\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 93, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/93\hfil Asymptotic behaviour of the solution] {Asymptotic behaviour of the solution for the singular Lane-Emden-Fowler equation with nonlinear convection terms} \author[Z. Zhang\hfil EJDE-2006/93\hfilneg] {Zhijun Zhang} \address{Zhijun Zhang \newline Department of mathematics and informational science, Yantai university, Yantai 264005, China} \email{zhangzj@ytu.edu.cn} \date{} \thanks{Submitted December 23, 2005. Published August 18, 2006.} \thanks{Supported by Grant 10071066 from the National Natural Science Foundation of China} \subjclass[2000]{35J65, 35B05, 35O75, 35R05} \keywords{ Semilinear elliptic equations; Dirichlet problem; singularity; \hfill\break\indent nonlinear convection terms; Karamata regular variation theory; unique solution; \hfill\break\indent exact asymptotic behaviour} \begin{abstract} We show the exact asymptotic behaviour near the boundary for the classical solution to the Dirichler problem $$-\Delta u =k(x)g(u)+\lambda |\nabla u|^q, \quad u>0,\; x\in \Omega,\quad u\big|_{\partial{\Omega}}=0,$$ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$. We use the Karamata regular varying theory, a perturbed argument, and constructing comparison functions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction and statement of the main results} Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb R^N$ ($N\geq 1$). Consider the singular Dirichlet problem for the Lane-Emden-Fowler equation $$-\Delta u=k(x)g(u)+\lambda |\nabla u|^q, \quad u>0, \; x \in \Omega, \quad u|_{\partial \Omega}=0, \label{e1.1}$$ where $\lambda\in\mathbb R$, $q\in [0,2]$, and the functions $g$, $k$ satisfy the hypotheses: \begin{itemize} \item[(H1)] $g\in C^1((0,\infty),(0,\infty))$, $g'(s)\leq 0$ for all $s>0$, $\lim_{s \to 0^+}g(s)=\infty$ \item[(H2)] $k \in C^{\alpha}(\bar{\Omega})$ for some $\alpha \in (0,1)$, is non-negative and non-trivial on $\Omega$. \end{itemize} The problem above arises in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrical materials \cite{c3,f1,g2,n1,s2}. The main feature of this paper is the presence of the three terms: the singularity term $g(u)$ which is regular varying at zero of index $-\gamma$ with $\gamma \in (0, 1)$, the weight $k(x)$ which may be vanishing at the boundary, the both of them include a large class of functions, and the nonlinear convection terms $\lambda |\nabla u|^q$. This problem was discussed in a number of works; see, for instance, \cite{b1,c1,c2,c3,c4,c5,f1,g1,g2,g3,g5,l1,l2,m1,s2,s3,u1,y1,z1,z2,z3,z4,z5}. When $\lambda =0$, i.e., problem \eqref{e1.1} becomes $$-\Delta u=k(x)g(u),\quad u>0, \; x\in\Omega,\quad u|_{\partial\Omega}=0. \label{e1.2}$$ For $k\equiv 1$ on $\Omega$. Fulks and Maybee \cite{f1}, Stuart \cite{s2}, Crandall, Rabinowitz and Tartar \cite{c3} showed that \eqref{e1.2} has a unique solution $u\in C^{2+\alpha}(\Omega) \cap C(\bar{\Omega})$. Moreover, Crandall, Rabinowitz and Tartar \cite[Theorems 2.2 and 2.5]{c3} showed that if $p\in C[0,a]\cap C^2(0,a]$ is the local solution to the problem $$-p''(s)=g(p(s)),\quad p(s)>0,\; 00 and t_0\geq 1 such that g(\xi t)\geq\xi^{-\theta}g(t) for all \xi \in (0,1) and 00 and a positive non-decreasing function h\in C(0,\delta_0) such that \item[(H6)] \lim_{d(x)\to 0}\frac{k(x)}{h(d(x))} =c_0 \item[(H7)] \lim_{t\to 0^+}h(t)g(t)=+\infty. \end{itemize} Then \eqref{e1.2} has a unique solution u\in C^{1,1-\alpha}(\bar{\Omega})\cap C^2(\Omega) satisfying $$\lim_{d(x)\to 0}\frac{u(x)}{p(d(x))}=\xi_0, \label{e1.4}$$ where T(\xi_0)=c_0^{-1}, and p\in C^1 [0,a]\cap C^2(0,a](a\in (0,\delta_0)) is the local solution to the problem -p''(s)=h(s)g(p(s)), \quad p(s)>0, \; 00, is called regular varying at zero with index \beta, written g \in RVZ_\beta if for each \xi>0 and some \beta \in \mathbb{R},$$ \lim_{t \to 0^+} \frac{g(\xi t)}{g(t)}= \xi^{\beta}. %\label{e1.6} $$\end{definition} Our main result is summarized in the following theorem. \begin{theorem} \label{thm1.1} Let g satisfy (H1) and g \in RVZ_{-\gamma} with \gamma \in (0,1) and k satisfy (H2), (H6) and h \in RVZ_\beta with \beta \in [0,1). If \beta<\gamma, then the unique solution u_\lambda \in C^{1,1-\alpha}(\bar{\Omega})\cap C^2(\Omega) to problem \eqref{e1.1} satisfies$$ \lim_{d(x)\to 0}\frac{u_\lambda (x)}{p(d(x))}=\xi_0, %\label{e1.7} $$where \xi_0=c_0 ^{1/(1+\gamma)}, and p\in C^1[0,a]\cap C^2(0,a] is the local solution to problem \eqref{e1.5}. \end{theorem} \begin{remark} \label{rmk1.1} \rm By (H1) and the proof of the maximum principle \cite[Theorems 10.1 and 10.2]{g4} we see that \eqref{e1.1} has at most one solution in  C^{2}(\Omega) \cap C(\bar{\Omega}) for each fixed \lambda. \end{remark} \begin{remark} \label{rmk1.2} \rm In section 2, we will see that g \in RVZ_{-\gamma} with \gamma >0  implies \lim_{s \to 0^+}g(s)=\infty and h \in RVZ_\beta with \beta >0 implies \lim_{t\to 0^+}h(t)=0. \end{remark} \begin{remark} \label{rmk1.3} \rm For the existence of solutions to \eqref{e1.5} with a\in (0, 1), see \cite[Corollary 2.1]{a1}. \end{remark} The outline of this article is as follows. In section 2, we recall some basic definitions and the properties to Karamata regular varying theory. In section 3, we prove the asymptotic behaviour of the unique solution in Theorem \ref{thm1.1}. \section{Karamata regular varying theory} Let us recall some basic definitions and the properties to Karamata regular varying theory, which is a basic tool in probability theory (see \cite{m2,r1,s1}). \begin{definition} \label{def2.1} \rm A positive measurable function f defined on [a,\infty), for some a>0, is called regular varying at infinity with index \rho, written f \in RV_\rho, if for each \xi>0 and some \rho \in \mathbb{R}, $$\lim_{t \to \infty} \frac{f(\xi t)}{f(t)}= \xi^\rho. \label{e2.1}$$ The real number \rho is called the index of regular variation. \end{definition} \begin{definition} \label{def2.2} \rm When \rho=0, a positive measurable function L defined on [a,\infty), for some a>0, is called slowly varying at infinity, if for each \xi>0 $$\lim_{t \to \infty }\frac{L(\xi t)}{L(t)}=1. \label{e2.2}$$ \end{definition} It follows by the definition that if f \in RV_\rho it can be represented in the form$$ f(t)=t^\rho L(t). % \label{e2.3} $$Some basic examples of slowly varying functions are: \begin{itemize} \item[(i)] \lim_{t\to \infty}L(t)=c\in (0, \infty); \item[(ii)]  L(t)=\prod _{m=1}^{m=n}(\log_{m}(t))^{\alpha_m}, \alpha_m\in \mathbb R; \item[(iii)]  L(t)=e^{(\prod _{m=1}^{m=n}(\log_{m}(t))^{\alpha_m})}, 0<\alpha_m<1; \item[(iv)] \displaystyle L(t)=\frac {1}{t}\int _a ^{t}\frac {ds}{\ln s}; \item[(v)] L(t)=\displaystyle e^{((\ln t)^{1/3}\cos((\ln t)^{1/3} ))}, where \lim_{t\to\infty}\inf L(t)=0, \lim_{t\to\infty}\sup L(t)=+ \infty. \end{itemize} \begin{lemma}[Uniform convergence theorem] \label{lem2.1} If f \in RV_\rho, then \eqref{e2.1} (and so \eqref{e2.2}) holds uniformly for \xi \in [a,b] with 00, where c(t) and y(t) are measurable and for t \to \infty, y(t) \to 0 and c(t) \to c, with c>0. \end{lemma} \begin{lemma} \label{lem2.3} If functions L, L_1 are slowly varying at infinity, then \begin{itemize} \item[(i)] L^\alpha for every \alpha\in \mathbb R, L(t)+L_1(t), L(L_1(t)) (if L_1(t)\to \infty as t\to\infty), are also slowly varying at infinity; \item[(ii)] for every \theta >0 and t\to \infty,$$ t^{\theta} L(t)\to \infty, \quad t^{-\theta} L(t)\to 0; %\label{e2.5} $$\item[(iii)] for t\to \infty, \ln (L(t))/{\ln t}\to 0. \end{itemize} \end{lemma} \begin{definition} \label{def2.3} \rm A positive measurable function H defined on some neighborhood (0,a) for some a>0, is called slowly varying at zero, if for each \xi>0$$ \lim_{t \to 0^+}\frac{H(\xi t)}{H(t)}=1. %\label{e2.6} $$\end{definition} It follows by Definitions \ref{def1.1} and \ref{def2.3} that if g \in RVZ_{-\gamma} it can be represented in the form g(t)=t^{-\gamma} H(t). % \label{e2.7} \begin{lemma} \label{lem2.4} Definition \ref{def1.1} is equivalent to saying that f^*(t)=g(1/t) is regular varying at infinity of index -\beta. \end{lemma} Thus we transfer our attention from infinity to the origin. \begin{corollary}[Representation theorem] \label{coro2.1} A function H is slowly varying at zero if and only if it may be written in the form$$ H(t)=c(t)\exp \big( \int_t^a \frac {y(s)}{s} ds \big), \quad 00$, where$c(t)$and$y(t)$are measurable and for$t \to 0^+$,$y(t) \to 0$and$c(t) \to c$, with$c>0$. \end{corollary} \begin{corollary} \label{coro2.2} If a function$H$is slowly varying at zero, then for every$\theta >0$and$t\to 0^+$,$t^{-\theta}H(t)\to \infty$,$t^{\theta} H(t)\to 0$. %\label{e2.9} \end{corollary} \begin{corollary} \label{coro2.3} If$g$satisfies (H1),$g \in RVZ_{-\gamma}$with$\gamma \in (0,1)$, and$k$satisfies (H2), (H6),$h \in RVZ_{\beta}$with$\beta\in (0, 1)$, then $$g(t)=t^{-\gamma} c_1(t)\exp \big( \int_t^a \frac{y_1(s)}{s}ds\big), \quad h(t)=t^{\beta} c_2(t)\exp \big(\int_t^a \frac{y_2(s)}{s}ds\big), % \label{e2.10}$$ where$y_1,y_2, c_1, c_2 \in C[0,a]$,$y_1(0)=y_2(0)=0$,$ c_1(0)>0$,$c_2(0)>0 $. \end{corollary} \section{Asymptotic behaviour} First we give some preliminary considerations. \begin{lemma} \label{lem3.1} If$g$satisfies (H1) and$g \in RVZ_{-\gamma}$with$\gamma \in (0,1)$, then $$\int_0^1 g(t)dt<\infty. % \label{e3.1}$$ \end{lemma} \begin{proof} We see by corollaries \ref{coro2.2} and \ref{coro2.3} that there exists$\gamma_1\in (\gamma,1)$such that $$\lim_{t\to 0^+}t^{\gamma_1}g(t)=\lim_{t\to 0^+}t^{\gamma_1-\gamma} c_1(t)\exp \big( \int_t^a \frac{y_1(s)}{s}ds\big)=0.$$ It follow that there exists$\delta\in (0,1)$such that$g(t)0$on$(0,a]$, so$p(s)$is increasing. Since$h$is non-decreasing, multiplying \eqref{e1.5} by$p'(s)$and integrating on$[t,a]$,$0\beta$, we see by corollaries \ref{coro2.2} and \ref{coro2.3} that $$\lim_{t\to 0^+} h(t)g(t)=\lim_{t\to 0^+} t^{-(\gamma-\beta)} c_1(t) c_2(t)\exp \big( \int_t^a \frac{y_1(s)}{s}ds\big) \exp \big( \int_t^a \frac{y_2(s)}{s}ds\big)=\infty,$$ and $$\lim_{t\to 0^+}\frac{g(bt)}{g(t)}=b^{-\gamma}.$$ Thus \begin{gather*} -p''(t)=h(t)g(p(t))\geq h(t)g(t)\frac {g(bt)}{g(t)}, \quad \forall t\in (0,a], \\ \lim_{s\to0^+}p''(s)=-\infty. \end{gather*} (iii) is follows by (i) and (ii). The proof is complete. \end{proof} \begin{proof}[Proof of the asymptotic behaviour in Theorem \ref{thm1.1}] Set$\xi_0=c_0^{1/(1+\gamma)}$and for a fix$\varepsilon\in(0,1/4)$let $$\xi_{1\varepsilon}=\big(\frac {c_0}{1-2\varepsilon}\big)^{1/(1+\gamma)},\quad \xi_{2\varepsilon} =\big(\frac {c_0}{1+2\varepsilon}\big)^{1/(1+\gamma)},$$ we see that $$\big(\frac {c_0}{2}\big)^{1/(1+\gamma)} <\xi_{2\varepsilon}<\xi_{1\varepsilon}< (2c_0)^{1/(1+\gamma)}.$$ For any$\delta > 0$, we define$\Omega_{\delta} = \{x\in\Omega: d(x)\leq \delta\}$. By the regularity of$\partial\Omega$and lemma \ref{lem3.2}, we can choose$\delta$sufficiently small such that \begin{itemize} \item[(i)]$d(x)\in C^2(\Omega_\delta)$; \item[(ii)]$|\frac{p'(s)}{p''(s)}\Delta d(x)+\lambda \xi_{i\varepsilon}^{q-1}\frac{(p'(s))^q}{p''(s)}|< \varepsilon$, for all$ (x,s) \in \Omega_\delta \times (0,\delta)$,$i=1, 2$and fixed$\lambda;$\item[(iii)]$\frac{\xi_{2\varepsilon}h(d(x))g(p(d(x)))}{g(p(d(x))\xi_{2\varepsilon})} (1+\varepsilon)