\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
\begin{equation}
\Delta u=k(x)g(u), \quad x \in \Omega, \quad u|_{\partial
\Omega}=+\infty.
\label{e1.1}
\end{equation}
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
\begin{equation}
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}
\end{equation}
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
\begin{equation}
-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{equation}
\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
\begin{equation}
-\Delta u=1, \quad u>0, \quad x \in \Omega ,\quad u
\big|_{\partial \Omega}=0.
\label{e1.4}
\end{equation}
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
\begin{equation}
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}
\end{equation}
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}
\end{equation}
where $p>1$, $C_0$ is a positive constant, $-2<\sigma$ and
$B_R=\{x\in \mathbb R^N: ||x||