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\markboth{ Behavior of positive radial solutions }
{ Marta Garc\'{\i}a-Huidobro}
\begin{document}
\setcounter{page}{173}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 173--187.\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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Behavior of positive radial solutions of a quasilinear equation
with a weighted Laplacian
%
\thanks{ {\em Mathematics Subject Classifications: 34B16.}
\hfil\break\indent
{\em Key words:} weighted Laplacian, singular solution, fundamental
solution.
\hfil\break\indent
\copyright 2001 Southwest Texas State University.
\hfil\break\indent Published January 8, 2001. \hfil\break\indent
Supported by FONDECYT grant 1990428. } }
\date{}
\author{ Marta Garc\'{\i}a-Huidobro }
\maketitle
\begin{abstract}
We obtain a classification result for positive radially
symmetric solutions of the semilinear equation
$$
-\mathop{\rm div}(\tilde a(|x|)\nabla u)=\tilde b(|x|)|u|^{\delta-1}u,
$$
on a punctured ball. The weight functions $\tilde a$ and $\tilde b$ are
$C^1$ on the punctured ball, are positive and measurable almost everywhere,
and satisfy certain growth conditions near zero.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
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\section{Introduction}
In this work we study the behavior of positive solutions to
\begin{equation}
-\mathop{\rm div}(\tilde a(|x|)\nabla u)=\tilde b(|x|)|u|^{\delta-1}u,
\quad x\in B_{r_0}^*(0),\quad r_0>0,\label{P}
\end{equation}
near an isolated singularity at the origin. Here $\delta>1$,
$B_{r_0}^*(0)$ is the
punctured ball $B_{r_0}(0)\setminus\{0\}$, and $\tilde a, \tilde b$
are weight functions which are positive and measurable.
Many authors have dealt with the non weighted case, i.e., with
positive solutions to the equation
\begin{equation}
-\mathop{\rm div}(\nabla
u)=|u|^{\delta-1}u, \mbox{in }\Omega\subseteq {\mathbb R}^N,\label{E}
\end{equation}
where $\delta>1$, $2\le N$, and $\Omega\subseteq {\mathbb R}^N$ or
$\Omega\subseteq {\mathbb R}^N\setminus\{0\}$ is a smooth
domain, bounded or unbounded.
When $N>2$, appear two critical values:
$\delta=\frac{N}{N-2}$, and $\delta= \frac{N+2}{N-2}$.
The first results for this case were obtained by Emden, and then
Fowler \cite{fow1,fow2, fow3}, where existence results are given as
well as a complete classification of
the global solutions in ${\mathbb R}^N$ or ${\mathbb R}^N\setminus\{0\}$,
in
the radial situation.
Lions \cite{L} studied the nonradial case for the behavior near 0 when
$\delta1$, and let the weight
functions $a$, $b$ satisfy (H1), (H2), (H3), and
\begin{description}
\item{\bf (H4)}
There exists a finite number $\tilde \delta > 1$, such
that $ \int_0b(r)(h(r))^{\tilde\delta}\,dr <\infty$.
\end{description}
Let $u$ be a positive singular solution to (\ref{Pr}). Then, there exists a
positive extended real number ${\mathcal S}$ (which we call the
generalized Serrin's number) such that
\begin{enumerate}
\item[(i)] If $1<\delta<{\mathcal S}$, then
$$\lim_{r\to 0^+}\frac{u(r)}{h(r)}>0.
$$
Also, if $\delta>{\mathcal S}$, then $u$ cannot be of fundamental type,
i.e.,
$$\lim_{r\to 0^+}\frac{u(r)}{h(r)}=0.
$$
\end{enumerate}
Assume next that ${\mathcal S}<\infty$.
\begin{enumerate}
\item[(ii)] If
$\int_0b(r)(h(r))^{{\mathcal S}}dr<\infty$,
then $u$ is also of fundamental type when
$\delta={\mathcal S}$.
\end{enumerate}
Assume further that there exist positive constants $c_1$ and $c_2$
such that
\begin{equation}\label{continuous-imbe}
c_1\le B(r)(h(r))^{{\mathcal S}}\le c_2\quad\mbox{for all $r\in(0,r_0)$},
\end{equation}
and the mapping
\begin{equation}\label{monotone-cond}
r\mapsto \frac{b(r)(h(r))^{{\mathcal S}+1}}{|h'(r)|}\quad\mbox{is monotone
in $(0,r_0)$}.
\end{equation}
\begin{enumerate}
\item[(iii)]
Then for ${\mathcal S}<\delta$ and $\delta\not=2{\mathcal S}-1$,
it holds that
$$\lim_{r\to 0^+}\frac{u(r)}{h(r)^{\frac{{\mathcal S}-1}{\delta-1}}}>0.
$$
\end{enumerate}
\end{theorem}
\paragraph{Remark 1.2} We recall that in the non weighted case, i.e.,
the case when
$a(r)=b(r)=r^{N-1}$, $N>2$, we have
$$h(r)=\int_r^1 s^{1-N}ds=r^{2-N}\Bigl(\frac{1-r^{N-2}}{N-2}\Bigr),\quad
B(r)=\frac{r^N}{N},\quad \mbox{
and }{\mathcal S}=\frac{N}{N-2}.$$
Also, in this case
$$B(r)(h(r))^{{\mathcal
S}}=\frac{1}{N}\Bigl(\frac{1-r^{N-2}}{N-2}\Bigr)^{\frac{N}{N-2}},\quad
\frac{b(r)(h(r))^{{\mathcal S}+1}}{|h'(r)|}=
\Bigl(\frac{1-r^{N-2}}{N-2}\Bigr)^{\frac{N}{N-2}+1}.$$ Thus
our assumptions (\ref{continuous-imbe}) and (\ref{monotone-cond}) are
satisfied
in that case. In fact, it can be easily shown that these assumptions
always hold when
the weights are powers near the origin.
\paragraph{Remark 1.3} We note here, that as a first striking difference
with the non
weighted case, the solutions can behave like the fundamental solution at
the critical
number ${\mathcal S}$, see example 1 in section \ref{examples}.
\paragraph{Remark 1.4} As it is stated in the theorem, the number
${\mathcal S}$ can be infinity. This of course happens in the non
weighted case when $N=2$. Nevertheless, in
this more general case, it can happen in different
situations, see example 2 in section \ref{examples}.
\medskip
To prove parts (i) and (ii) of the theorem, as in \cite{gmy}, we think of
the critical number ${\mathcal S}$ as the limiting value of $\delta$ so
that a singular solution
behaves like the fundamental solution. (We can show that thanks to
assumption $(H_4)$,
there exists at least one value of $\delta$ with that property). Then, we
make an
appropriate change of variable, (which corresponds to the one used in
\cite{gv} in the
non weighted case) to study the behavior when it is not of the fundamental
type.
The organization of this paper is as follows. In section
\ref{preliminaries} we prove
some preliminary results concerning a priori bounds for the positive
solutions to
(\ref{Pr}), some of them can also be found in \cite{cm}, where the authors
establish
nonexistence results for an equation containing a more general
non-homogeneous operator.
In section \ref{defpserrin-(i)-(ii)}, we find the critical number
${\mathcal S}$ and we
prove parts (i) and (ii) of Theorem \ref{mainclass}. Then in section
\ref{proof-(iv)}
we prove part (iii). We do this by following the idea in \cite{bellman}
and a regularity result proved in \cite{gmy}, see also \cite{gkmy}. Finally
in section
\ref{examples} we give some examples to illustrate the main differences
with respect to
the non weighted case.
\section{Preliminary Results}\label{preliminaries}
We start this section by proving some basic facts concerning positive
solutions to (\ref{Pr}). As we pointed out in the introduction, if $u$ is a
positive singular solution to (\ref{Pr}), then $u'(r)< 0$ for
$r\in(0,r_0)$,
$\lim_{r\to 0}a(r)|u'(r)|=\ell$ exists and $\ell\ge 0$.
Therefore, by L'Hospital's rule, also $\lim_{r\to 0}u(r)/h(r)$ exists (and
it is equal to $\ell$). Moreover, we will prove next that $u/h$ is in
fact monotone increasing in some right neighborhood of zero (see also
\cite{cm}).
\begin{lemma}\label{u/hmonotone}
Let the weight functions $a, b$ satisfy assumptions (H1), (H2), and (H3),
and let
$u$ be a positive singular solution to (\ref{Pr}) such that
$$\lim_{r\to 0}a(r)|u'(r)|=0.$$ Then, there exists $r_*\in(0,r_0)$
such that $u/h$ is monotone increasing on $(0,r_*)$.
\end{lemma}
\paragraph{Proof.}
The result follows easily by making the change of variable
$$s=\frac{1}{h(r)},\quad v(s):=su(r).$$
We observe that $v$ turns out to be concave with $v(0)=0$, and
thus, since it is a positive function, it has to be increasing near zero.
\hfill$\Box$
Next we find an a-priori estimate for the growth of $u$ near zero. We
have.
\begin{lemma}\label{ubounds}
Let the weight functions $a, b$ satisfy assumptions (H1), (H2), and (H3),
and let
$u$ be a positive singular solution to (\ref{Pr}) such that
$\lim\limits_{r\to
0}a(r)|u'(r)|=0.$ Then
\begin{equation}\label{a-prioriest1}
u^{\delta-1}(r)\le (B(r))^{-1}(h(r))^{-1}\quad\mbox{for all }r\in(0,r_*),
\end{equation}
where $r_*$ is given in Lemma \ref{u/hmonotone}.
\end{lemma}
\paragraph{Proof.}
Let $u$ be a positive singular solution to (\ref{Pr}). By Lemma
\ref{u/hmonotone}, we have that $|u'|\le |h'|(u/h)$ on $(0,r_*)$, and thus,
using that
$u$ is decreasing on $(0,r_0)$, we find that
$$a(r)|h'|\frac{u(r)}{h(r)}\ge
a(r)|u'|=\int_0^rb(t)u^{\delta}(t)dt\ge B(r)u^{\delta}(r),
$$
and the result follows by observing that $a(r)|h'|\equiv 1$. \hfill$\Box$
\paragraph{Remark 2.1} Note that in the non weighted case this last lemma
establishes the well known result
$$u(r)\le C\ r^{\frac{-2}{\delta-1}}\quad\mbox{for small $r>0$.}$$
We finish this section by recalling a regularity result from \cite{gkmy}.
\begin{lemma}\label{supersollema}
Assume that the weight functions $a, b$ satisfy assumptions
(H1)-(H4) and let $1< \delta$ be such that (\ref{continuous-imbe})
holds. Moreover, assume that there exists a nonnegative
function $\nu\in C^1(0,r_0)$ such that $a\nu^{\delta} \in L^1(0,r_0)$
and
\begin{equation}\label{supersol1}
-a(r)\nu'(r)\ge K\int_0^rb(t)(\nu(t))^{\delta}dt,\quad
r\in(0,\overline{r}_0),
\end{equation}
for some positive constant $K$, and some $\overline{r}_0\in (0, r_0).$
Let $u$ be a positive solution to the equation in (\ref{Pr}) which is
defined
on a right neighborhood of 0
and satisfies
\begin{equation} \label{opeque}
\lim_{r\to 0}\frac{u(r)}{\nu(r)}=0.
\end{equation}
Then, $u$ is a bounded solution.
\end{lemma}
\paragraph{Proof.}
The proof is rather technical and it consists in proving that there exist
an interval $(0,r_*)$, a positive constant $C$, and a sequence
$\{\epsilon_n\}$ tending to 0 as $n\to\infty$, such that
$$u(r)\le \epsilon_n \nu(r)+C\quad \mbox{for all $r\in(0,r_*)$},$$
from where the result follows by letting $n\to\infty$. Since it is lengthy
and a
similar version can be found also in \cite{gmy}, where the non weighted
case,
but for a non-homogeneous operator is treated, we omit it.
\section{Definition of ${\mathcal S}$ and proof
of Theorem \ref{mainclass}} \label{defpserrin-(i)-(ii)}
We first observe that a necessary condition for a positive singular
solution to (\ref{Pr}) to behave like the fundamental solution $u$ is that
\begin{equation}\label{neccond}
\int_0b(t)(h(t))^{\delta}dt<\infty.
\end{equation}
Indeed, this comes from the fact that
$$a(r)|u'(r)|\ge\int_0^r b(t)u^{\delta}(t)dt,
$$
and thus, if $u(r)\ge Ch(r)$ for $r$ small enough, then
(\ref{neccond}) follows.
Let us set
$${\mathcal W}:=\{\delta>1\ :\ \int_0b(t)(h(t))^{\delta}dt<\infty\}.$$
Thanks to hypothesis $(H_4)$, we have that
${\mathcal W}\not=\emptyset$,
and thus we may define
\begin{equation}\label{serrin}
{\mathcal S}:=\sup{\mathcal W}.
\end{equation}
\subsection*{Proof of Theorem \ref{mainclass} parts (i)-(ii).}
Let ${\mathcal S}$ be defined as in (\ref{serrin}). Since
$\lim_{r\to0}\frac{u(r)}{h(r)}$ exists,
we only have to prove that if $u$ is
a positive singular solution to (\ref{Pr}), then this limit cannot be 0. We
will argue by
contradiction and thus assume that $a(r)|u'(r)|\to 0$ as $r\to 0$. Then, by
lemma
\ref{u/hmonotone} there is $r_*\in(0,r_0)$ such that $u/h$ is increasing on
$(0,r_*)$.
Let us first prove $(i)$, i.e., assume $\delta\in(1,{\mathcal S})$. Then by
the definition of ${\mathcal S}$, it
holds that
$\int_0 b(t)h^{\delta}(t)dt<\infty$ and thus, given $\varepsilon>0$, there
exists $r_1\in(0,r_*)$ such that
\begin{equation}\label{choice1}
\int_0^rb(t)h^{\delta}(t)dt<\varepsilon\quad\mbox{for all }r\in(0,r_1),
\end{equation}
and since we are assuming that $u/h\to 0$ as $r\to 0$, we may assume that
$r_1$ is small enough to have
\begin{equation}\label{choice2}
\frac{u(r_1)}{h(r_1)}<\varepsilon^{1-\delta}.
\end{equation}
Then, by the monotonicity of $u/h$ and (\ref{choice1}) we have that
$$a(r)|u'(r)|=\int_0^rb(t)h^{\delta}(t)\Bigl(\frac{u(t)}{h(t)}\Bigr)^{\delta}dt
< \Bigl(\frac{u(r)}{h(r)}\Bigr)^{\delta}\varepsilon,$$
implying that
$$|u'(r)|u^{-\delta}(r)\le
\varepsilon|h'(r)|h^{-\delta}(r).$$
By integrating this inequality over $(r,r_1)$, we find that
$$u^{1-\delta}(r_1)-u^{1-\delta}(r)<
\varepsilon h^{1-\delta}(r_1),\quad r\in(0,r_1),$$
which is equivalent to
$$u^{1-\delta}(r_1)-\varepsilon h^{1-\delta}(r_1)<
u^{1-\delta}(r),\quad r\in(0,r_1).$$
Hence, from (\ref{choice2}) we deduce that
$$u^{\delta-1}(r)<
(u^{1-\delta}(r_1)-\varepsilon h^{1-\delta}(r_1))^{-1},$$
contradicting the unboundedness of $u$ near 0.
Hence, we must have that
$$\lim_{r\to 0^+}\frac{u(r)}{h(r)}>0.
$$
Next we observe that as we mentioned at the beginning of this section, a
necessary
condition for $u$ to behave like the fundamental solution near 0 is that
$\int_0b(t)h^{\delta}(t)dt<\infty$ and thus $\delta\le {\mathcal S}$, i.e.,
if $\delta>{\mathcal S}$,
then $\lim_{r\to 0}(u/h)(r)=0$.
To prove (ii), we note that the assumption is equivalent to saying
${\mathcal S}\in{\mathcal W}$, and
thus the result follows in the same way as above.
\section{Proof of Theorem \ref{mainclass} part {\rm(iii)}}
\label{proof-(iv)}
In this section we treat the case $\delta>{\mathcal S}$. We use a similar
argument to the one used
in \cite{bellman}. We do this by considering the following
change of variable
$$t=\mu\log(h(r)),\quad v(t)=\frac{u(r)}{(h(r))^{\frac{{\mathcal
S}-1}{\delta-1}}},$$
where $\mu$ could be any positive constant but will be chosen later to
compare with the
non weighted case.
\paragraph{Proof of {\rm(iii)}.}
>From (\ref{a-prioriest1}) in lemma \ref{ubounds}, and the assumption
$c_1\le
B(r)h^{{\mathcal S}}(r)$ for all $r$ sufficiently small in
(\ref{continuous-imbe}), we have that
$$u(r)\le C h^{\frac{{\mathcal S}-1}{\delta-1}}(r) \quad \mbox{for $r$
small enough},$$
and hence $v$ is bounded by an absolute constant not depending on $u$.
Also, it can be readily verified that
\begin{equation}\label{first-v-der}
\dot{v}+\frac{\theta}{\mu} v=\frac{u'(r)h^{(1-\theta)}(r)}{\mu h'(r)}>0,
\quad\mbox{for all $r\in(0,r_0)$},
\end{equation}
where
$\dot{ }=\frac{d}{dt}$, $'=\frac{d}{dr}$, and $\theta:= \frac{{\mathcal
S}-1}{\delta-1}$.
We conclude then, by using again the monotonicity of $u/h$, i.e., that
$\frac{|u'(r)|}{|h'(r)|}\le \frac{u(r)}{h(r)}$ for $r$ small, that
$$|\dot{v}(t)|\le \frac{\theta+1}{\mu} v(t)$$
and thus $|\dot{v}|$ is also bounded by an absolute constant.
Finally, by differentiating (\ref{first-v-der}) with respect to $r$ and
using the equation in
$(P_{r})$, we see that (\ref{Pr}) transforms into
$$\ddot{v}+(\frac{2\theta}{\mu}-\frac{1}{\mu})\dot
v-\frac{(1-\theta)\theta}{\mu^2}v=
-\frac{b(r)(h(r))^{{\mathcal S}+1}}{\mu^2|h'(r)|}v^{\delta},\quad t\ge
t_0.$$
To simplify our writing we will set $q=\frac{\theta}{\mu}$ and re-write
this
equation as
\begin{equation}
\ddot{v}+(2q-\frac{1}{\mu}) \dot v-(\frac{1}{\mu}-q)q v=
-\frac{b(r)(h(r))^{{\mathcal S}+1}}{\mu^2|h'(r)|}v^{\delta},\quad t\ge
t_0. \label{Pinfty}
\end{equation}
By multiplying the equation in $(P_{\infty})$ by $\dot v$, we find that
\begin{eqnarray*}
\lefteqn{ \frac{d}{dt}\frac{\dot v^2}{2}+(2q-\frac{1}{\mu})
\dot v^2-(\frac{1}{\mu}-q)q \frac{d}{dt} \frac{v^2}{2} } \\
&=&
-\frac{d}{dt}\frac{b(r)(h(r))^{{\mathcal S}+1}}
{\mu^2|h'(r)|}\frac{v^{\delta+1}}{\delta+1}+
\frac{v^{\delta+1}}{\delta+1}\frac{d}{dt}\frac{b(r)(h(r))^{{\mathcal S}+1}}
{\mu^2|h'(r)|},
\end{eqnarray*}
or equivalently,
\begin{eqnarray} \label{dotv2=}
(2q-\frac{1}{\mu}) \dot v^2
&=& \frac{d}{dt}\Bigl((\frac{1}{\mu}-q)q \frac{v^2}{2}-\frac{\dot v^2}{2}
-\frac{b(r)(h(r))^{{\mathcal
S}+1}}{\mu^2|h'(r)|}\frac{v^{\delta+1}}{\delta+1}\\
&&+\int_{t_0}^t\frac{v^{\delta+1}}{\delta+1}\frac{d}{ds}\frac{b(r)(h(r))
^{{\mathcal S}+1}}{\mu^2|h'(r)|}ds\Bigr).\nonumber
\end{eqnarray}
Note that
\begin{equation}\label{defA}
C:=2q-\frac{1}{\mu}=\frac{1}{\mu}\Bigl(\frac{2{\mathcal
S}-(\delta+1)}{\delta-1}\Bigr)\not=0
\end{equation}
by assumption.
We will prove next that $\frac{bh^{{\mathcal S}+1}}{|h'|}$ is bounded.
Indeed, since by
assumption (\ref{monotone-cond}) this function is monotone, its limit as
$r\to 0$ exists.
Let $\{r_n\}$ be a
sequence of positive numbers such that $r_n\to 0$ as $n\to\infty$ and such
that
\begin{equation}\label{liminf1}
\liminf_{r\to 0}\frac{b(r)h(r)}{B(r)|h'(r)|}=
\lim_{n\to\infty}\frac{b(r_n)h(r_n)}{B(r_n)|h'(r_n)|}.
\end{equation}
Then, by using assumption (\ref{continuous-imbe}), (\ref{liminf1}), and by
L'Hospital's rule,
we have that
\begin{eqnarray*}
\lim_{r\to 0}\frac{b(r)h^{{\mathcal S}+1}(r)}{|h'(r)|}
&=& \lim_{n\to\infty}\frac{b(r_n)h^{{\mathcal S}+1}(r_n)}{|h'(r_n)|} \\
&=& \lim_{n\to\infty}\frac{b(r_n)h(r_n)}{B(r_n)|h'(r_n)|}B(r_n)
h^{{\mathcal S}}(r_n)\\
&\le& c_2\lim_{n\to\infty}\frac{b(r_n)h(r_n)}{B(r_n)|h'(r_n)|}\\
&=& c_2\liminf_{r\to 0}\frac{b(r)h(r)}{B(r)|h'(r)|}\\
&\le& c_2 \liminf_{r\to 0}\frac{|\log(B(r))|}{\log(h(r))}.
\end{eqnarray*}
We claim that it must be that
\begin{equation}\label{liminfbdd}
\liminf_{r\to 0}\frac{|\log(B(r))|}{\log(h(r))}<\infty.
\end{equation}
Indeed, assume on the contrary that this $\liminf$ is equal to $\infty$.
Then, given any $M>0$, and in particular $M>{\mathcal S}$, there is
$r_*>0$
such that $|\log B(r)|>\log h^M(r)$ for all $r\in(0,r^*)$, hence
$B(r)h^M(r)\le 1$ for all $r\in(0,r^*)$. But from the left hand side
inequality in
(\ref{continuous-imbe}), we conclude that
$$c_1(h(r))^{M-{\mathcal S}}\le 1\quad \mbox{for all $r\in(0,r^*)$},$$
contradicting (H3).
Thus we find that there is a positive constant $K$ such that
\begin{equation}\label{quotientbounded}
\frac{b(r)h^{{\mathcal S}+1}(r)}{|h'(r)|}\le K <\infty
\quad \mbox{for $r$ small
enough},
\end{equation}
proving our assertion and implying in particular that
\begin{equation}\label{pedazo}
(\frac{1}{\mu}-q)q \frac{v^2}{2}-\frac{\dot v^2}{2}-
\frac{b(r)(h(r))^{{\mathcal S}+1}}{\mu^2|h'(r)|}\frac{v^{\delta+1}}
{\delta+1}
\end{equation}
is bounded.
Next we observe that also from (\ref{dotv2=}),
\begin{eqnarray}
F(t)&:=&(\frac{1}{\mu}-q)q \frac{v^2}{2}-\frac{\dot v^2}{2}-
\frac{b(r)(h(r))^{{\mathcal S}+1}}{\mu^2|h'(r)|}
\frac{v^{\delta+1}}{\delta+1} \nonumber \\
&&+\int_{t_0}^t
\frac{v^{\delta+1}}{\delta+1}\frac{d}{ds}
\frac{b(r)(h(r))^{{\mathcal S}+1}}{\mu^2|h'(r)|}\,ds \label{finitelimit}
\end{eqnarray}
is monotone
(increasing or decreasing according to whether $C$ is negative or
positive)
and thus it has a limit as $t\to\infty$.
We will prove next that this limit is finite.
Clearly, from (\ref{pedazo}), we only have to establish the
convergence of the integral
$$\int_{t_0}^t
\frac{v^{\delta+1}}{\delta+1}\frac{d}{ds}\frac{b(r)(h(r))^{{\mathcal
S}+1}}{\mu^2|h'(r)|}ds.$$
But this follows directly from (\ref{monotone-cond}), the boundedness
of $v$, and
the monotonicity of the change of variables $r=r(t)$, hence we conclude
that
\begin{equation}\label{dotvL2}
|\dot{v}|^{2}\in L^1(t_0,\infty).
\end{equation}
Finally, we will prove that
\begin{equation}\label{dotvto0}
\lim_{t\to\infty}\dot{v}(t)=0.
\end{equation}
From
(\ref{Pinfty}) and the boundedness of $v$ and $\dot v$, we have
that $|\ddot{v}|$ is bounded and thus (\ref{dotvto0}) easily follows from
(\ref{dotvL2}).
We conclude then from the existence of the limit of $F$ defined in
(\ref{finitelimit})
that $\lim_{t\to\infty}v(t)$ exists. It only
remains to prove that this limit cannot be zero. To this end, we will
prove that
due to the assumption $\delta>{\mathcal S}$, the
function
$$\nu(r)=h^{\frac{{\mathcal S}-1}{\delta-1}}(r)$$
satisfies (\ref{supersol1}), and thus by Lemma \ref{supersollema}, if
$\lim_{t\to\infty}v(t)=0$,
then $u$ is bounded, a contradiction.
Indeed,
$$a(r)|\nu'(r)|=\theta h^{\theta-1},$$
where as before, $\theta=\frac{{\mathcal S}-1}{\delta-1}$, and thus
$$( a(r)|\nu'(r)|)'= - \theta (\theta-1)
h^{\theta-1}\frac{|h'(r)|}{h(r)}.
$$
Hence, from L'Hospital's rule and using that since $\delta>{\mathcal S}$,
it is $\theta-1<0$,
we have that
$$\liminf_{r\to 0}\frac{a(r)|\nu'(r)|}{\int_0^rb(t)(\nu(t))^{\delta}dt}\ge
\theta(1-\theta)\liminf_{r\to 0}\frac{|h'(r)|}{b(r)h^{{\mathcal S}+1}},
$$
and the result follows from (\ref{quotientbounded}).
\hfill$\Box$
We end this section with a partial result concerning the case
$\delta={\mathcal S}$. This
case, as well as the subcritical and supercritical case for the $p$-Laplace
operator
is treated in detail in a forthcoming paper, see \cite{bvgh}.
\begin{prop}\label{critical}
Let $a, b$ satisfy assumptions (H1)-$(H_4)$, and assume that
(\ref{continuous-imbe})
holds. Let $u$ be a positive singular solution to (\ref{Pr}) with
$\delta={\mathcal S}$, and
suppose that $\int_0b(t)(h(t))^{\mathcal S}dt=\infty$. Then, there is $\bar
r_0>0$ and
$C>0$ such that
$$u(r)\le Ch(r)(\log(h(r)))^{-1/({\mathcal S}-1)}\quad\mbox{for all
$r\in(0,\bar
r_0)$}.$$
\end{prop}
\paragraph{Proof.}
Since the convergence of the integral $\int_0b(t)(h(t))^{\mathcal S}dt$
is a necessary
condition to have $\lim\limits_{r\to 0}u(r)/h(r) >0$, we have that in this
case
$\lim\limits_{r\to 0}a(r)|u'(r)|=0$. Hence, by Lemma \ref{u/hmonotone}
we have that
$u/h$ is monotone increasing near 0, that is,
$$\frac{|u'(r)|}{|h'(r)|}\le \frac{u(r)}{h(r)}\quad\mbox{for $r$
sufficiently small.}$$
Hence, from the left hand side inequality in (\ref{continuous-imbe}),
we have
\begin{eqnarray*}
(a(r)|u'(r)|)'&\ge& b(r)\Bigl(\frac{u(r)}{h(r)}\Bigr)^{\mathcal
S}(h(r))^{\mathcal S}\\
&\ge& b(r)\Bigl(\frac{|u'(r)|}{|h'(r)|}\Bigr)^{\mathcal
S}(h(r))^{\mathcal S}\\
&\ge& c_0\frac{b(r)}{B(r)} \Bigl(\frac{|u'(r)|}{|h'(r)|}\Bigr)
^{\mathcal S},
\end{eqnarray*}
and thus, using that $a(r)|u'(r)|=\frac{|u'(r)|}{|h'(r)|}$, we find that
$$ \Bigl(\frac{|u'(r)|}{|h'(r)|}\Bigr)'\ge
c_0\frac{b(r)}{B(r)} \Bigl(\!\frac{|u'(r)|}{|h'(r)|}\!\Bigr)^{\mathcal
S}$$
implying that
$$ \Bigl(\!\frac{|u'(r)|}{|h'(r)|}\!\Bigr)^{-\mathcal
S}\Bigl(\frac{|u'(r)|}{|h'(r)|}\Bigr)'\ge c_1\frac{b(r)}{B(r)} .$$
Integrating this last inequality over $(r,r_*)$, with $r_*$ sufficiently
small we
conclude that
$$ \Bigl(\!\frac{|u'(r)|}{|h'(r)|}\!\Bigr)^{\mathcal
S-1}\le \bar c_0|\log B(r)|^{-1}\quad \mbox{for all $r\in(0,r_*)$.}$$
Hence, for all $r\in(0,r_*)$, it holds that
$$u(r)\le u(r_*)+\bar c_0\int_r^{r_*}|h'(t)||\log
B(t)|^{\frac{-1}{{\mathcal S}-1}}dt\le
Kh(r)|\log B(r)|^{\frac{-1}{{\mathcal S}-1}}$$
for some positive constant $K$. The result follows now by using the right
hand side
inequality in (\ref{continuous-imbe}).
\section{Examples} \label{examples}
We finish this paper by giving some examples to illustrate our results as
well
as the main differences with the non-weighted case.
\paragraph{Example 1.}
We consider first a case for which ${\mathcal S}<\infty$ and it is such
that when
$\delta={\mathcal S}$, any
singular
solution behaves like the fundamental solution.
Let
$$a(r)=r^{N-1},\quad N>2,\quad
B(r)=r^{\theta(N-2)}(\log(r^{-1}))^{-2}\quad \mbox{near zero},\quad
\theta>1.$$
Then it can be easily verified that $h(r)=Cr^{2-N}$ near zero.
Also, if $\delta<{\mathcal S}$, we have that
$$\int_0^rb(t)(h(t))^\delta dt \ge B(r)(h(r))^\delta
=C^{\delta}r^{(\theta-\delta)(N-2)}(\log(r^{-1}))^{-2},$$
and
thus it follows that
${\mathcal S}\le\theta$.
Next, by integrating by parts we find that
\begin{eqnarray*}
\int_s^rb(t)(h(t))^\theta dt&\le & B(r)(h(r))^\theta +
\theta\int_s^tB(t)(h(t))^{\theta-1}|h'(t)|dt\\
&\le & B(r)(h(r))^\theta + C_1 \theta\int_s^r\Bigl(\log
(t^{-1})\Bigr)^{-2}\frac{1}{t}dt\\
&\le & B(r)(h(r))^\theta +C_1 \theta(\log(r^{-1})-\log(s^{-1})),
\end{eqnarray*}
where $C_1$ is some positive constant. Hence, ${\mathcal S}=\theta$ and
$\theta\in{\mathcal W}$, and the claim follows
from Theorem \ref{mainclass}(i)-(ii), that is, any radially symmetric
singular solution to
$$ -\Delta u=
|x|^{\theta(N-2)-N}\log|x|^{-1}(\theta(N-2)\log|x|^{-1}+2)|u|^{\delta-1}u,\quad
x\in
B_{r_0}^*(0),
$$
behaves like $|x|^{2-N}$ near zero for $1<\delta\le\theta$ and satisfies
$$\lim_{|x|\to 0}|x|^{N-2}u(x)=0\quad\mbox{for $\delta>\theta$}.$$
Next we give an example for which any positive singular solution is
of the fundamental type.
\paragraph{Example 2.}
Let $a(r)=r^{N-2}$, $N\ge 2$, and set
$ b(r)=r^{-\theta-1}e^{-1/r^{\theta}}$, $\theta>0$.
When $N>2$, the fundamental solution is $h(r)=Cr^{2-N}$, and clearly the
integral
$$\int_0r^{-\theta-1-L(N-2)}e^{-1/r^{\theta}}dr<\infty$$
for any $L>0$. Thus ${\mathcal S}=\infty$.
In the case that $N=2$, the fundamental solution is $h(r)=\log(r^{-1})$,
and we also have
that
$$\int_0r^{-\theta-1}e^{-1/r^{\theta}}(\log(r^{-1}))^Ldr\quad\mbox{converges
for any
$\theta>0$.}$$
Hence in this case we also have ${\mathcal S}=\infty$.
We conclude that any positive radially symmetric solution to
$$ -\Delta u=
|x|^{-\theta-2-N}\exp(|x|^{-\theta})|u|^{\delta-1}u,\quad x\in
B_{r_0}^*(0),
$$
behaves like the fundamental solution, for any $N\ge 2$.
Finally, we give a general example to which all our results apply.
\paragraph{Example 3.}
Let $m(r)$ be any continuous monotone function satisfying
$$m_0\le m(r)\le m_1\quad\mbox{for all }r\in[0,1],$$
let $a\in C^1(0,1)$ be a positive function such that
$1/a\not\in L^1(0,1)$,
and set
$$b(r):=m(r)\frac{(h(r))^{-L-1}}{a(r)}.$$
Then $a, b$ satisfy all the assumptions in Theorem \ref{mainclass} and it
can be easily
shown that ${\mathcal S}=L$. Indeed, let $1<\deltaL$, $\delta\not= 2L-1,$ then any positive
singular
solution to (\ref{ex3}) behaves like
$(h(r))^{\frac{L-1}{\delta-1}}.$
\item{\bf (iii)} If $\delta=L$, then
$$\lim_{r\to 0}\frac{u(r)}{h(r)}=0,\quad \mbox{and}\quad
u(r)\le Ch(r)(\log(h(r)))^{-1/({\mathcal S}-1)}$$
for all $r$ sufficiently small.
\end{description}
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\noindent{\sc Marta Garc\'{\i}a-Huidobro}\\
Departamento de Matem\'atica \\
Pontificia Universidad Cat\'olica de Chile,\\
Casilla 306, Correo 22, Santiago, Chile\\
e-mail: mgarcia@mat.puc.cl
\end{document}