\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 240, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/240\hfil Asymptotic behavior of positive solutions] {Asymptotic behavior of positive solutions for the radial p-Laplacian equation} \author[S. Ben Othman, H. M\^aagli\hfil EJDE-2012/240\hfilneg] {Sonia Ben Othman, Habib M\^aagli} % in alphabetical order \address{Sonia Ben Othman \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis, Campus universitaire, 2092 Tunis, Tunisia} \email{Sonia.benothman@fsb.rnu.tn} \address{Habib M\^aagli \newline King Abdulaziz University, College of Sciences and Arts, Rabigh Campus, Department of Mathematics, P.O. Box 344, Rabigh 21911, Saudi Arabia} \email{habib.maagli@fst.rnu.tn} \thanks{Submitted September 23, 2012. Published December 28, 2012.} \subjclass[2000]{34B15, 35J65} \keywords{p-Laplacian; asymptotic behavior; positive solutions; \hfill\break\indent Schauder's fixed point theorem} \begin{abstract} We study the existence, uniqueness and asymptotic behavior of positive solutions to the nonlinear problem \begin{gather*} \frac{1}{A}(A\Phi _p(u'))'+q(x)u^{\alpha}=0,\quad \text{in }(0,1),\\ \lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0, \end{gather*} where $\alpha 0$, $\frac{1}{c}\leq q(x)(1-x)^{\beta }\exp \Big( -\int_{1-x}^{\eta }\frac{z(s)}{s}ds\Big)\leq c.$ Our arguments combine monotonicity methods with Karamata regular variation theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} Let $p>1$ and $\alpha 0$, such that $$\lim_{t\to 0}\frac{tL'(t)}{L(t)}=0.$$ For two nonnegative functions $f$ and $g$ defined on a set $S$, we write $f(x)\approx g(x)$, if there exists a constant $c>0$ such that $\frac{1}{c}g(x)\leq f(x)\leq cg(x)$, for each $x\in S$. Furthermore, we refer to $G_pf$, as the function defined on $(0,1)$ by $G_pf(x):=\int_{x}^{1}\Big(\frac{1}{A(t)} \int_0^{t}A(s)f(s)ds\Big)^{\frac{1}{p-1}}dt,$ where $f$ is a nonnegative measurable function in $(0,1)$. We point out that if $f$ is a nonnegative continuous function such that the mapping $x\mapsto A(x)f(x)$ is integrable in a neighborhood of $0$, then $G_pf$ is the solution of the problem $$\label{e1.3} \begin{gathered} -\frac{1}{A}(A\Phi _p(u'))'=f,\quad \text{in } (0,1), \\ A\Phi _p(u')(0)=0,\quad u(1)=0. \end{gathered}$$ As it is mentioned above, our main purpose in this paper is to establish existence and global behavior of a positive solution for problem \eqref{eP}. Let us introduce our hypotheses. The function $A$ is continuous in $[0,1)$, differentiable and positive in $(0,1)$ such that $A(x)\approx x^{\lambda }(1-x)^{\mu }$ with $\lambda \geq 0$ and $\mu 1)$ such that $\int_0^{\eta }t^{\frac{1-\beta }{p-1}}( L(t))^{\frac{1}{p-1}}dt<+\infty .$ \end{itemize} We need to verify the condition $\int_0^{\eta }t^{\frac{1-\beta }{p-1}}(L(t))^{ \frac{1}{p-1}}dt<+\infty$ in hypothesis (H1), only if $\beta =p$ (See Lemma \ref{lem2} below). As a typical example of function $q$ satisfying (H1), we have $q(x):=(1-x)^{-\beta }(\log \frac{2}{1-x})^{-\nu }, \quad x\in [ 0,1).$ Then for $\beta p-1$, the function $q$ satisfies (H1). Our main result is as follows. \begin{theorem} \label{thm1} Assume {\rm (H1)}. Then problem \eqref{eP} has a unique positive and continuous solution $u$ satisfying, for $x\in (0,1)$, $$\label{e1.4} u(x)\approx \theta _{\beta }(x),$$ where $\theta _{\beta }$ is the function defined on $[0,1)$ by \label{e1.5} \theta _{\beta }(x):= \begin{cases} \Big(\int_0^{1-x}\frac{(L(s))^{\frac{1}{p-1}} }{s}ds\Big)^{\frac{p-1}{p-1-\alpha }}, & \text{if } \beta =p \\ (1-x)^{\frac{p-\beta }{p-1-\alpha }}(L(1-x))^{\frac{1}{ p-1-\alpha }}, & \text{if } \frac{(\mu +1)(p-1-\alpha )+\alpha p}{p-1}<\beta 0$. Then$L_1L_2\in \mathcal{K}$,$L_1^{m}\in \mathcal{K}$, and$\lim_{t\to 0^{+}}t^{\epsilon }L_1(t)=0$. \end{lemma} \begin{lemma}[\cite{m3,s1}] \label{lem2} Let$L\in \mathcal{K}$and$\delta \in\mathbb{R}$. Then we have the following: \begin{itemize} \item[(i)] If$\delta <2$, then$\int_0^{\eta}t^{1-\delta }L(t)dt$converges and $\int_0^{s}t^{1-\delta}L(t)dt\sim \frac{s^{2-\delta }L(s)}{2-\delta }\quad \text{as } s\to 0^{+}.$ \item[(ii)] If$\delta >2$, then$\int_0^{\eta }t^{1-\delta }L(t)dt$diverges and $\int_{s}^{\eta }t^{1-\delta }L(t)dt \sim \frac{s^{2-\delta }L(s)}{\delta -2}\quad \text{as } s\to 0^{+}.$ \end{itemize} \end{lemma} \begin{lemma}[\cite{c1}] \label{lem3} Let$L\in \mathcal{K}$be defined on$(0,\eta ]$, then we have $t\mapsto \int_{t}^{\eta }\frac{L(s)}{s}ds\in \mathcal{K}.$ If further$\int_0^{\eta }\frac{L(s)}{s}ds$converges, then $t\mapsto \int_0^{t}\frac{L(s)}{s}ds\in \mathcal{K}.$ \end{lemma} \begin{proposition} \label{prop1} Assume$q$satisfies (H1). Then for$x\in (0,1)$, we have $G_pq(x)\approx \Psi (1-x),$ where$\psi $is the function defined on$(0,1]$by \label{e2.1} \Psi (t)=\begin{cases} \int_0^{t}\frac{(L(s))^{\frac{1}{p-1}}}{s}ds, &\text{if }\beta =p, \\ t^{\frac{p-\beta }{p-1}}(L(t))^{\frac{1}{p-1}},&\text{if }\mu +1<\beta \beta >\mu +1$, then by Lemma \ref{lem2}, $\int_{1-t}^{1/2}s^{\mu -\beta }L(s)ds\approx (1-t)^{\mu +1-\beta }L(1-t).$ So, $(1+\int_{1-t}^{1/2}s^{\mu -\beta }L(s)ds)^{\frac{1}{p-1}}\approx (1-t)^{\frac{\mu +1-\beta }{p-1}}L^{ \frac{1}{p-1}}(1-t).$ Thus, using the fact that $\beta u_2(x_0)$ for some $x_0\in (0,1)$. Then there exists $x_1,x_2\in [ 0,1]$, such that $0\leq x_1u_2(x)$ with $u_1(x_2)=u_2(x_2)$, $u_1(x_1)=u_2(x_1)$ or $x_1=0$. We deduce that $$\label{e3.4} A\Phi _p(u_2')(x_1)\leq A\Phi _p(u_1')(x_1).$$ On the other hand, since $\alpha <0$, we have $u_1^{\alpha }(x)1$, then \begin{gather*} -\frac{1}{A}(A\Phi _p(v'))'+\frac{1}{A}(A\Phi _p(c^{ \frac{-\alpha }{p-1}}u'))'=q(x)(v^{\alpha }-c^{-\alpha }u^{\alpha }),\quad \text{in }(0,1), \\ \lim_{x\to 0^{+}}(A\Phi _p(v')-A\Phi _p(c^{ \frac{-\alpha }{p-1}}u'))(x)=0, \\ (v-c^{\frac{-\alpha }{p-1}}u)(1)=0. \end{gather*} So, we have $-\frac{1}{A}(A\Phi _p(v'))'+\frac{1}{A} (A\Phi _p(c^{\frac{-\alpha }{p-1}}u'))'\geq 0\quad\text{in }(0,1),$ which implies that the function $\theta (x):=(A\Phi _p(c^{ \frac{-\alpha }{p-1}}u')-A\Phi _p(v'))(x)$ is nondecreasing on $(0,1)$ with $\lim_{x\to 0^{+}}\theta (x)=0$. Hence from the monotonicity of $\Phi _p$, we obtain that the function $x\mapsto (c^{\frac{-\alpha }{p-1}}u-v)(x)$ is nondecreasing on $[0,1)$ with $(c^{-\frac{\alpha }{p-1}}u-v)(1)=0$. This implies that $c^{\frac{-\alpha }{p-1}}u\leq v$. On the other hand, we deduce by symmetry that $v\leq c^{\frac{\alpha }{p-1}}u$. Hence $c^{\frac{\alpha }{p-1}}\in J$. Now, since $\alpha 1$, we have $c^{\frac{\alpha }{p-1}}p-1$. \end{itemize} Using Theorem \ref{thm1}, we deduce that problem \eqref{eP} has a positive continuous solution\ $u$ in $[0,1]$ satisfying \begin{itemize} \item[(i)] If $\beta <\frac{(\mu +1)(p-1-\alpha )+\alpha p}{p-1}$, then for $x\in (0,1)$, $u(x)\approx (1-x)^{\frac{p-1-\mu }{p-1}}.$ \item[(ii)] If $\beta =\frac{(\mu +1)(p-1-\alpha )+\alpha p}{ p-1}$ and $\sigma =1$, then for $x\in (0,1)$, $u(x)\approx (1-x)^{\frac{p-1-\mu }{p-1}}\Big(\log \log \frac{3}{1-x} \Big)^{\frac{1}{p-1-\alpha }}.$ \item[(iii)] If $\beta =\frac{(\mu +1)(p-1-\alpha )+\alpha p}{ p-1}$ and $\sigma <1$, then for $x\in (0,1)$, $u(x)\approx (1-x)^{\frac{p-1-\mu }{p-1}}\Big(\log \frac{3}{1-x}\Big)^{ \frac{1-\sigma }{p-1-\alpha }}.$ \item[(iv)] If $\beta =\frac{(\mu +1)(p-1-\alpha )+\alpha p}{ p-1}$ and $\sigma >1$, then for $x\in (0,1)$, $u(x)\approx (1-x)^{\frac{p-1-\mu }{p-1}}.$ \item[(v)] If $\frac{(\mu +1)(p-1-\alpha )+\alpha p}{p-1} <\beta p-1$, then for $x\in (0,1)$, $u(x)\approx \Big(\log \frac{3}{1-x}\Big)^{\frac{p-1-\sigma }{p-1-\alpha}}$ \end{itemize} \subsection*{Second application} Let $q$ be a function satisfying (H1) and let \$\alpha ,\gamma