\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 02, 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-2006/02\hfil Existence of large solutions] {Existence of large solutions for a semilinear elliptic problem via explosive sub- supersolutions} \author[Z. Zhang\hfil EJDE-2006/02\hfilneg] {Zhijun Zhang} \address{Zhijun Zhang \hfill\break Department of Mathematics and Informational Science, Yantai University, Yantai, Shandong, 264005, China} \email{zhangzj@ytu.edu.cn} \date{} \thanks{Submitted May 21, 2005. Published January 6, 2006.} \thanks{Supported by grant 10071066 from the National Natural Science Foundation of China.} \subjclass[2000]{35J60, 35B25, 35B50, 35R05} \keywords{Semilinear elliptic equations; explosive subsolutions; \hfill\break\indent explosive superbsolutions; existence; global optimal asymptotic behaviour} \begin{abstract} We consider the boundary blow-up nonlinear elliptic problems $\Delta u\pm\lambda |\nabla u|^q=k(x)g(u)$ in a bounded domain with boundary condition $u|_{\partial \Omega}=+\infty$, where $q\in [0, 2]$ and $\lambda\geq0$. Under suitable growth assumptions on $k$ near the boundary and on $g$ both at zero and at infinity, we show the existence of at least one solution in $C^2(\Omega)$. Our proof is based on the method of explosive sub-supersolutions, which permits positive weights $k(x)$ which are unbounded and/or oscillatory near the boundary. Also, we show the global optimal asymptotic behaviour of the solution in some special cases. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \section{Introduction} The purpose of this paper is to investigate existence and global optimal asymptotic behaviour of solutions to the problems \begin{gather} \Delta u + \lambda |\nabla u|^q=k(x)g(u), \quad x \in \Omega, \quad u|_{\partial \Omega}=+\infty, \label{Pp}\\ \Delta u - \lambda |\nabla u|^q=k(x)g(u), \quad x \in \Omega, \quad u|_{\partial \Omega}=+\infty, \label{Pm} \end{gather} where the boundary condition means $u(x) \to +\infty$ as $d(x)=\mathop{\rm dist}(x, \partial \Omega)\to 0$, $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 1)$, $q\in [0, 2]$ and $\lambda \geq 0$. The solutions to the above problems are called large solutions' or explosive solutions'. Our assumptions on the function $g$ are as follows: \begin{itemize} \item[(G1)] $g\in C^1([0,\infty))$ is non-decreasing on $[0, \infty)$, $g(s)\leq C_1 s^{p_1}$, for all $s \in (0, \infty)$ and $g(s)\geq C_2 s^{p_2}$ for large $s$, with $p_1\geq p_2>1$ and $C_1, C_2$ are positive constants. \item[(G2)] $g\in C^1(\mathbb R)$ is non-decreasing on $\mathbb R$, $g(s)\leq C_1 e^{p_1 s}$, for all $s \in \mathbb R$ and $g(s)\geq C_2 e^{p_2s}$ for large $|s|$ with $p_1\geq p_2>0$ and $C_1, C_2$ are positive constants. \end{itemize} We assume that $k \in C_{\rm loc}^\alpha(\Omega)$ for some $\alpha \in (0,1)$, is positive in $\Omega$, and satisfies \begin{itemize} \item[(K1)] There exist constants $C_1, C_2$ such that $C_1 (d(x))^{\gamma_2} \leq k(x)\leq C_2 (d(x))^{\gamma_1}$, for all $x\in \Omega$ with $-2<\gamma_1\leq\gamma_2$. \end{itemize} When $\lambda=0$, problems \eqref{Pp}, \eqref{Pm} become $$\Delta u=k(x)g(u), \quad x \in \Omega, \quad u|_{\partial \Omega}=+\infty. \label{e1.1}$$ For $k(x) \equiv 1$ on $\Omega$, $f(u)=e^u$ and $N=2$, problem \eqref{e1.1} was first considered by Bieberbach \cite{l3} in 1916. In this case, problem \eqref{e1.1} plays an important role in the theory of Riemannian surfaces of constant negative curvatures and in the theory of automorphic functions. Rademacher \cite{l3}, using the ideas of Bieberbach, showed that if $\Omega$ is a bounded domain in $\mathbb{R}^3$ with $C^2$ boundary, then problem \eqref{e1.1} has a unique solution $u \in C^2 (\Omega)$ such that $|u(x)+2 \ln d(x)|$ is bounded on $\Omega$. In this case, this problem arises in the study of an electric potential in a glowing hollow metal body. For general increasing nonlinearities $f(u)$, $k(x) \equiv 1$ on $\Omega$ and a bounded smooth domain $\Omega$, Keller \cite{k1} and Osserman \cite{o1} supplied a necessary and sufficient condition $\int^\infty 1/ \sqrt{G(s)}\,ds <\infty$ where $G'(s)=g(s)$ for the existence of large solutions to problem \eqref{e1.1}. Later, Loewner and Nirenberg \cite{l5} showed that if $g(u)=u^{p_0}$ with $p_0=(N+2)/(N-2)$, $N>2$, then problem \eqref{e1.1} has a unique positive solution $u$ satisfying $\lim_{d(x)\to 0}u(x)(d(x))^{(N-2)/2}=(N(N-2)/4)^{(N-2)/4}$. In this case, the problem arises in the differential geometry. The asymptotic behaviour and uniqueness of solutions to \eqref{e1.1} have been established in \cite{b1,b3,c1,c2,c3,c4,c5,c6,d1,d2,d3,g2,g5,l4,l6,m1,m2,v1}, where the uniqueness was derived through an analysis of the asymptotic behaviour of solutions near the boundary. For $\lambda\neq 0$ and $k(x) \equiv 1$ on $\Omega$, Bandle and Giarrusso \cite{b2} and Giarrusso \cite{g3,g4} established the asymptotic behaviour and uniqueness of solutions of \eqref{Pp} and \eqref{Pm}. For more investigations of explosive problems for elliptic equations, we refer the reader to \cite{c2,l1,l2,t1,z1,z2,z3,z4}. Recently, the author \cite{z3} established an explosive sub-supersolution method for the existence of solutions to general elliptic problems with nonlinear gradient terms. Garcia-Melian \cite{g1} also established an explosive sub-supersolution method for the existence of solutions to \eqref{e1.1}. By constructing explosive subsolutions and explosive supersolutions, he showed the following results. \begin{itemize} \item[(I)] If (K1) and (G1) are satisfied, then \eqref{e1.1} has at least one positive solution $u\in C^2 (\Omega)$ and satisfies $$m [d(x)]^{-(2+\gamma_1)/(p_1-1)} \leq u(x)\leq M [d(x)]^{-(2+\gamma_2)/(p_2-1)}, \quad \forall x\in \Omega; \label{e1.2}$$ where $m, M$ are positive constants with $m\leq M$. \item[(II)] If (K1) and (G2) are satisfied, then \eqref{e1.1} has at least one solution $u\in C^2 (\Omega)$ and satisfies $$-m -(2+\gamma_1)/{p_1} \ln d(x))\leq u(x)\leq M-(2+\gamma_2)/{p_2}\ln d(x) , \quad \forall x\in \Omega.\label{e1.3}$$ \end{itemize} In this paper, we extended the above results to problems \eqref{Pp} and \eqref{Pm}. Let $w \in C^{2+\alpha} (\Omega) \bigcap C^{1}(\overline {\Omega})$ be the unique solution of the problem $$-\Delta u=1, \quad u>0, \quad x \in \Omega ,\quad u \big|_{\partial \Omega}=0. \label{e1.4}$$ As is well known, $\nabla w(x) \neq 0$, for all $x \in \partial \Omega$ and $C_1 d(x)\leq w(x) \leq C_2 d(x)$, for all $x \in \Omega$, where $C_1$, $C_2$ are positive constants. Thus (K1) is equivalent to \begin{itemize} \item[(K2)] $c_1 (w(x))^{\gamma_2} \leq k(x)\leq c_2 (w(x))^{\gamma_1}$, for $x\in \Omega$ with $-2<\gamma_1\leq\gamma_2$. \end{itemize} For convenience in the following, we denote \begin{gather*} |u|_{\infty}=\max_{x\in \bar{\Omega}}|u(x)|; \quad u\in C(\bar{\Omega});\quad \beta_1=\frac {2+\gamma_1}{p_1-1};\quad \beta_2=\frac {2+\gamma_2}{p_2-1}; \\ c_0= \min_{x \in \overline \Omega}[|\nabla w(x)|^2+w(x)];\quad C_0=\max _{x \in \overline \Omega}[|\nabla w(x)|^2+w(x)]; \\ c_\beta= \min_{x \in \overline \Omega}[(1+\beta)|\nabla w(x)|^2+w(x)];\quad C_\beta=\max _{x \in \overline \Omega}[(1+\beta) |\nabla w(x)|^2+w(x)] \end{gather*} for $\beta>0$. Our main results are summarized in the following theorems. \begin{theorem} \label{thm1.1} Under assumptions (G1) and (K2), if $0\leq q<\min \{p_2,\frac {2p_2+\gamma_2}{p_2+\gamma_2+1}\}$, then problem \eqref{Pp} has at least one positive solution $u_\lambda\in C^2(\Omega)$ for each $\lambda \geq 0$, and satisfies $$m [w(x)]^{-(2+\gamma_1)/(p_1-1)} \leq u_\lambda(x)\leq M [w(x)]^{-(2+\gamma_2)/(p_2-1)}, \quad \forall x\in \Omega; \label{e1.5}$$ where $m, M$ are positive constants with $m\leq M$. \end{theorem} \begin{theorem} \label{thm1.2} Under assumptions (G1) and (K2), if $10, \quad x\in B_R, \quad u\big| _{\partial B_R}=+\infty, \label{e3.1} where$p>1$,$C_0$is a positive constant,$-2<\sigma$and$B_R=\{x\in \mathbb R^N: ||x||