\documentclass[twoside]{article} \usepackage{amsfonts} % used for R in Real numbers \pagestyle{myheadings} \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)} \vspace{\bigskipamount} \\ % 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}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} In this work we study the behavior of positive solutions to $$-\mathop{\rm div}(\tilde a(|x|)\nabla u)=\tilde b(|x|)|u|^{\delta-1}u, \quad x\in B_{r_0}^*(0),\quad r_0>0,\label{P}$$ 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 $$-\mathop{\rm div}(\nabla u)=|u|^{\delta-1}u, \mbox{in }\Omega\subseteq {\mathbb R}^N,\label{E}$$ 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 $$\label{continuous-imbe} c_1\le B(r)(h(r))^{{\mathcal S}}\le c_2\quad\mbox{for all r\in(0,r_0)},$$ and the mapping $$\label{monotone-cond} r\mapsto \frac{b(r)(h(r))^{{\mathcal S}+1}}{|h'(r)|}\quad\mbox{is monotone in (0,r_0)}.$$ \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 $$\label{a-prioriest1} u^{\delta-1}(r)\le (B(r))^{-1}(h(r))^{-1}\quad\mbox{for all }r\in(0,r_*),$$ 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 $$\label{supersol1} -a(r)\nu'(r)\ge K\int_0^rb(t)(\nu(t))^{\delta}dt,\quad r\in(0,\overline{r}_0),$$ 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 $$\label{opeque} \lim_{r\to 0}\frac{u(r)}{\nu(r)}=0.$$ 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 $$\label{neccond} \int_0b(t)(h(t))^{\delta}dt<\infty.$$ 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 $$\label{serrin} {\mathcal S}:=\sup{\mathcal W}.$$ \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 $$\label{choice1} \int_0^rb(t)h^{\delta}(t)dt<\varepsilon\quad\mbox{for all }r\in(0,r_1),$$ 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 $$\label{choice2} \frac{u(r_1)}{h(r_1)}<\varepsilon^{1-\delta}.$$ 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 $$\label{first-v-der} \dot{v}+\frac{\theta}{\mu} v=\frac{u'(r)h^{(1-\theta)}(r)}{\mu h'(r)}>0, \quad\mbox{for all r\in(0,r_0)},$$ 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 $$\ddot{v}+(2q-\frac{1}{\mu}) \dot v-(\frac{1}{\mu}-q)q v= -\frac{b(r)(h(r))^{{\mathcal S}+1}}{\mu^2|h'(r)|}v^{\delta},\quad t\ge t_0. \label{Pinfty}$$ 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 $$\label{defA} C:=2q-\frac{1}{\mu}=\frac{1}{\mu}\Bigl(\frac{2{\mathcal S}-(\delta+1)}{\delta-1}\Bigr)\not=0$$ 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 $$\label{liminf1} \liminf_{r\to 0}\frac{b(r)h(r)}{B(r)|h'(r)|}= \lim_{n\to\infty}\frac{b(r_n)h(r_n)}{B(r_n)|h'(r_n)|}.$$ 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 $$\label{liminfbdd} \liminf_{r\to 0}\frac{|\log(B(r))|}{\log(h(r))}<\infty.$$ 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 $$\label{quotientbounded} \frac{b(r)h^{{\mathcal S}+1}(r)}{|h'(r)|}\le K <\infty \quad \mbox{for r small enough},$$ proving our assertion and implying in particular that $$\label{pedazo} (\frac{1}{\mu}-q)q \frac{v^2}{2}-\frac{\dot v^2}{2}- \frac{b(r)(h(r))^{{\mathcal S}+1}}{\mu^2|h'(r)|}\frac{v^{\delta+1}} {\delta+1}$$ 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 $$\label{dotvL2} |\dot{v}|^{2}\in L^1(t_0,\infty).$$ Finally, we will prove that $$\label{dotvto0} \lim_{t\to\infty}\dot{v}(t)=0.$$ 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} \begin{thebibliography}{00} \bibitem{av} {\sc P. Avil\'es} On isolated singularities in some non linear partial differential equations, Indiana Univ. Math. J. {\bf 35} (1983), 773 -791. \bibitem{bellman} {\sc R. Bellman}, Stability Theory of Differential Equations, Dover Publications Inc, New York, 1969. \bibitem{bvgh} {\sc M. Bidaut-V\'eron,} and {\sc M. Garc\'\i a-Huidobro}, Classification of the singular solutions of a quasilinear equation with weights. {\em In preparation.} \bibitem{cm} {\sc G. Caristi and E. Mitidieri}, Nonexistence of positive solutions of quasilinear equations, Advances in Diff. 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Serrin} , Isolated Singularities of solutions of Quasilinear Equations, Acta Math. {\bf 113} (1965), 219-240. \end{thebibliography} \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}