdot,+\infty;l_s)$ is increasing we obtain
 $\bar{y}_1(\tilde{s})<P_1^+$, and this gives an absurd conclusion. Thus,
 $0<\bar{y}_1(s)<P_1^+$ for $s<s_0$.

 Now, we show that $s_0=+\infty$, so that \eqref{eq:g} holds for any
 $s \in \mathbb R$ and the Lemma is proved. Assume by contradiction that
 $s_0<+\infty$. Consider the curve $\bar{\mathbf{y}}(s) =(\bar{y}_1(s),
 \bar{y}_2(s))$ defined for $s \le s_0$. Since
 $\bar{y}_2(s)=\dot{\bar{y}}_1(s)>0$ for $s \le s_0$, it follows that
 $\bar{\mathbf{y}}(s)$ is a graph on the $y_1$-axis, and we can
 parametrize it by $\bar{y}_1$. Hence, we set
 $Q(\bar{y}_1):=\dot{\bar{y}}_1(\bar{y}_1)$ so that $\bar{\mathbf{y}}(s)$
 for $s \le s_0$ and $\mathbf{\Gamma}:=\mathbf{\Gamma}(y_1)=(y_1,Q(y_1))$ for $y_1
 \in (0,\bar{y}_1(s_0)]$ are reparametrization of the same curve. As a consequence
 we have
 \begin{equation}\label{graph}
 \frac{\partial Q}{\partial \bar{y}_1}=\frac{\partial Q}{\partial s}
 \frac{\partial s}{\partial
 \bar{y}_1}=\frac{\ddot{\bar{y}}_1}{\dot{\bar{y}}_1}=
 -A(l_s)+\frac{B(l_s)\bar{y}_1-g(\bar{y}_1,s;l_s)}{Q(\bar{y}_1)}.
 \end{equation}
 In the phase plane, consider the line $r(\mu)$ passing through
 $\mathbf{R}=(\bar{y}_1(s_0),0)$ with angular coefficient $-\mu$, i.e.
 \[
 r(\mu):= \big\{ (y_1,y_2) \mid y_2= \mu
 (\bar{y}_1(s_0)-y_1) \big\}.
 \]
 Since $\bar{y}_2(s_0)=\dot{\bar{y}}_1(s_0)>0$, we see that
 $\mathbf{\Gamma}(\bar{y}_1(s_0))=(\bar{y}_1(s_0), \bar{y}_2(s_0))$ lies
 above $\mathbf{R}$. By construction $r(\mu)$ intersects $\mathbf{\Gamma}$
 at least in a point, for any $\mu>0$: We denote by
 $\big(Y_1(\mu),\mu (\bar{y}_1(s_0)-Y_1(\mu))\big)$ the intersection
 with the smallest $Y_1$. Then, it follows that $Y_1<\bar{y}_1(s_0)$
 and $\frac{\partial Q}{\partial \bar{y}_1}(Y_1) \ge -\mu$. From
 these inequalities, along with \eqref{graph}, and using the fact
 that
 \begin{equation} \label{eq:rapporto-B}
 B(l_s)\bar{y}_1(s_0)=  g(\bar{y}_1(s_0),s_0;l_s)
 \end{equation}
 we obtain
\begin{equation} \label{graph2}
 \begin{aligned}
&-\mu \le  \frac{\partial Q}{\partial \bar{y}_1}(Y_1)  \\
& =  -A(l_s)+\frac{ B(l_s) [Y_1- \bar{y}_1(s_0)] +  [
 g(\bar{y}_1(s_0),s_0;l_s) -g(Y_1,s;l_s)]}{\mu
 [\bar{y}_1(s_0)-Y_1]}  \\ 
&\le  -A(l_s)-\frac{B(l_s)}{\mu}+\frac{g(\bar{y}_1(s_0),s_0;l_s)
 -g(Y_1,s_0;l_s)}{\mu [\bar{y}_1(s_0)-Y_1]}  \\ 
& \le  -A(l_s)+\frac{1}{\mu } \Big[ - B +\partial_{y_1} g(C,s_0;l_s) \Big]  \\
 & \le  -A+\frac{1}{\mu } \Big[ - B +\frac{(\bar{q}-1) g(C,s_0;l_s)}{C} \Big] \\
 & <  -A+\frac{1}{\mu } \Big[ - B +\frac{(\bar{q}-1)
 g(\bar{y}_1(s_0),s_0;l_s)}{\bar{y}_1(s_0)} \Big] 
= -A+\frac{B(\bar{q}-2)}{\mu } 
\end{aligned}
\end{equation}
 where $C \in (Y_1,\bar{y}_1(s_0))$ and we used the mean value
 theorem. Further $\bar{q}$ stands for $q$ if $f$ is of type
 \eqref{eq:kmodel-1} and it stands for $q_2$ if $f$ is of type
 \eqref{eq:potential-1}. Therefore, using \eqref{graph2} along with
 \eqref{eq:rapporto-B}, we obtain
 \[
 \mu^2 - A \mu +B(\bar{q}-2)= \mu^2 - A \mu- B+ \partial_{y_1}g(P_1^+,+\infty, l_s)
 > 0, \quad \text{for any }\mu>0.
 \]
 But this is verified if and only if
 \[
 A^2 - 4 B(\bar{q}-2) = A^2-4[ \partial_{y_1}g(P_1^+,+\infty, l_s)-B]
 < 0 ,
 \]
 which is equivalent to $l_s \in (\sigma_*,\sigma^*)$, cf
 \eqref{defsigma}, so we have found a contradiction. Hence
 $s_0=+\infty$. In particular, it follows that
 $\bar{y}_1(s)<P_1^{+}$, $\dot{\bar{y}}_1(s)>0$, for any $s \in \mathbb R$,
 and \eqref{eq:g} holds.
\end{proof}

\begin{lemma}\label{ord1}
 Assume the hypotheses of Lemma \ref{ord0} are verified. Also, assume
 that condition {\rm (A5)} holds. Then
 \begin{equation} \label{eq:g1}
\frac{\partial{g}}{\partial  y_1}(\bar{y}_1(s),s;l_s)
 < \frac{\partial{g}}{\partial  y_1}(P_1^+,+\infty;l_s)
 \end{equation}
\end{lemma}

\begin{proof}
 From a straightforward computation we see that, when $f$ is as in
 \eqref{eq:kmodel-1}, then \eqref{eq:g} implies \eqref{eq:g1}. When
 $f$ is as in \eqref{eq:potential-1},
 \[
 \partial_{y_1}g(y_1,s,l_s)=(q_1-1) k_1(\mathrm{e}^s) y_1^{q_1-2} + (q_2-1)
 k_2(\mathrm{e}^s) y_1^{q_2-2} \le (q_2-1) g(y_1,s,l_s)/y_1.
 \]
 So, let $\bar{\mathbf{y}}(s)$ be a trajectory corresponding to a GS of
 \eqref{radsta} as above; If (A5) holds, from \eqref{eq:g} we obtain
 \begin{equation}\label{bruttastoria}
 \begin{split}
 \frac{\partial g}{\partial y_1}(\bar{y}_1,s,l_s)
& \le (q_2-1) \frac{ g(\bar{y}_1(s),s,l_s)}{\bar{y}_1(s)} \\
& \le (q_2-1) \frac{ g(P_1^+,+\infty;l_s)}{P_1^+} \\
&\le \frac{\partial g}{\partial y_1}(P_1^+,+\infty;l_s),
 \end{split}
 \end{equation}
 so \eqref{eq:g1} follows and the proof is complete.
\end{proof}

We emphasize that Lemma \ref{ord1} is already contained in the last lines
of the proof of Lemma \ref{ord0}.
We decided to restate and prove it in details because this is the only point where
the assumption (A5) is explicitly required.
Such a condition is in fact needed also in \cite{YZ} where the stability
problem for an $f$ of the form
\begin{equation}\label{compareYZ}
 f(u,r)= r^{\delta_1}u^{q_1-1}+r^{\delta_2}u^{q_2-2}
\end{equation}
is discussed. Notice that \eqref{compareYZ} is a special case of
\eqref{eq:potential-1} considered in this paper.
In \cite{YZ} condition (A5), i.e.\ \eqref{condizione}, is omitted,
but in fact it is needed to prove \cite[Proposition 2.3]{YZ}.
To be more precise: In the important case
$\delta_1=\delta_2=0$, for which \eqref{condizione} does not hold,
we find that $l_s=q_1$, and $m(l_s)=2/(q_1-2)$.
In general if \eqref{condizione} does not hold we find that $m(l_s)$
(hence the asymptotic behavior of positive solutions
for $r$ large, which is $m_1$ in the notation of \cite{YZ}) depends on $q_1$,
while if \eqref{condizione} does not hold we find that $m(l_s)$
depends on $q_2$.
Consequently in the former case we find $g(y_1,+\infty;l_s)= y_1^{q_1-2}$
and in the latter $g(y_1,+\infty;l_s)= y_1^{q_2-2}$.
Hence, the last estimate in \eqref{bruttastoria} holds in the latter case
but not in the former. Analogously,
in the proof of \cite[Proposition 2.3]{YZ} (in the last two lines of page 112),
using the notation of \cite{YZ} it is required that $p_1$
is the largest exponent, which is indeed equivalent to \eqref{condizione}.

We believe that condition (A5), i.e.\ \eqref{condizione} is just technical,
and that it might be removed using the methods
introduced by Bae and Naito in \cite{BaeN}.

\begin{proposition}\label{ord1-}
 Assume {\rm (A1)--(A3), (A5)}.
Then $U(r,\alpha_1)<U(r,\alpha_2)$ for any $r>0$, whenever
 $\alpha_1<\alpha_2$.
\end{proposition}

We emphasize that if $g(y_1,s;l)$ is $s$-independent, as in \cite{W},
Lemma \ref{ord0} implies Proposition~\ref{ord1-}. This fact follows
directly by noticing that $M^u$ is a graph on the $y_1$-axis, since
$y_1(s)=U(\mathrm{e}^s,\alpha) \mathrm{e}^{m(l_s)s}$ is increasing in $s$, for any
$\alpha>0$. In view of Lemma~\ref{ord0}, we can parametrize the
manifold $M^u$ by $\alpha$, then the ordering of the regular solutions
$U(r,\alpha)$ is preserved as $s$ varies (i.e. as $r$ varies), since
they all move along a $1$-dimensional object.

When we turn to consider an $s$-dependent function $g(y_1,s; l)$,
Proposition \ref{ord1-} needs a separate proof, which can be obtained
by adapting the ideas developed in \cite{DLL,YZ}. In fact, in such a
case $W^u(\tau;l_s)$ is still one dimensional but may not be a graph
on the $y_1$-axis, so a priori we may lose the ordering property.

\begin{proof}[Proof of Proposition \ref{ord1-}]
 Let us set $Q(s)=\mathrm{e}^{\lambda_1 s}$ and observe that
 \begin{equation}\label{linear}
 \ddot{Q}+A\dot{Q}+[\partial_{y_1}g (P_1^+,+\infty;l_s)-B] Q=0.
 \end{equation}
Denote by $W(s):= [U(\mathrm{e}^{s},\alpha_2)- U(\mathrm{e}^{s},\alpha_1)]
 \mathrm{e}^{m(l_s) s }$, and observe that
 \begin{equation}\label{difference}
 \ddot{W}+A\dot{W}-B W+ D(s)=0,
 \end{equation}
 where
\begin{equation} \label{D-nuovo}
D(s):=g(U(\mathrm{e}^{s},\alpha_2)] \mathrm{e}^{m(l_s) s }, s;l_s)-
 g(U(\mathrm{e}^{s},\alpha_1)] \mathrm{e}^{m(l_s) s }, s;l_s).
\end{equation}
Using continuous dependence on initial data we see that
 $U(r,\alpha_2)>U(r,\alpha_1)$ for $r$ small enough, so that $D(s)>0$
 for $s \ll 0$. Assume by contradiction that there is
 $\bar{r}=\mathrm{e}^{\bar{s}}>0$ such that $U(r,\alpha_2)- U(r,\alpha_1)>0$
 for $0\le r<\bar{r}$, and $U(\bar r, \alpha_2)- U(\bar r,
 \alpha_1)=0$. Then, $W(s)$, and $D(s)$ are positive for $s<\bar{s}$
 and they are null for $s=\bar{s}$.

 Setting $Z(s) :=\dot{W}(s)Q(s)-W(s)\dot{Q}(s)$, by direct
 calculation we can easily see that
 $\dot{Z}(s)=\ddot{W}(s)Q(s)-W(s)\ddot{Q}(s)$. Then from
 \eqref{linear} and \eqref{difference} we obtain
 \begin{equation}\label{ODE}
 \dot{Z}=-A Z(s)+Q(s)[\partial_{y_1}g
 (P_1^+,+\infty;l_s)W(s)-D(s)].
 \end{equation}
 Observe now that $W(s) \sim (\alpha_2-\alpha_1) \mathrm{e}^{m(l_s)s}$, as
 $s \to -\infty$, and also that
 \[
 \dot{W}(s)=m(l_s)W(s)+
 [U'(\mathrm{e}^{s},\alpha_2)-U'(\mathrm{e}^{s},\alpha_1)]\mathrm{e}^{[1+m(l_s)]s}\sim
 m(l_s) (\alpha_2-\alpha_1) \mathrm{e}^{m(l_s)s},
 \]
 as $s \to -\infty$. Hence, we obtain
 \begin{equation}\label{zero}
 Z(s)\sim
 (m(l_s)-\lambda_1(l_s))(\alpha_2-\alpha_1)(\mathrm{e}^{[m(l_s)+\lambda_1(l_s)]s})
 \to 0 \quad \text{ as $s \to -\infty$ .}
 \end{equation}
 Moreover $\lambda_1(l_s)+A(l_s)=-\lambda_2(l_s)>0$ and $D(s) \to 0$ as
$s \to -\infty$, hence $\mathrm{e}^{A s} Q(s)D(s)\in
 L^1(-\infty,\bar{s}]$. Since $Z(s)$ is the unique solution of
 \eqref{ODE} satisfying \eqref{zero} we find
 \begin{equation}\label{integral}
 Z(\bar{s})= \int_{-\infty}^{\bar{s}} \mathrm{e}^{-A (\bar{s}-s)}
 Q(s)[\partial_{y_1}g (P_1^+,+\infty;l_s)W(s)-D(s)] ds.
 \end{equation}
 From the mean value theorem we find that (see \eqref{D-nuovo})
 \[
 \partial_{y_1}g (P_1^+,+\infty;l_s)W(s)-D(s)= [\partial_{y_1}g
 (P_1^+,+\infty;l_s)- \partial_{y_1}g (U(s),s;l_s)]W(s).
 \]
 where $U(s)$ lies between $U(r,\alpha_1)r^{m(l_s)}$ and
 $U(r,\alpha_2)r^{m(l_s)}$.

Since $\partial_{y_1}g (y_1,s;l_s)$ is
 increasing in $y_1$, and using \eqref{eq:g1}, for $s <\bar{s}$ we
 find
 \begin{equation}\label{convex}
 \begin{split}
 &\partial_{y_1}g (P_1^+, +\infty;l_s)W(s)-D(s) \\
& \ge  [\partial_{y_1}g (P_1^+,+\infty;l_s)- \partial_{y_1}g
 (U(\mathrm{e}^s,\alpha_2)\mathrm{e}^{m(l_s)s},s;l_s)]W(s)>0
 \end{split}
 \end{equation}
Hence, from \eqref{integral} and \eqref{convex} we obtain
 \[
 0<Z(\bar{s})=\dot{W}(\bar{s})Q(\bar{s})-W(\bar{s})\dot{Q}(\bar{s})
= \dot{W}(\bar{s})Q(\bar{s}) \, ,
 \]
 which gives $\dot{W}(\bar{s})>0$, but this contradicts the initial assumption
that $W(s)>0$ for  $s<\bar{s}$ and $W(\bar{s})=0$:
 Hence $U(r,\alpha_2)- U(r,\alpha_1)>0$ for any $ r \ge 0$.
\end{proof}

Now, we consider the singular solution $U(r,\infty)$.

\begin{proposition}\label{ord3-}
 Under the hypotheses of Proposition \ref{ord1-},  
 $U(r,\infty)r^{m(l_s)}$ is non-decreasing for any $r>0$, and
 $U(r,\alpha)<U(r,\infty)$ for any $r>0$, $\alpha>0$.
\end{proposition}
%
This proposition is new even for $f$ of type $f(u,r)=K(r)
u^{q-1}$ or of type $f(u,r)= u^{q_1-1}+u^{q_2-1}$, which are
considered, respectively, in \cite{DLL,YZ}.

\begin{proof}
 The result is well known when the system is autonomous: In fact in
 this case $U(r, \infty) r^{m(l_s)}\equiv P_1^{+}$ and
 $W^u_{l_s}=W^u_{l_u}$ is a graph on the $y_1$-axis connecting the
 origin and $\mathbf{P}^+$.

From the  previous discussion we know that the manifold $M^u$ of the
 autonomous system \eqref{si.na}, where $l=l_u$ and
 $g=g(y_1,-\infty;l_u)$, is a graph on the $y_1$-axis connecting the
 origin and the critical point $\mathbf{P}^-$. Now, we turn to consider the
$s$-dependent setting.
 Let us recall first that $\mathbf{y}^*(s; l_u)$ is the trajectory
 corresponding to the unique singular solution $U(r,\infty)$, and that
 $\lim_{s \to -\infty}\mathbf{y}^*(s;
 l_u)=\mathbf{P}^-$.
 Observe that
 for any $\tau \in \mathbb R$ the manifold $W^u(\tau;l_u)$ is a graph
 connecting the origin and $\mathbf{y}^*(s; l_u)$.

 We claim that $W^u(\tau;l_u)$ is a graph on the $y_1$-axis, for any
 $\tau\in \mathbb R$. In fact let $\mathbf{Q}, \mathbf{R} \in W^u(\tau;l_u)$, with
 $\mathbf{Q}=(Q_1,Q_2), \mathbf{R}=(R_1,R_2)$, and let $U(r, \alpha_Q)$ and $U(r,
 \alpha_R)$ be the corresponding solution of \eqref{radsta}. From
 Proposition \ref{ord1-} we know that if $\alpha_Q<\alpha_R$, then
 \begin{equation}\label{abo}
 Q_1=U(\mathrm{e}^{\tau}, \alpha_Q)\mathrm{e}^{m(l_u)\tau}<U(\mathrm{e}^{\tau},
 \alpha_R)\mathrm{e}^{m(l_u)\tau}=R_1 \, ,
 \end{equation}
 so the claim follows.

Moreover, we also get $Q_1< y_1^*(\tau;l_u)$. Assume by
 contradiction that $Q_1 > y_1^*(\tau;l_u)$. Then we can choose $\mathbf{R}$
 in the branch of $W^u(\tau;l_u)$ between $\mathbf{Q}$ and $\mathbf{y}^*(\tau;
 l_u)$, so that $\alpha_{R}>\alpha_{Q}$ and $Q_1 >R_1> y_1^*(\tau;
 l_u)$; but this contradicts \eqref{abo}. Similarly if $Q_1 =
 y_1(\tau,*; l_u)$, then $\mathbf{R} \in W^u(\tau;l_u)$ is such that
 $\alpha_R>\alpha_Q$, and $R_1>Q_1= y_1(\tau,*; l_u)$. But again we
 can choose $\tilde{\mathbf{R}}$ in the branch of $W^u(\tau;l_u)$ between
 $\mathbf{R}$ and $\mathbf{y}^*(\tau; l_u)$, and reasoning as above we find again
 a contradiction. Therefore $U(r,\alpha)<U(r,\infty)$ for any $r>0$,
 and any $\alpha>0$.

 Further, since $W^u(\tau;l_u)$ and $W^u(\tau;l_s)$ are homothetic,
 cf \eqref{cambioL}, then $W^u(\tau;l_s)$ is a graph on the
 $y_1$-axis, which connects the origin and the trajectory
 $\mathbf{y}^*(s; l_s)$ corresponding to $U(r,\infty)$. Further
 $W^u(\tau;l_s) \subset \{(y_1,y_2) \mid 0<y_1<P_1^+ , \; y_2>0 \}$
 (see Lemma~\ref{ord0}). Therefore $y_2^*(s; l_s) \ge 0$ for any $s
 \in \mathbb R$. Hence $U(r,\infty)r^{m(l_s)}$ is non-decreasing for any
 $r>0$, and the proof is concluded.
\end{proof}

\begin{remark}\label{ord3-bis} \rm
Proposition~\ref{ord3-} is interpreted in terms of system \eqref{si.na}:
Under the hypotheses of Proposition \ref{ord1-} (hence of
 Proposition~\ref{ord3-}) we have that $W^u(\tau;l_u)$,
 $W^u(\tau;l_s)$, and $W^u(\tau;l_*)$ are graphs on the $y_1$-axis
 respectively for any $\tau \in \mathbb R$. Further they are contained in
 $y_2 \ge 0$ and connect the origin respectively with
 $\mathbf{y}^*(\tau;l_u)$, $\mathbf{y}^*(\tau;l_s)$, and
 $\mathbf{y}^*(\tau;l_*)$.
\end{remark}

 \subsection{Asymptotic estimates for slow decay solutions}\label{subsec:asympotic}

 In this subsection we state the asymptotic estimates for slow decay
 solutions of \eqref{radsta}, which are crucial to prove our main
 results: We always assume (A1), (A2), and (A4).

In fact, we generalize the results obtained in \cite[\S 3]{DLL}
for $f(u,r)=k(r)u^{q-1}$
with $q>\sigma^*$, and in \cite{Dnew}, for the same potential, in the
case of $q=\sigma^*$. The main argument in \cite{DLL} has been reused in
 \cite{YZ}, and it is an adaptation to the non-autonomous context of
 the scheme introduced by Li in \cite{L} (and developed in
 \cite{GNW1}). Here, we follow a different approach: We give an
 interpretation of the main argument behind \cite[\S\,~3] {DLL} in
 terms of some general facts of the ODE theory. This approach
 contributes to make the scheme used in \cite[\S 3] {DLL} clearer.

From assumption (A4) we can now set $\zeta= \mathrm{e}^{-\gamma s}$
 in \eqref{si.naas}, and obtain a smooth system which has
 $\mathbf{\mathcal{P}}:=(P_1^+, 0,0)$ as critical point. For the remainder
 of this subsection we consider this system and its linearization around
 $\mathbf{\mathcal{P}}$ so we leave the explicit dependence on $l_s$
 unsaid. Hence, we consider \eqref{si.naas} where $\varpi=\gamma$ and
 the following system
 \begin{equation}\label{si.lin}
 \begin{pmatrix}
 \dot{y}_{1} \\ \dot{y}_{2} \\ \dot{\zeta}
 \end{pmatrix} =
 \begin{pmatrix}
 0 & 1 & 0 \\ B- \partial_{y_1} g^{+\infty}(P_1^+) & -A & 0 \\ 0 &
 0 & -\gamma
 \end{pmatrix}
 \begin{pmatrix}
 y_1 \\ y_2 \\ \zeta
 \end{pmatrix}
 \end{equation}
 Let us denote by $\mathcal{A}$ the matrix in \eqref{si.lin}: It has
 $3$ negative eigenvalues $\lambda_2 \le \lambda_1 <0$ and
 $-\gamma<0$. Observe that (A4) is needed  to
guarantee smoothness of the system
 \eqref{si.naas} for $\zeta=0$. Therefore the critical point
 $\mathbf{\mathcal{P}}$ of \eqref{si.naas} is a stable node.

 Assume first that the $3$ eigenvalues are simple, then we have $3$
 eigenvectors, respectively $v_{1}=(1,-m+\lambda_1,0)$,
 $v_2=(1,-m+\lambda_2,0)$, and $v_z:=v_3=(0,0,1)$. Any solution $\ell(t)$
 of \eqref{si.lin} can be written as
 \begin{equation}\label{lin}
 \ell(s)=\bar{a} v_1 \mathrm{e}^{\lambda_1 s}+ \bar{b} v_2 \mathrm{e}^{\lambda_2 s}+z v_z
 \mathrm{e}^{-\gamma s}
 \end{equation}
 for some $\bar{a},\bar{b},z \in \mathbb R$.

 By standard facts in invariant manifold theory (see, e.g.,
\cite[\S 13]{CoLe}), any trajectory $(\mathbf{y}(s), \zeta(s))$ of \eqref{si.naas}
 converging to $\mathbf{\mathcal{P}}$ can be seen as a non-linear
 perturbation of a solution $\ell(s)$ of \eqref{si.lin}. More
 precisely set $\mathbf{n}(s)=(n_1(s),n_2(s))= (y_1(s)- P_1^+,y_2(s))$,
 then $\mathbf{N}(s):=(n_1(s),n_2(s),\zeta(s))=\ell(s)+O(|\ell(s)|^2)$.
 Therefore
 \[
 n_1(s)= \bar{a} \mathrm{e}^{\lambda_1 s}+ \bar{b} \mathrm{e}^{\lambda_2 s}+z \mathrm{e}^{-\gamma
 s} + O(\mathrm{e}^{2\lambda_1 s}+ \mathrm{e}^{2\lambda_2 s}+ \mathrm{e}^{-2\gamma s})
\]
In \cite [Appendix]{BF-arxiv} we prove that the expansion can be
 continued to an arbitrarily large order: This is the content of
 Proposition~\ref{sintetizzo2} and of its general form containing
 resonances, i.e. Proposition~\ref{sintetizzo}. Let us rewrite
 \eqref{si.na} as
 \begin{equation}\label{xxx}
 \dot{\vec{x}}= \mathcal{A} \vec{x}+ \vec{N}(\vec{x})
 \end{equation}
 where $\vec{x}=(y_1,y_2,\zeta)$, and $\mathcal{A}$ is as the matrix
 in \eqref{si.lin}.

 \begin{proposition}\label{sintetizzo2}
 Assume for simplicity $\vec{N} \in C^{\infty}$ and that the
 eigenvalues of $\mathcal{A}$ are real, negative and simple and are
 rationally independent, i.e there is no
 $\chi=(\chi_1,\chi_2,\chi_3) \in \mathbb{Z}^3 \backslash \{(0,0,0) \}$
such that $ \chi_1 |\lambda_1|+\chi_2 |\lambda_2|+\chi_3 \gamma=0$, so
 that no resonances are possible. Further assume for definiteness
 that $|\lambda_1|<\gamma$.

 Then for any $k \in \mathbb{N}$ we can find a polynomial $P$ of
 degree $k$ in $3$ variables such that
 \[
 y_1(t)= P(\mathrm{e}^{\lambda_1 t}, \mathrm{e}^{\lambda_2 t},
 \mathrm{e}^{-\gamma t})+o(\mathrm{e}^{[(k+1)\lambda_1+\varepsilon] t})
 \]
 as $t \to +\infty$, for $\varepsilon>0$ small enough.
\end{proposition}

We refer the interested reader to \cite[Appendix]{BF-arxiv}}
for further details.

Now, we rephrase the result in a more suitable form for our
purpose. Let us set
\begin{equation}\label{Itheta}
 I_\theta=\big\{\chi=(\chi_1,\chi_2,\chi_3) \in \mathbb{N}^3 \, \colon \, \chi_1
 |\lambda_1|+\chi_2 |\lambda_2|+\chi_3 |\gamma| \le \theta \big\}.
\end{equation}
Then, we can expand $n_1(s)$ as
\begin{equation}\label{expand0}
 \begin{array}{c}
 n_1(s)= a \mathrm{e}^{\lambda_1 s}+ b \mathrm{e}^{\lambda_2 s}+z \mathrm{e}^{-\gamma s} +
 P_{\theta}(s) +o( \mathrm{e}^{-\theta s}),
 \end{array}
\end{equation}
where the function $P_{\theta}(s)$ is completely determined by the
values of the coefficients $a,b,z$.

As a first case, assume that $\gamma, |\lambda_1|, |\lambda_2|$ are rationally
independent. Then, there are constants $c^\chi \in \mathbb R$ such that
\begin{equation}\label{Ptheta}
P_{\theta}(s)=\sum_{ \chi\in I_\theta ,\, |\chi| \ge 2 }
  c^\chi \mathrm{e}^{(\chi_1 \lambda_1+\chi_2 \lambda_2-\chi_3 \gamma)s} \quad
 \text{with } \chi=(\chi_1,\chi_2,\chi_3)
\end{equation}
and $|\chi|=\chi_1+\chi_2+\chi_3$.

Let us now consider the resonant cases, i.e. when there are
$M^0, M^1, \ldots , M^j$, (a $j$-ple resonance)
$M^i=(\chi^i_1,\chi^i_2,\chi^i_3) \in I_\theta$, $|M^i|>0$ for $i=1,
\ldots, j$, such that
\[
 \chi^i_1 |\lambda_1| + \chi^i_2 |\lambda_2| + \chi^i_3 \gamma= \bar{\theta} \le
 \theta \,.
\]
Then, we have to replace $\sum_{i=0}^j c_{M^i} \mathrm{e}^{(\chi^i_1
 \lambda_1+\chi^i_2 \lambda_2-\chi^i_3 \gamma)s}$ by
\begin{equation}\label{sost}
 \sum_{i=0}^j c_{M^i} s^{i} \mathrm{e}^{(\chi^i_1 \lambda_1+\chi^i_2 \lambda_2-\chi^i_3
 \gamma)s} \quad \text{in the function $P_{\theta}$},
\end{equation}
(notice that we have included the possible case of resonances with the
linear terms, e.g., $\chi_2$ multiple of $\chi_1$ etc.). The same
happens when we have resonances within the linear terms,
e.g.\ $|\lambda_1|=|\lambda_2|$ (i.e. $l_s=\sigma^*$), or $|\lambda_1|=\gamma$: We
replace the terms as done in \eqref{sost}.

Before collecting all these facts in Proposition \ref{sintetizzo}
below, we need some further notation. Let us introduce the following
sets:
\begin{gather}
 J_{|\lambda_1|} = \{\chi=(0,0,\chi_3) \in \mathbb{N}^3 : 0< \chi_3 \gamma < |\lambda_1| \}
 \,, \\
J_{|\lambda_2|} = \{\chi=(\chi_1,\chi_2,\chi_3) \in \mathbb{N}^3
 : |\lambda_1| < \chi_1|\lambda_1|+ \chi_3 \gamma < |\lambda_2| \}.
\end{gather}
Observe that $ J_{|\lambda_1|}$ is empty if $|\lambda_1| \le \gamma$, and
$J_{|\lambda_2|}$ is empty if $|\lambda_2| < 2|\lambda_1|$ and $|\lambda_2|\le \gamma$. We denote
\begin{equation}
 \Psi(s) =\sum_{\chi=(0,0,\chi_3) \in J_{|\lambda_1|}} c^\chi \mathrm{e}^{-\chi_3
 \gamma s} + \chi_r(s) \mathrm{e}^{\lambda_1 s} \label{defPsi}
\end{equation}
where $\chi_r(s)=0$ if $|\lambda_1|/\gamma \not\in \mathbb{N}$, and
$\chi_r(s)=\chi_r s$ if $|\lambda_1|/\gamma \in \mathbb{N}$ and
$l_s>\sigma^*$, while $ \chi_r(s)=\chi_r s^2$ if $|\lambda_1|/\gamma \in
\mathbb{N}$ and $l_s=\sigma^*$, for a certain $\chi_r \in \mathbb R$.

 \begin{proposition}\label{sintetizzo}
 Assume {\rm (A1), (A2), (A4)}. Let
 $\bar{I}_{\theta}=I_{\theta} \backslash [\{ (1,0,0) , (0,1,0) \}
 \cup J_{|\lambda_1|} \cup J_{|\lambda_2|}]$. Any trajectory
 $(y_1(s),y_2(s),\zeta(s))$ converging to $\mathbf{\mathcal{P}}$ is
 such that $y_1(s)$ has the following expansion if $l_s> \sigma^*$:
 \begin{equation}\label{expandy}
y_1(s)= P_1^{+}+ \Psi(s) +a
 \mathrm{e}^{\lambda_1 s}+ Q^1_{\theta}(s)+b \mathrm{e}^{\lambda_2 s}+
 Q^2_{\theta}(s) +o( \mathrm{e}^{-\theta s}) ,
\end{equation}
 where
\begin{gather*}
 Q_{1,\theta}(s)=\sum_{\chi \in
 J_{|\lambda_2|}} c^\chi \mathrm{e}^{(\chi_1 \lambda_1+\chi_2 \lambda_2-\chi_3
 \gamma)s}, \quad \text{with $\chi=(\chi_1,\chi_2,\chi_3)$, and } \\
 Q_{2,\theta}(s)=\sum_{\chi \in
 \bar{I}_{\theta}} c^{\chi} \mathrm{e}^{(\chi_1 \lambda_1+\chi_2 \lambda_2-\chi_3  \gamma)s}
 \end{gather*}
 as $s \to +\infty$, if we do not have resonances; otherwise we
 need to replace the resonant terms in $Q_{1,\theta}(s)$ according
 to \eqref{sost}.

 If $l_s=\sigma^*$ so that $\lambda_1=\lambda_2$ we have
 \begin{equation}\label{expandybis}
 y_1(s)= P_1^{+}+ \Psi(s)+a s\mathrm{e}^{\lambda_1 s}+b \mathrm{e}^{\lambda_1
 s}+ Q_{2,\theta}(s) +o( \mathrm{e}^{-\theta s}) \,
 \end{equation}
 as $s \to +\infty$, again if we do not have resonances, otherwise
 we need to replace the resonant terms in $ Q_{2,\theta}(s)$
 according to \eqref{sost}.
 \end{proposition}

 \begin{remark}\label{importante} \rm
 We emphasize that $Q_{1,\theta}(s)$ contains terms which are
 negligible with respect to $a \mathrm{e}^{\lambda_1 s}$ while
 $Q_{2,\theta}(s)$ contains terms which are negligible with respect
 to $b \mathrm{e}^{\lambda_2 s}$. Further if $|\lambda_1|<\gamma$ then $\Psi(s)$
 is identically null by definition.
 \end{remark}

 The proof is developed in \cite[Appendix]{BF-arxiv}
by means of an asymptotic expansion result for ODEs,
 which seems to be new to the best of our knowledge. In fact, we
 borrow some of the ideas from \cite{DLL,YZ}.

 \begin{remark}\label{defa} \rm
 Fix $\mathbf{Q}$ and $\tau \in \mathbb R$; then $y_1(t,\tau,\mathbf{Q};l_s)$ admits an
 expansion either of the form \eqref{expandy} or of the form
 \eqref{expandybis}. All the coefficients in the expansions are
 determined by the choice of $a,b$, which are in fact smooth
 functions of $\mathbf{Q}$, i.e.\ $a=a(\mathbf{Q})$, $b=b(\mathbf{Q})$.

 In fact, all the coefficients in $\Psi(s)$ are determined when the
 non-linearity $g$ and $\tau$ are fixed; the coefficients in
 $Q_{1,\theta}$ are assigned (and can be determined) once $a$ is
 fixed, while $Q_{2,\theta}$ is assigned once $a$ and $b$ are
 assigned.
 \end{remark}

 \begin{remark}\label{datogliere} \rm
 Fix $\mathbf{Q}$ and $\tau$, the coefficients $a=a(\mathbf{Q})$, $b=b(\mathbf{Q})$ may be
 evaluated through the method explained in \cite{DLL}. However from
 the previous discussion we have the following. Let $a_1,b_1, z_1$
 be such that $(\mathbf{Q}- \mathbf{P^+},\mathrm{e}^{-\gamma \tau})=a_1
 v_1+b_1v_1+z_1v_z$. Then $a=a_1+O(|\mathbf{Q}- \mathbf{P^+}|^2)$ and
 $b=b_1+O(|\mathbf{Q}- \mathbf{P^+}|^2)$.
 \end{remark}

 The proof of what is stated in these two remarks is provided in
 \cite[Appendix]{BF-arxiv}. For further details about these
 points see \cite[Remarks~4.12, 4.16]{BF-arxiv}.

 Now, we translate Proposition \ref{sintetizzo} for the original
 equation \eqref{radsta}.

 \begin{lemma}\label{asinteqna}
 Assume {\rm (A1), (A2)} with $l_u \ge l_s \ge \sigma^*$,
 {\rm (A3), (A4)}. Consider either a GS $U(r,\alpha)$ for $\alpha>0$,
 or the SGS $U(r,\infty)$; Then there are continuous functions
 $\mathcal{A}\colon (0,+\infty] \to \mathbb R$, $\mathcal{B}\colon (0,+\infty]
 \to \mathbb R$, such that $\mathcal{A}$ is monotone increasing, and if
 $l_s>\sigma^*$
 \begin{equation}
 \begin{aligned}
 U(r,\alpha)&= \frac{P_1^{+}}{r^{m}}+
 \frac{\Psi(\ln(r))}{r^m}+ \mathcal{A}(\alpha) r^{\lambda_1-m}+
 \frac{Q_{1,\theta}(\ln(r))}{r^m}\\
&\quad+\mathcal{B}(\alpha)  r^{\lambda_2-m} + \frac{Q_{2,\theta}(\ln(r))}{r^m} +o(
 r^{-\theta-m})
 \end{aligned} \label{expandU}
\end{equation}
as $r \to +\infty$. If
 $l_s=\sigma^*$ we have
\begin{equation}
\begin{aligned}
 U(r,\alpha)&=  \frac{P_1^{+}}{r^{m}}+
 \frac{\Psi(\ln(r))}{r^m}+\mathcal{A}(\alpha)\ln(r)
  r^{\lambda_1-m} +\mathcal{B}(\alpha) r^{\lambda_2-m}\\
&\quad + \frac{Q_{2,\theta}(\ln(r))}{r^m} +o( r^{-\theta-m}).
 \end{aligned} \label{expandUbis}
 \end{equation}
 \end{lemma}

 \begin{remark} \rm
If we replace (A3) with the weaker assumption
 (A6) in Lemma \ref{asinteqna}, then we still have the
 expansions in \eqref{expandU}, \eqref{expandUbis}, but we cannot
 ensure that $\mathcal{A}$ is monotone decreasing.
 \end{remark}

 \begin{proof}[Proof of Lemma~\ref{asinteqna}]
Fix $\tau \in \mathbb R$; let $\mathbf{y}(s,\tau, \mathbf{Q}(\alpha);l_s)$ be the
 trajectory of \eqref{si.na} corresponding to $U(r,\alpha)$, so
 that $\mathbf{Q}(\alpha) \in W^u_{l_s}(\tau)$. Then we can apply
 Proposition \ref{sintetizzo} to $y_1(s,\tau, \mathbf{Q}(\alpha);l_s)$ and
 we find the expansions \eqref{expandU}, \eqref{expandUbis}, where,
 according to Remark \ref{defa}, the coefficients $a,b$ are
 $a=a(\mathbf{Q}(\alpha))$ and $b=b(\mathbf{Q}(\alpha))$. We set
 \begin{equation}\label{defAA}
 \mathcal{A}(\alpha)=a(\mathbf{Q}(\alpha)), \, \text{ and }\,
 \mathcal{B}(\alpha)=b(\mathbf{Q}(\alpha)).
 \end{equation}
 It follows that $\mathcal{A}\colon (0,+\infty) \to \mathbb R$ and
 $\mathcal{B}:(0,+\infty) \to \mathbb R$ are continuous functions.
 Finally if (A3) holds, then $U(r,\alpha_1)< U(r,\alpha_2)$ if
 $\alpha_1<\alpha_2$ for any $r>0$, and in particular for $r$
 large, so $\mathcal{A}(\alpha)$ is monotone increasing.
 \end{proof}

 \section{Main results: Stability and asymptotic
 stability} \label{sec:stability}

Let us state Theorems~\ref{stabile} and \ref{asint.stabile} from which
 Theorems~\ref{thmD}, \ref{main-1}, \ref{main-2} follow directly.
 Let $r>0$, we denote by $[[r]]:=\{ k \in \mathbb{N}  k-1<r \le k \}$.
 We have the following results

\begin{theorem}\label{stabile}
 Suppose $f$ is $C^k$ where $k=[[|\lambda_1|/\gamma]]$. Assume
{\rm (A1)--(A5)}. Then any radial GS
 $U(r,\alpha)$ of \eqref{parab} is stable with respect to the norm
 $\|\cdot\|_{m(l_s)+\lambda_1}$ if $l_s> \sigma^*$, and with respect
 to the norm $\norm{\cdot }_{m(l_s)+|\lambda_1|}$ if
 $l_s=\sigma^*$.
\end{theorem}

\begin{theorem}\label{asint.stabile}
 Assume the hypotheses of Theorem \ref{stabile}. Then any radial GS
 $U(r,\alpha)$ of \eqref{parab} is weakly asymptotically stable with
 respect to the norm $\|\cdot\|_{m(l_s)+|\lambda_2|}$ if $l_s> \sigma^*$,
 and with respect to the norm $\| \cdot  \|_{m(l_s)+|\lambda_1|}$
if $l_s=\sigma^*$.
\end{theorem}

Let us recall that the stability of positive GS $U(|x|, \alpha)$ of
\eqref{parab} has been analyzed in a number of papers, (see
\cite{DLL,GNW1, GNW2, W}). In \cite{GNW1}, when $f(u, |x|)=u^{q-1}$
and $q>\sigma^\ast$, the authors proved that the positive GS of
\eqref{parab} are stable in the norm $\|\cdot  \|_{m+
 |\lambda_1|}$, and weakly asymptotically stable with respect to $\|
\cdot  \|_{m+ |\lambda_2|}$. These results have been subsequently
extended in \cite{DLL} to functions $f(u, |x|)$ of the form
$k(|x|)r^{\delta}|u|^{q-1}$ where $K$ is a monotone decreasing
uniformly positive and bounded function. Here we drop the assumption that
 $k$ is bounded:
This will allow us to consider potentials giving rise to singular
solutions $U(r,\infty)$ having two different behaviors as $r \to 0$
(i.e. $U(r,\infty) \sim P^- r^{-m(l_u)}$) and as $r \to \infty$
(i.e. $U(r,\infty) \sim P^+ r^{-m(l_s)}$).

\subsection{Proof of Theorem \ref{stabile}}

We first introduce some standard definitions.

\begin{definition} \rm
 We say that $\overline{\phi}$ is a \emph{super-solution} of \eqref{laplace} if
 $\Delta \overline{\phi} + f(\overline{\phi},|x|) \le 0$; analogously
 $\underline{\phi}$ is a \emph{sub-solution} of \eqref{laplace} if $\Delta \underline{\phi} +
 f(\underline{\phi},|x|) \ge 0$.
\end{definition}

We refer to \cite{W} or to \cite[\S 3]{BF} for an extension of these
definitions to weak and mild solutions. Also, depending on a number of
very relevant factors (for instance, the type of domain and of the
boundary conditions, the regularity of the forcing term, etc.) the
notion of weak solution for parabolic equations can change
considerably as described, e.g., in \cite{BH, Friedman, Gazzola,
 QS}. In particular, we mention that, a dynamical approach to study a
generalized parabolic equation on an unbounded strip-like domain is
given in \cite{Polat}: In this case a suitable definition of weak
solutions, on weighted Sobolev (and Bochner) spaces, is considered and
the author proved the existence of a global attractor. Then, this
situation is further generalized in \cite{BC}.

Both Theorems \ref{stabile}, \ref{asint.stabile} depend strongly on
the following well known fact, proved in \cite[Theorem 2.4]{W}, see
also \cite[Theorem 3.10]{BF}.

\begin{lemma} \label{keyintrobis}
Assume {\rm (A1), (A2)} and let
 $U_1(r)$ and $U_2(r)$ be positive solutions of \eqref{radsta}
 respectively for $r \le R_1$ and for $r \ge R_2$, where $R_1>R_2$,
 and let $R \in (R_2,R_1) $ be such that $U_1(R)=U_2(R)$. Consider
 \[
 \phi(x)=\begin{cases}
 U_1(r) & \text{if }  0<|x| \le R, \\
 U_2(r) & \text{if }  |x| \ge R.
 \end{cases}
 \]
 We have
 \begin{itemize}
 \item  If $U'_1(R)\ge U'_2(R)$, then $\phi(x)$ is a continuous weak
 super-solution of \eqref{laplace}.\smallskip

 \item  If $U'_1(R)\le U'_2(R)$, then $\phi(x)$ is a continuous weak
 sub-solution of \eqref{laplace}.
 \end{itemize}
\end{lemma}

\begin{lemma} \label{keyintro}
Assume {\rm (A1), (A2)}:
 \begin{itemize}
 \item [(i)] If the initial value $\phi$ in \eqref{data} is
 a continuous weak super-(sub-) solution of \eqref{laplace}, then
 the solution $u(t,x; \phi)$ of \eqref{parab}-\eqref{data} is
 non-increasing (non-decreasing) in $t$ as long as it exists, for
 any $x$;
 strictly if $\phi$ is not a solution.

 \item[(ii)] If $\phi$ is radial, then $u(t,x; \phi)$ is
 radial in the $x$ variable for any $t>0$.
 \end{itemize}
\end{lemma}

To prove Theorem \ref{stabile} we adapt the main ideas developed in
\cite{GNW1,DLL,YZ}.

As a consequence of the proof of Proposition \ref{ord1-} we obtain the
following result (see Lemma~\ref{strict} below) which will be useful
to prove the stability of the solutions, and replaces a longer
elliptic estimate performed in \cite[Lemma 4.3]{DLL} and adapted in \cite{YZ,Dnew}.
We stress that in fact the proof in the critical case, considered in
\cite{Dnew}, suffers from a flaw.

\begin{lemma}\label{strict}
 Assume {\rm (A1)--(A5)}. Assume
 $\beta>\alpha$ then $\mathcal{A}(\beta)>\mathcal{A}(\alpha)$.
\end{lemma}

\begin{proof}
 Since $U(r,\beta)>U(r,\alpha)$ for any $r>0$ (see
 Lemma~\ref{ord1-}), we already know that $\mathcal{A}(\beta) \ge
 \mathcal{A}(\alpha)$, so we just need to prove that the inequality
 is strict. Set
 $h(s)=[U(\mathrm{e}^{s},\alpha_2)-U(\mathrm{e}^{s},\alpha_1)]\mathrm{e}^{(m(l_s)-\lambda_1)s}$,
 and, following the notation of Proposition \ref{ord1-},
 $Q(s)=\mathrm{e}^{\lambda_1 s}$. Following the main line in the proof of
 Proposition \ref{ord1-} we see that $\dot{h}(s)=Z(s)/Q^2(s)$. In
 particular, from \eqref{integral} and \eqref{convex}, $\dot{h}(s)>0$
 for any $s \in \mathbb R$. Since $\lim_{s \to -\infty} h(s)=0$ we see
 that $h(s)>0$ for any $s \in \mathbb R$, and $\lim_{s \to +\infty}
 h(s)>0$.

 If $l_s>\sigma^*$, then $\lim_{s \to +\infty}
 h(s)=\mathcal{A}(\beta)-\mathcal{A}(\alpha)>0$, and the proof is
 complete.

 Assume now $l_s=\sigma^*$, and also assume by contradiction that
 $\mathcal{A}(\beta)= \mathcal{A}(\alpha)$. In this case we see that
 $\lim_{s \to +\infty} h(s)=\mathcal{B}(\beta)-\mathcal{B}(\alpha)
 \in (0, +\infty)$. However, from \eqref{integral}, since
 $A=-2\lambda_1$, for any $\bar{s}\in \mathbb R$ we find
\[
 \dot{h}(\bar{s})= \int_{-\infty}^{\bar{s}} \mathrm{e}^{A s}
 Q(s)[\partial_{y_1}g (P_1^+,+\infty;l_s)W(s)-D(s)]>0 \, .
\]
 Therefore $\liminf_{s \to +\infty} \dot{h}(s)\ge \dot{h}(0) >0$,
 hence $\mathcal{B}(\beta)-\mathcal{B}(\alpha) =\lim_{s \to +\infty}
 h(s)=+\infty$, but this is a contradiction. Hence
 $\mathcal{A}(\beta)> \mathcal{A}(\alpha)$.
\end{proof}

\begin{lemma}\label{vicinanza}
 Assume {\rm (A1)--(A5)} and  $l_u \ge l_s$. If $l_s>\sigma^*$.\\
Then
 $\|U(r,\beta)-U(r,\alpha)\|_{m+ |\lambda_1|} \to 0$ as $\beta \to
 \alpha$, \\
while if $l_s= \sigma^*$, then 
$ \norm{ U(r,\beta)-U(r,\alpha) }_{m+ |\lambda_1|}\to 0$ as $\beta \to
 \alpha$.
\end{lemma}

\begin{proof}
 We develop the proof assuming $l_s>\sigma^*$, the case
 $l_s=\sigma^*$ is completely analogous. It is well known that, for
 any fixed $R>0$ and any $\varepsilon>0$, there is $\delta_1(\varepsilon,\alpha,R)>0$
 such that
 \begin{equation}\label{azero}
 \sup \{|U(r,\beta)-U(r,\alpha)| \mid 0 \le r \le R \} < \varepsilon
 \end{equation}
 whenever $|\beta-\alpha|<\delta_1$ (this is a continuous dependence
 on initial data argument for the singular equation \eqref{radsta}).
 Further from \eqref{expandU} we see that for $r$ large enough we
 have
\begin{equation}\label{dentro}
 \big| \big(U(r,\beta)-U(r,\alpha)\big)\big|
 (1+r^{m-\lambda_1}) \cong
 |\mathcal{A}(\beta)-\mathcal{A}(\alpha)|+o(r^{|\lambda_2-\lambda_1|/2})
\end{equation}
Thus, for any $\varepsilon>0$ there exists $M(\varepsilon)$ such that $
 |o(r^{|\lambda_2-\lambda_1|/2})|\leq C\varepsilon $, when $r \ge M(\varepsilon)$. Further
 from Lemma \ref{asinteqna} we see that for any $\varepsilon>0$ we can find
 $\delta_2(\varepsilon,\alpha)>0$ such that $
 |\mathcal{A}(\beta)-\mathcal{A}(\alpha)| \le \varepsilon$ if
 $|\beta-\alpha|<\delta_2$. Therefore
\begin{equation}\label{fuori}
 \left| \big(U(r,\beta)-U(r,\alpha)\big)\right|
 (1+r^{m-\lambda_1}) \le \varepsilon \,, \quad \text{for $r \ge M$}\,.
 \end{equation}
The proof follows from \eqref{dentro}--\eqref{fuori}, choosing
 $M=R$ and $\delta(R,\alpha,\varepsilon)= \min \{\delta_1, \delta_2 \}$.
\end{proof}

We are now ready to prove Theorem~\ref{stabile}.

\begin{proof}[Proof of Theorem~\ref{stabile}]
 We give the proof just in the case $l_s>\sigma^*$, when
 $l_s=\sigma^*$ the calculations are completely analogous and we omit them. Fix
 $\varepsilon>0$ (small) and
 $\alpha>0$; let $\phi(x)$ be such that $\|U(|x|,
 \alpha)-\phi(x)\|_{m+|\lambda_1|} =\delta$, where $\delta>0$ will be
 chosen below.

Let $|\eta|<\alpha$ and set
 \begin{equation}
 z(r,\eta)= [U(r, \alpha+ \eta)-U(r,\alpha)](1+r^{m -\lambda_1})
 \end{equation}
Observe that $z(0,\eta)=\eta$ and $\lim_{r\to +\infty} z(r,\eta)=
 \mathcal{A}(\alpha+\eta)- \mathcal{A}(\alpha)$. So we can set
 \begin{equation}
\underline{z}(\eta)= \min \{ |z(r,\eta)| \mid r>0 \} \quad \text{and}\quad
 \overline{z}(\eta)= \max \{ |z(r,\eta)| \mid r>0 \}.
 \end{equation}
Moreover $z(r,\eta)$ is uniformly positive (respectively negative)
 for any $r>0$ if $\eta>0$ (resp. $\eta<0$), so
 $\underline{z}(\eta)>0$ if $\eta \ne 0$: This follows from Lemmas
 \ref{ord1-} and \ref{strict}.

Finally, from Lemma \ref{vicinanza}, we know that $\lim_{\eta \to
 0} \underline{z}(\eta)= \lim_{\eta \to 0} \overline{z}(\eta)=0$.
 Then, for any $\varepsilon>0$ we can find $d=d(\varepsilon)>0$ such that
 $\overline{z}(-d) < \varepsilon$, and $\overline{z}(d) < \varepsilon$. Set
 $\alpha_1=\alpha-d$, $\alpha_2=\alpha+d$, and choose 
$\delta= \min  \{ \underline{z}(-d)\,, \underline{z}(d) \}$. Then
 \begin{equation}
 \begin{gathered}
 U(|x|,\alpha_1) < \phi(x) < U(|x|,\alpha_2) , \\
 \|U(|x|,\alpha_i) -U(|x|,\alpha)\|_{m-\lambda_1} \le \varepsilon \quad
 \text{for $i=1,2$}
 \end{gathered}
 \end{equation}
Therefore, from the comparison principle (see, e.g.,
 \cite[Appendix]{Gazzola}), we have that
 \begin{equation}
 U(|x|,\alpha_1) < u(t,x;\phi) < U(|x|,\alpha_2), \quad
\text{for  any $t \ge 0$, $x \in \mathbb R^n$},
 \end{equation}
 and the proof is complete.
\end{proof}

\subsection{Weak asymptotic stability}

To prove weak asymptotic stability we follow the outline of the proof
of \cite[Theorem 4.1]{GNW1} and adapted in \cite{DLL,YZ}.

\begin{proposition}\label{4.1}
 Assume the hypotheses of Theorem \ref{stabile} and
 consider the stationary problem \eqref{radsta}. Then, for any
 radial GS $U(\cdot,d)$ of \eqref{radsta}, there is a sequence of
 radial strict super-solutions
 $\overline{U}^{(1)}(\cdot,e^1)>\overline{U}^{(2)}(\cdot,e^2)>\ldots>U(\cdot,d)$
 of \eqref{laplace} and a sequence of radial strict sub-solutions
 $\underline{U}^{(1)}(\cdot,c^1)<\underline{U}^{(2)}(\cdot,c^2)
 <\ldots<U(\cdot,d)$ such that $U(\cdot,d)$ is the only solution of
 \eqref{laplace} satisfying
 $\underline{U}^{(k)}(\cdot,c^k)<U(\cdot,d)<\overline{U}^{(k)}(\cdot,e^k)$,
 for every $k$. Moreover
 \begin{equation}\label{limit}
 \lim_{k \to \infty}\underline{U}^{(k)}(\cdot,c^k)=U(\cdot,d)=
 \lim_{k \to \infty}\overline{U}^{(k)}(\cdot,e^k)
 \end{equation}
\end{proposition}

 \begin{proof}
 Let $h:[0,+\infty) \to [0,1]$ be a monotone decreasing
 $C^{\infty}$ function such that $h(0)=1$ and $h(r)\equiv 0$ for $r
 \ge 1$. Let  $\mathcal{G}(y_1,s;l_s)=g(y_1,s;l_s)-g(y_1,+\infty;l_s)$ and
 observe that $\mathcal{G}(y_1,s;l_s) \ge 0$ and it is decreasing
 in $s$  for any $y_1,s$.

 Assume first $\mathcal{G}(y_1,s) \not\equiv  0$, i.e. consider the generic 
case, and denote
 \begin{gather*}
 \overline{g}^{(k)}(y_1,s)= g(y_1,s;l_s)+\frac{h(\mathrm{e}^s)}{2k}  \mathcal{G}(y_1,s;l_s) \\
\underline{g}^{(k)}(y_1,s)=
 g(y_1,s;l_s)-\frac{h(\mathrm{e}^s)}{2k} \mathcal{G}(y_1,s;l_s)
 \end{gather*}
and let $\overline{f}^{(k)}$, $\underline{f}^{(k)}$ be the corresponding
 functions obtained via \eqref{transf1}. Notice that by
 construction $\overline{g}^{(k)}(y_1,s)$, and $\underline{g}^{(k)}(y_1,s)$ are
 both decreasing in $s$ for any $k \ge 1$; Hence 
$\overline{f}^{(k)} \ge f \ge \underline{f}^{(k)}$ satisfy
(A1)--(A5) so that Lemma~\ref{ord0}, and
 Proposition~\ref{ord1-} hold. In particular all the regular
 solutions of the respective problem \eqref{radsta}, say
 $\overline{U}^{(k)}(r,\alpha)$, $U(r,\alpha)$,
 $\underline{U}^{(k)}(r,\alpha)$, are GSs. Further the corresponding
 trajectories of \eqref{si.na}, say $\mathbf{\overline{y}^{(k)}}(s,\alpha)$,
 $\mathbf{y}(s,\alpha)$, $\mathbf{\underline{y}^{(k)}}(s,\alpha)$ are monotone
 increasing in their first component and converge to $\mathbf{P^+}$,
 and have the asymptotic expansion as described in Proposition
 \ref{sintetizzo}. More precisely they both have either the
 expansion \eqref{expandU} or \eqref{expandUbis}, where the
 function $\Psi(\ln(r))$ coincide for $r \ge 1$, while the
 coefficients $a=\overline{\mathcal{A}}^{(k)}(\alpha)$,
 $a=\underline{\mathcal{A}}^{(k)}(\alpha)$ and
 $b=\overline{\mathcal{B}}^{(k)}(\alpha)$,
 $b=\underline{\mathcal{B}}^k(\alpha)$ are different, see
 Lemma~\ref{asinteqna}. Further by construction,
 $\overline{U}^{(k)}(r,\alpha)$, $\underline{U}^{(k)}(r,\alpha)$ are
 respectively super- and sub-solutions for the original problem
 \eqref{radsta}.

 We divide our argument in several steps. 
\smallskip

\noindent \textbf{Step 1.} If there is $R>0$ such that
 $U(R,d)=\overline{U}^{(k)}(R,c)$ (respectively
 $U(R,d)=\underline{U}^{(k)}(R,e)$), then $U(r,d) \ge
 \overline{U}^{(k)}(r,c)$ (respectively $U(r,d)\le \underline{U}^{(k)}(r,e)$)
 for  any $r \ge R$.

 Let $\tau(\xi)\colon(0,2) \to \mathbb R$ be the inverse of the function
 $\xi(\tau)$ defined in \eqref{manistella}. We consider
 \begin{equation}\label{si.nastella}
 \begin{pmatrix}
 \dot{y}_{1} \\ \dot{y}_{2} \\ \dot{\xi}
 \end{pmatrix}
 = \begin{pmatrix} 0 & 1 &0 \\ B(l_*) & -A(l_*) & 0 \\ 0 & 0 & C
 \end{pmatrix}
 \begin{pmatrix}
 y_1 \\ y_2 \\ \xi
 \end{pmatrix}-
 \begin{pmatrix}
 0 \\ g(y_1,\tau(\xi);l_*)\\ 0
 \end{pmatrix},
 \end{equation}
 where $A(l_*)$, $B(l_*)$, $C$ coincide with $A(l_u)$, $B(l_u)$, $\varpi$ for
 $s\le 0$ and with $A(l_s)$, $B(l_s)$, $\gamma$ for $s\ge 0$, and similarly
 $g(y_1,\tau(\xi);l_*)$ equals $g(y_1,\frac{\ln(\xi)}{\varpi};l_u)$
 for $\xi \le 1$ (i.e. $s \le 0$) and
 $g(y_1,\frac{\ln(2-\xi)}{\varpi};l_s)$ for $\xi \ge 1$ (i.e. $s
 \ge 0$). Notice that \eqref{si.nastella} coincides with
 \eqref{si.naa} when $\xi \le 1$ (i.e. $s \le 0$) and it is
 equivalent to \eqref{si.naas} when $\xi \ge 1$ (i.e. $s \le 0$ and
 $\zeta \le 1$, it differs from \eqref{si.naas} just in the fact
 that $\xi=2-\zeta$). Further we recall that the unstable manifold
 $\mathbf{W}^u(l_*)$ defined in \eqref{manistella} has dimension $2$
 and connects the $\xi$-axis and the graph of $\mathbf{y^{*}}(s,l_*)$;
 Further it is a graph on the $y_2=0$ plane, see
 Remark~\ref{ord3-bis}.
 Let us denote 
 \[
 E=\{(y_1,y_2,\xi) : 0<y_1<y_1^u(\tau(\xi),l_*) \,, \;
 0<\xi< 2 \}
 \]
 and by $E_0:=\mathbf{W}^u(l_*)$. It follows that $E_0 \subset E$ and $E_0$ splits $E$
 in $2$ open components, say $E^+$ and $E^-$ (the one with larger
 and smaller $y_2$).

 By construction the flow of the modified system
 \eqref{si.nastella} where $g$ is replaced respectively by
 $\overline{g}^{k}$ and by $\underline{g}^{k}$ on $\mathbf{W}^u(l_*)$ points
 towards $E^-$ and $E^+$ respectively for $s \le 0$, and it is
 tangent to $E^0$ for $s \ge 0$. So the corresponding manifolds
 $\mathbf{\overline{W}^{u,(k)}}(l_*)$ and $\mathbf{\underline{W}^{u,(k)}}(l_*)$ lie
 respectively in $E^-$ and $E^+$.

 Now assume $U(R,d)=\overline{U}^{(k)}(R,c)$ and consider the
 corresponding trajectories $\mathbf{y}(s;l_*)$, and $\mathbf{\overline{y}^{(k)}
 }(s;l_*)$: Then $y_{1}(\ln(R); l_*)=\overline{y}^{(k)}_{1}(\ln(R);l_*)$
 and $y_{2}(\ln(R); l_*) \ge \overline{y}^{(k)}_2(\ln(R);l_*)$. Hence
 $y_{1}(s; l_*)\ge \overline{y}^{(k)}_{1}(s;l_*)$ for $s$ in a right
 neighborhood of $\ln(R)$. Then the claim in Step 1 concerning
 $\overline{U}^{(k)}(r,c)$ follows. The claim concerning
 $\underline{U}^{(k)}(r,e)$ is analogous.

 We continue the discussion for later purposes. We know that
 $y_{2}(\ln(R); l_*) \ge \overline{y}^{(k)}_2(\ln(R);l_*)$, assume first
 $y_{2}(\ln(R); l_*) > \overline{y}^{(k)}_2(\ln(R);l_*)$. Then $y_{1}(s;
 l_*) > \overline{y}^{(k)}_1(s;l_*)$ for $s$ in a right neighborhood of
 $\ln(R)$.

 Assume now $y_{2}(\ln(R); l_*) = \overline{y}^{(k)}_2(\ln(R);l_*)$: Then
 $R \ge 1$. In fact assume for contradiction that $0<R<1$, then
 $\mathbf{\overline{y} }( \ln(R);l_*)=\mathbf{Q}=\mathbf{\overline{y}^{(k)} }( \ln(R);l_*)$
 is such that $(\mathbf{Q},\xi(\ln(R))\in E^0$, but from \eqref{si.na}
 we obtain $\dot{\overline{y}}_{2}(\ln(R); l_*) < \dot{y}_2^{(k)}(\ln(R);l_*)$.
 Hence $\mathbf{\overline{y}^{(k)} }( r;l_*)$
 crosses transversally $E^0$ at $s=\ln(R)$, going from $E^+$ to
 $E^-$, in particular it is in $E^+$ when $s$ is in a sufficiently
 small left neighborhood of $\ln(R)$. But
 $(\mathbf{\overline{y}^{(k)}}(s;l_*), \xi(s)) \in
 \mathbf{\overline{W}^{u,(k)}}(l_*)\subset E^-$, and this is a contradiction,
 so $R>1$.

 Observe that if $R \ge 1$ then $\mathbf{\overline{y}^{(k)}}(s;l_*)$ and
 $\mathbf{y}(s;l_*)$ are solutions of the same equation \eqref{si.na}
 for $s \ge 0$ which coincide for $s=\ln(R)$, so they coincide for
 $s \ge 0$.

 We have already proved the following result, i.e.
 \smallskip

 \noindent \textbf{Step 2.} For any $0<r<1$ we have that
 \begin{equation}\label{ineq}
 \overline{U}^{(k)}(r,d) < U(r,d) < \underline{U}^{(k)}(r,d)
 \end{equation}
 and either \eqref{ineq} holds for any $r>0$ or the functions
 coincide for any $r \ge 1$. Moreover $\overline{\mathcal{A}}^{(k)}(d)
 \le \mathcal{A}(d) \le \underline{\mathcal{A}}^{(k)}(d)$.
 \smallskip

\noindent \textbf{Step 3.} Fix $d$ and the corresponding
 coefficient $\mathcal{A}(d)$. It is possible to choose $c^k\le
 d \le e^k$ so that $\underline{\mathcal{A}}^{(k)}(e^k)=
 \overline{\mathcal{A}}^{(k)}(c^k)=\mathcal{A}(d)$.

 Fix $\tau >0$ and $0<c<d<e$; let $\mathbf{y}(s,\tau,\mathbf{P}; l_s)$,
 $\mathbf{y}(s,\tau,\mathbf{Q}; l_s)$, $\mathbf{y}(s,\tau,\mathbf{R}; l_s)$ be the
 trajectories of \eqref{si.na} corresponding to the solutions
 $U(r,c)$, $U(r,d)$, $U(r,e)$ of \eqref{radsta}. It follows that
 $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ are points in $W^u(\tau,l_s)$ and $\mathbf{P}$,
 $\mathbf{R}$ are respectively the closest to and the farthest from the
 origin. Let us consider the lines $\ell^l, \ell^r$ parallel to
 the $y_2$-axis and passing through $\mathbf{P}$ and $\mathbf{R}$
 respectively: We denote by $\overline{\mathbf{P}^{(k)}}$ and
 $\overline{\mathbf{R}^{(k)}}$, the intersections of
 $\overline{W}^{u,(k)}(\tau,l_s)$ respectively with $\ell^l$ and with
 $\ell^r$. Using continuous dependence on initial data of ODE we
 see that $\overline{\mathbf{P}^{(k)}} \to \mathbf{P}$ and $\overline{\mathbf{R}^{(k)}} \to
 \mathbf{R}$ as $k \to \infty$. Since $a(\mathbf{Q})$ is continuous, see
 Remark \ref{defa} and \eqref{defAA}, we see that
 $a(\overline{\mathbf{P}^{(k)}}) \to
 a(\mathbf{P})=\mathcal{A}(c)<\mathcal{A}(d)$, while
 $a(\overline{\mathbf{R}^{(k)}}) \to
 a(\mathbf{R})=\mathcal{A}(e)>\mathcal{A}(d)$. Therefore we can choose
 $N$ large enough so that
 $a(\overline{\mathbf{P}^{(k)}})<\mathcal{A}(d)<a(\overline{\mathbf{R}^{(k)}})$ for any
 $k \ge N$. Hence we can find $\mathbf{\overline{Q}^{(k)}}\in
 \overline{W}^{u,(k)}(\tau,l_s)$ between $\overline{\mathbf{P}^{(k)}}$ and
 $\overline{\mathbf{R}^{(k)}}$ such that
 $a(\mathbf{\overline{Q}^{(k)}})=\mathcal{A}(d)$. Correspondingly we find
 $e^k$ such that
 $\overline{\mathcal{A}}^{(k)}(e^k)=a(\mathbf{\overline{Q}^{(k)}})=\mathcal{A}(d)$.
 Note that in view of \emph{Step 2} we have $e^k \ge d$. The proof
 for $\underline{\mathcal{A}}^{(k)}(c^k)$ is analogous.
\smallskip

 Then, putting together
 Step 1 and Step 3, we see that $U(r,d)$ is the unique solution
 of the original equation \eqref{radsta} such that
 \begin{equation}\label{eq:step3}
 \underline{U}^{(k)}(r,c^k) \le U(r,d) \le \overline{U}^{(k)} (r,e^k) , \,\,
 \text{for any $r \ge 0$}
 \end{equation}

\noindent\textbf{Step 4.} Formula \eqref{limit} and the
 following Remark hold.

 \begin{remark}\label{estimateL} \rm
 $\overline{\mathcal{B}}^{(k)}(e^k)$ and
 $\underline{\mathcal{B}}^{(k)}(c^k)$ are respectively strictly
 decreasing and increasing in $k$ and they both converge to
 $\mathcal{B}(d)$.
 \end{remark}

 \begin{proof}
 To prove \eqref{limit} it is sufficient to observe that, by
 construction, the functions $\underline{U}^k(r,c^k)$ and
 $\overline{U}^k(r,e^k)$ are bounded and monotonically respectively
 increasing and decreasing in $k$. Then, from standard
 elliptic estimates we see that they both converge to solutions of
 the original problem \eqref{radsta} as $k \to +\infty$.
 Then, from \emph{Step 3} we see that the limit
 of both $\underline{U}^k(r,c^k)$ and
 $\overline{U}^k(r,e^k)$ is the same solution $U(r,d)$ of the original problem
 \eqref{radsta}.

 Now, we consider Remark \ref{estimateL}. From Step 3 we know that
 $\underline{\mathcal{A}}^{(k)}(e^k)=\mathcal{\mathcal{A}}(d)=
 \overline{\mathcal{A}}^{(k)}(c^k)$.
 Further from the previous argument we also infer that
 $\overline{\mathcal{B}}^{(k)}(e^k)$ and
 $\underline{\mathcal{B}}^{(k)}(c^k)$ are respectively decreasing
 and increasing and converge to $\mathcal{B}(d)$. As next
 step, we show that $\underline{\mathcal{B}}^{(k)}(c^k)<
 \underline{\mathcal{B}}^{(k-1)}(c^{k-1})<\mathcal{B}(d)<
 \overline{\mathcal{B}}^{(k-1)}(e^{k-1})<\overline{\mathcal{B}}^{(k)}(e^k)$,
i.e.\ $\underline{\mathcal{B}}^{(k)}(c^k)$ and
 $\overline{\mathcal{B}}^{(k)}(e^k)$ are strictly increasing and decreasing.
 As usual we just prove the last inequality, the others being
 analogous. Let $\overline{u}^j(x)$ be the radial function defined
 by $\overline{u}^j(x)=\overline{U}^j(|x|,e^j)$. Observe that $\Delta
 [\overline{u}^k(x))-\overline{u}^{k-1}(x)] \le 0$, hence from standard
 arguments (see \cite[Theorem 3.8]{Ni}), we see that there is
 $C>0$ such that 
$\overline{U}^k(r,e^k)-\overline{U}^{k-1}(r,e^{k-1})> C r^{-(n-2)}$. Assume
 $\overline{\mathcal{B}}^{(k)}(e^k)=\overline{\mathcal{B}}^{(k-1)}(e^{k-1})$
 for contradiction. Since
 $\overline{\mathcal{A}}^{(k)}(e^k)=\mathcal{\mathcal{A}}(d)$ for
 any $k$, from the construction in Lemma \ref{asinteqna} it
 follows that $\mathbf{\overline{y}^{(k)}}(s, e^k;l_s) \equiv
 \mathbf{\overline{y}^{(k-1)}}(s, e^{k-1};l_s)$ for any $s \ge 0$, i.e.
 $\overline{U}^k(r,e^k)= \overline{U}^{k-1}(r,e^{k-1})$ for $r \ge 1$,
 but this is a contradiction and the Remark is proved.
 \end{proof}

 From Remark \ref{estimateL} we see that the inequalities in
 \eqref{eq:step3} are strict for $r$ large. Then, from
 \emph{Step 1} we conclude the proof of Proposition \ref{4.1}
 in the case $G(y_1,s) \not\equiv 0$. 

 Assume now $G(y_1,s) \equiv 0$, this is the case,
 e.g., when $f(u,r)= c u|u|^{q-2}$. Following \cite{GNW1} we
 denote by $\underline{f}^{(k)}(u,r):= [1- \mu h(r)/k]f(u,r)$
 and $\overline{f}^{(k)}(u,r):= [1+ \mu h(r)/k]f(u,r)$, for $k
 \in \mathbb{N}$ and where $\mu>0$ is chosen small enough so
 that $\underline{f}^{(1)}(u,r)$ satisfies (A6); then it is
 easy to check that $\underline{f}^{(k)}(u,r)$ and
 $\overline{f}^{(k)}(u,r)$ satisfy (A6) for any $k \in
 \mathbb{N}$. So Proposition~\ref{super} holds, and in all
 the $3$ cases all the regular solutions of \eqref{radsta},
 denoted respectively by $\underline{U}^{(k)}(r,\alpha)$,
 $U(r,\alpha)$, $\overline{U}^{(k)}(r,\alpha)$, are GSs, but a a
 priori they might not be ordered. However repeating the
 argument of \emph{Step 1} in \cite[Theorem 4.1]{GNW1}, it it
 easy to prove that
 \begin{equation}\label{pre-step-2}
 \overline{U}^{(k)}(r,\alpha) \le U(r,\alpha) \le
 \underline{U}^{(k)}(r,\alpha)
 \end{equation}
 for any $r >0$ and any $\alpha > 0$. The proof might be
 concluded arguing as in \cite[Theorem 4.1]{GNW1}. However
 notice that we can also repeat the argument at the end of
 Step 1 of this proof to get \eqref{ineq} for any $r>0$,
 and then carry on through Step 2,3,4, of this proof and
 conclude also in this case, with no further changes.
 \end{proof}


 \subsection{Proof of the weak asymptotic stability}
 
 Now we consider $d>0$ fixed, and we use the shorthand notation
 $\overline{U}^{(1)}(r,e^1)= \overline{U}(r)$,
 $\underline{U}^{(1)}(r,c^1)= \underline{U}(r)$,
 $\overline{u}(t,x)=u(t,x; \overline{U}(|x|))$,
 $\underline{u}(t,x)=u(t,x; \underline{U}(|x|))$.

 \begin{lemma}\label{forteasintotico}
 Under the hypotheses of Theorem \ref{stabile}, we have
 $\overline{u}(t,x) \searrow U(|x|,d)$ and $\underline{u}(t,x)
 \nearrow U(|x|,d)$ as $t \to +\infty$, with the norm $\| \cdot
 \|_{ l}$, for any $0 \le l<m+|\lambda_2|$.
 \end{lemma}
 Notice that if $l_s=\sigma^*$ then $\| \cdot \|_{ m+|\lambda_1|}=\|
 \cdot \|_{ m+|\lambda_2|}$.
 \begin{proof}

 Let us set $B:=\lim_{|x| \to
 +\infty}[\overline{U}(|x|)-\underline{U}(|x|)]|x|^{m+|\lambda_2|}$
 and notice that $B>0$ is finite, see Proposition \ref{4.1} and
 Remark \ref{estimateL}. Fix $0 \le l<m+|\lambda_2|$ and observe that
 for any $\varepsilon>0$ we can find $\rho>0$ such that
 \begin{equation}\label{Due}
 [\overline{U}(|x|)-\underline{U}(|x|)]|x|^{l}<2 B |x|^{l-m-|\lambda_2|}< \varepsilon/2
 \end{equation}
 for $\|x \| \ge \rho$.

 Since $\overline{U}(|x|)$ and $\underline{U}(|x|)$ are
 respectively a radial super- and sub-solution of \eqref{laplace},
 then $\overline{u}(t,x)$ and $\underline{u}(t,x)$ are respectively
 radially symmetric super- and sub-solution of \eqref{parab}.
 Further they are resp. monotone decreasing and increasing in $t$,
 so they converge to a radial solution of \eqref{laplace}, see
 Lemma \ref{keyintro}. From Lemma \ref{4.1} we know that $U(r,d)$
 is the unique solution of \eqref{radsta} between $\overline{U}(r)$
 and $\underline{U}(r)$, so $\overline{u}(t,x)$ and
 $\underline{u}(t,x)$ converge monotonically to $U(|x|,d)$ as $t
 \to +\infty$, for any fixed $x \in \mathbb R^n$. Then, from the
 equiboundedness of the functions involved and of their derivatives
 we see that the convergence is uniform in any ball of radius $R>0$
 fixed. Hence setting $R=\rho>0$, for any $\varepsilon>0$ we find
 $T(\varepsilon)>0$ such that
 \begin{equation}\label{Uno}
 [\overline{u}(x,t)-\underline{u}(x,t)] |x|^l \le \varepsilon/2
 \end{equation}
 for any $|x| \le \rho$. Further from \eqref{Due} and the
 comparison principle we easily find that
 \begin{equation}\label{Tre} 
[\overline{u}(x,t)-\underline{u}(x,t)]
 |x|^l
 \le [\overline{U}(|x|)-\underline{U}(|x|)]|x|^{l} \le \varepsilon/2
 \end{equation}
 for $|x| \ge \rho$. Hence the Lemma follows from \eqref{Uno} and
 \eqref{Tre}.
 \end{proof}

 \begin{proof}[Proof of Theorem~\ref{asint.stabile}]
 Assume for definiteness $l_s>\sigma^*$, the case 
$l_s \ge \sigma^*$ being analogous. Fix $d>0$ and denote
 \begin{equation}\label{www}
 \begin{gathered}
 \overline{W}(r,d)= [\overline{U}(r)-U(r,d)] (1+r^{m+|\lambda_2|})\,, 
 \quad  \overline{\delta}= \inf_{r>0}\overline{W}(r,d) \\ 
\underline{W}(r,d)= [U(r,d)-\underline{U}(r)] (1+r^{m+|\lambda_2|}) \,, 
 \quad  \underline{\delta}= \inf_{r>0}\underline{W}(r,d)
 \end{gathered}
 \end{equation}
Observe that $\overline{W}(r,d)$, $\underline{W}(r,d)$ are both
 positive for any $r>0$, see Proposition \ref{4.1}. Further
 $\overline{W}(0,d)= e^1-d>0$, $\underline{W}(0,d)= d-c^1>0$,
 $\lim_{r\to +\infty} \overline{W}(r,d)= \overline{B}^{(1)}(e^1)-B(d)>0$, $\lim_{r\to +\infty}
 \underline{W}(r,d)=B(d)-\underline{B}^{(1)}(c^1)>0$, see Remark
 \ref{estimateL}. It follows that $\delta= \min \{
 \overline{\delta}, \underline{\delta} \}>0$.

Now let us consider $\phi$ such that 
$\|\phi-U(| \cdot |, d)\|_{m+|\lambda_2|} < \delta$: by construction we have
 $\underline{U}(|x|) \le \phi(x) \le \overline{U}(|x|)$, for any
 $x \in \mathbb R^n$. Therefore
 $$ 
\underline{u}(t,x) \le u(t,x ; \phi) \le \overline{u}(t,x)
$$
for any $t>0$ and any $x \in \mathbb R^n$. So from Lemma
 \ref{forteasintotico} we easily complete the proof.
 \end{proof}

 \subsection*{Acknowledgments} 
The authors are members of the Gruppo Nazionale per l'Ana\-lisi Matematica,
la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto
Nazionale di Alta Matematica (INdAM).

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\end{document}