\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
$1<q\leq \frac {2p_1+\gamma_1}{p_1+\gamma_1+1}$, then problem
\eqref{Pm}  has
at least one positive solution $u_\lambda\in C^2 (\Omega)$ for
each $\lambda \geq 0$, and  satisfies \eqref{e1.5}.
\end{theorem}

\begin{theorem} \label{thm1.3}
Under assumptions (G2) and (K2), if $0\leq q\leq 2$, then problem
\eqref{Pp} has at least one solution $u_\lambda\in C^2 (\Omega)$
for each $\lambda \geq 0$, and satisfies
\begin{equation}
- m -(2+\gamma_1)/{p_1} \ln w(x))\leq u_\lambda(x)\leq
M-(2+\gamma_2)/{p_2}\ln w(x) , \quad \forall x\in
\Omega,\label{e1.6}
\end{equation}
\end{theorem}

\begin{theorem} \label{thm1.4}
Assume (G2) and (K2).
\begin{itemize}
\item[(I)]  If  $1<q\leq 2$, then problem \eqref{Pm}
has at least one solution $u_\lambda\in C^2 (\Omega)$ for each
$\lambda \geq 0$, and satisfies
\begin{equation}
-m -\beta \ln d(x)\leq u(x)\leq M-(2+\gamma_2)/{p_2}\ln d(x) ,
\quad \forall x\in \Omega;\label{e1.7}
\end{equation}
where $\beta \in (0, (2+\gamma_2)/{p_2})$ is small enough.

\item[(II)]  If $0<q\leq 1$, then problem \eqref{Pm}
 has at least one solution
$u_\lambda\in C^2 (\Omega)$ for each $\lambda \in [0, \lambda_0]$,
and  satisfies \eqref{e1.6},  where
\[
\lambda_0=((2+\gamma_1)/{p_1})^{1-q}\frac { cc_0 }{
|w|_\infty^{2-q}|\nabla w|_\infty^q},
\]
 with $c\in (0, 1)$.
\end{itemize}
\end{theorem}



The outline of this article is as follows. In section 2,  we prove
Theorems \ref{thm1.1}--\ref{thm1.4}. In the final section,
 we give two examples.


 \section{Proofs of theorems}
First we introduce an explosive  sub - supersolution
method.
We consider the  following  general problem
\begin{equation}
-\Delta u=f(x,u, \nabla u), \quad x\in\Omega,\quad
u|_{\partial\Omega}=+\infty,
 \label{e2.1}
\end{equation}
where  $f(x,s,\eta)$ satisfies the following  conditions:
\begin{itemize}
\item[(F1)]  $f(x,s,\eta)$ is locally H\"{o}lder continuous in $\Omega
\times I\times \mathbb R^N$ and continuously differentiable with
respect to the variables $s$ and $\eta$

\item[(F2)]  There exists increasing function $h\in C^1([0, \infty),
[0, \infty))$ such that
 $$
 |f(x,s,\eta)|\leq h(|s|)(1+|\eta|^2),\quad \forall (x, s, \eta)\in
 \Omega\times I\times \mathbb R^N
 $$

\item[(F3)]   $f$ is nondecreasing in $s$ for each
$(x,\eta)\in \Omega \times \mathbb R^N$;
where $I=[0, \infty)$ or  $I=\mathbb R$.
\end{itemize}

\noindent {\bf Definition.} %2.1
 A function $\underline {u}\in C^2 (\Omega)$ is called an
 explosive subsolution of \eqref{e2.1}  if
\begin{equation}
\Delta \underline {u}\geq f(x, \underline {u}, \nabla  \underline
{u}),  \; x\in \Omega, \quad  \underline {u} \big|
_{\partial \Omega}=+\infty.
\label{e2.2}
\end{equation}

\noindent {\bf Definition.} %2.2
 A function $\overline {u} \in C^2(\Omega)$ is called an explosive
  supersolution of \eqref{e2.1} if
\begin{equation}
\Delta \overline {u} \leq  f(x, \overline {u}, \nabla\overline
{u}),  \;   x\in \Omega, \quad  \overline {u} \big|
_{\partial \Omega}=+\infty.
\label{e2.3}
\end{equation}



\begin{lemma}[{\cite[Theorem 4.1]{z3}}] \label{lem2.1}
Suppose that \eqref{e2.1} has an explosive supersolution $\overline{u}$
and an explosive subsolution $\underline{u}$ such that
$\underline{u}\leq \overline{u}$ on $\Omega$, then \eqref{e2.1}
 has at least one solution $u\in C^2(\Omega)$ satisfying
 $\underline{u}\leq u\leq\bar{u}$ on $\Omega$.
\end{lemma}

For proving our main results, we use the above lemma; i.e.,
we construct an explosive supersolution $\overline {u}$ and an explosive
subsolution $\underline {u}$  such that  $\underline {u} \leq
\overline {u}$ on $\Omega$.

 \begin{proof}[Proof of Theorem \ref{thm1.1}]
  For  $0\leq q<\min \{p_2$,  $\frac {2p_2+\gamma_2}{p_2+\gamma_2+1}\}$,
let $\underline{u} =m (w(x))^{-\beta_1}$, where   $m$ is a positive
constant satisfying $ C_1C_2 m^{p_1-1}\leq \beta_1 c_ {\beta_1}$.
Then
\begin{align*}
\Delta \underline{u}
&=  m\beta_1 [ (1+\beta_1)|\nabla
w(x)|^2+w(x)](w(x))^{-2-\beta_1} \\
&\geq   m\beta_1 c_{\beta_1}(w(x))^ {-2-\beta_1 }\\
&\geq C_1C_2 m^{p_1} (w(x))^{\gamma_1} (w(x))^ {-\beta_1 p_1}\\
&\geq k(x)g(\underline {u}(x)), \quad x\in \Omega;
\end{align*}
i.e,  $\underline{u}  =m (w(x))^{-\beta_1}$ is an explosive
subsolution of (1.1).

Let $\bar{u} =M (w(x))^{-\beta_2}$,   where $M$ is a positive
constant satisfying
$$
C_1C_2 M^{p_2-1}\geq \beta_2 C_{\beta_2} +\lambda M^{q-1}
{\beta_2}^q {|w|_\infty}^{2+\beta_2-(1+\beta_2)q }|\nabla
w|_\infty ^q .
$$
We  see that
$$
\Delta \bar{u}+\lambda |\nabla \bar{u}|^q \leq C_1 (w(x)^{\gamma_2}
 C_2 \bar{u}^{p_2}\leq
  k(x)g(\bar{u})+\lambda |\nabla \bar{u}|^q,\quad \forall x\in \Omega;
$$
i.e., $\bar{u} =M (w(x))^{-\beta_2}$ is an explosive subsolution
of (1.1), Clearly $M\geq m$, i.e., $\overline u \geq \underline u$
on $\Omega$. Hence the desired conclusion follows by Lemma
\ref{lem2.1}.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.2}]
 For   $1<q\leq \frac {2p_1+\gamma_1}{p_1+\gamma_1+1}$. Let
$\overline u =M (w(x))^{-\beta_2}$, where   $M$ is a positive
constant satisfying $ M^{p_2-1}\geq \frac{\beta_2C_{\beta_2}}{C_1C_2}$.
 Then
\begin{align*}
 \Delta \overline u
&=  M\beta_2 [(1+\beta_2)|\nabla w(x)|^2+w(x)](w(x))^{-2-\beta_2} \\
&\leq   C_1C_2  M^{p_2} (w(x))^{\gamma_2} (w(x))^ {-\beta_2 p_2}\\
&\leq k(x)g(\bar{u}(x)), \quad x\in \Omega;
\end{align*}
i.e.,  $\overline u =M (w(x))^{-\beta_2}$ is an explosive
supersolution of (1.2). Let $\underline{u} =m (w(x))^{-\beta_1}$,
where $m$ is a positive constant satisfying
\[
C_1C_2 m^{p_1-1}-\lambda m^{q-1}  \beta_1^q
|w|_\infty^{2+\beta_1-(1+\beta_1)q}|\nabla w|_\infty ^q \leq
\beta_1 c_{\beta_1}.
\]
We  see that
$$
\Delta \underline{u}+\lambda |\nabla \underline{u}|^q\geq C_2 (w(x)^{\gamma_1}
 C_1 \underline{u}^{p_1}\geq
  k(x)g(\underline{u}),\quad \forall x\in \Omega;
$$
i.e., $\underline{u} =m (w(x))^{-\beta_1}$ is an explosive
subsolution of (1.2). Clearly $M\geq m$, i.e., $\overline u \geq
\underline u$ on $\Omega$. Hence the desired conclusion follows by
Lemma \ref{lem2.1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
 For    $0\leq q\leq 2$. Let $\underline{u} =-m-(2+\gamma_1)/{p_1}\ln w(x)$,
where  $m$ is a positive constant satisfying
$$
e^{-mp_1}\leq \frac {(2+\gamma_1)c_0}{C_1 C_2p_1}.
$$
Then
\begin{align*}
 \Delta \underline{u}
& = (2+\gamma_1)/{p_1} (|\nabla w(x)|^2+w(x))(w(x))^{-2}\\
& \geq  C_2 (w(x))^{\gamma_1} C_1 e^{-mp_1}
(w(x))^{-2-\gamma_1}\\
&\geq k(x)g(\bar {u}(x)), \quad \forall x\in \Omega;
\end{align*}
i.e,, $\underline{u} =-m-(2+\gamma_1)/{p_1} \ln w(x)$ is an
explosive subsolution of (1.1). Let  $\bar {u} =
M-(2+\gamma_2)/{p_2} \ln w(x)$, where $M$ is a positive constant
satisfying
$$
C_1C_2e^{Mp_2}\geq C_0 (2+\gamma_2)/{p_2}+ \lambda
((2+\gamma_2)/{p_2})^q |w|_\infty^{2-q}|\nabla w|_\infty^q.
$$
Then
\[
 \Delta \bar {u} \leq k(x)g(\underline{u} (x))+\lambda |\nabla
\bar{u}|^q, \quad \forall x\in \Omega;
\]
i.e., $\bar {u} = M-(2+\gamma_2)/{p_2} \ln w(x)$ is an explosive
superbsolution of \eqref{Pp}.
Clearly,   $\overline u \geq \underline u$ on $\Omega$. Hence the
desired conclusion follows by Lemma \ref{lem2.1}.
\end{proof}

 \begin{proof}[Proof of Theorem \ref{thm1.4}]
 For  $0<q\leq 2$.
Let $\overline u =M-(2+\gamma_2)/{p_2}\ln w(x)$, where  $M$ is a
positive constant satisfying
$$
C_1 C_2 e^{Mp_2}\geq C_0 (2+\gamma_2)/{p_2}.
$$
Then
\begin{align*}
\Delta \overline u &=  (|\nabla
w(x)|^2+w(x))(2+\gamma_2)/{p_2}(w(x))^{-2}\\
& \leq  C_1 (w(x))^{\gamma_2} C_2 e^{Mp_2}
(w(x))^{-2-\gamma_2}\\
&\leq k(x)g(\bar {u}(x)), \quad \forall x\in
\Omega;
\end{align*}
i.e, $\overline u =M-(2+\gamma_2)/{p_2} \ln w(x)$ is an explosive
supersolution of \eqref{Pm}.

We need to construct an explosive subpersolution of \eqref{Pm}.

\noindent Case (I) $1<q\leq 2$. Let  $\underline {u} = -m-\beta \ln w(x)$,
where $\beta\in (0, (2+\gamma_1)/{p_1})$ is small enough such that
$$
c_0\beta/2\geq \lambda \beta^q |w|_\infty^{2-q}|\nabla
w|_\infty^q
$$
and $m$ is a positive constant satisfying
$$
c_0\beta/2 \geq C_1C_2e^{-mp_1}|w|_\infty^{2+\gamma_1-p_1\beta}.
$$
Then
\begin{align*}
\Delta \underline{u}
&\geq \beta   (|\nabla w(x)|^2+w(x))(w(x))^{-2}\\
& \geq  C_1 (w(x))^{\gamma_1} C_2 e^{-mp_1}
(w(x))^{-p_2\beta}+\lambda \beta ^q w^{-q}|\nabla w|^q\\
&\geq k(x)g(\underline{u} (x)), \quad \forall x\in \Omega;
\end{align*}
i.e., $\overline u =-M-\beta \ln w(x)$ is an explosive subsolution
of \eqref{Pm}.
Clearly,   $\overline u \geq \underline u$ on $\Omega$.
\smallskip

\noindent Case (II) $0<q\leq 1$ and $\lambda \in [0, \lambda_0]$.
Let $\underline {u} = -m-(2+\gamma_1)/{p_1} \ln w(x)$, where  $m$
is a positive constant satisfying
$$
(1-c) c_0(2+\gamma_1)/{p_1} \geq C_1C_2e^{-m}.
$$
Since $\lambda \in [0, \lambda_0]$, 1.e.,
$$
cc_0(2+\gamma_1)/{p_1}\geq \lambda ((2+\gamma_1)/{p_1})^q
|w|_\infty^{2-q}|\nabla w|_\infty^q;
$$
we see that
\begin{align*}
 \Delta \underline{u}
& \geq \beta   (|\nabla w(x)|^2+w(x))(w(x))^{-2}\\
& \geq C_1 (w(x))^{\gamma_1} C_2 e^{-m}
(w(x))^{-p_2\beta}+\lambda \beta ^q w^{-q}|\nabla w|^q\\
&\geq k(x)g(\underline{u} (x)), \quad \forall x\in \Omega;
\end{align*}
i.e., $\underline {u} = -m-(2+\gamma_1)/{p_1} \ln w(x)$ is an
explosive subsolution of \eqref{Pm}. Hence the desired conclusion
follows by Lemma \ref{lem2.1}.
\end{proof}

\section{Examples}

As an applications of Lemma \ref{lem2.1}, we  give the following two
examples.

\begin{example} \label{ex3.1} \rm
Consider the problem
\begin{equation}
\Delta u =C_0 (R^2-r^2)^{\sigma}u^p,  \quad  u>0, \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
$B_R=\{x\in \mathbb R^N: ||x||<R\}$ with $N\geq2$.

Note that in the case of $\Omega=B_R$, $w(x)=w(r)=(R^2-r^2)/{2N}$.
We see that  $\overline u =M (R^2-r^2)^{-(2+\sigma)/(p-1)}$ is an
explosive supersolution to  \eqref{e3.1} and $\underline{u}
=m(R^2-r^2)^{-(2+\sigma)/(p-1)}$ is an explosive subsolution to
\eqref{e3.1}, where
$$
C_0M^{p-1}=\frac {2(2+\sigma)R^2}{p-1}
\max \big\{N, \frac {2(p+\sigma+1)}{p-1}\big\};
$$
and
$$
C_0m^{p-1}=\frac {2(2+\sigma)R^2}{p-1}
\min \big\{N, \frac {2(p+\sigma+1)}{p-1}\big\}.
$$
Thus  \eqref{e3.1} has at least one positive solution $u$
satisfying
$$
m(R^2-r^2)^{-(2+\sigma)/(p-1)}\leq u(x)\leq M
(R^2-r^2)^{-(2+\sigma)/(p-1)}.
$$
In  particular, if $N=\frac {2(p+\sigma+1)}{p-1}$, i.e., $p=\frac
{N+2(\sigma+1)}{N-2}$ with $N\geq  3$, $M=m$, then  $u =m
(R^2-r^2)^{-(2+\sigma)/(p-1)}$ is an exact solution to
\eqref{e3.1}.
\end{example}

We remark that $u(x)$ is radially symmetric, as shown in \cite{c1}.

\begin{example} \label{ex3.2} \rm
Consider the  problem
\begin{equation}
\Delta u =C_0 (R^2-r^2)^{\sigma}e^u,  \quad   x\in B_R , \quad u
\big|_{\partial B_R}=+\infty,
\label{e3.2}
\end{equation}
where  $C_0$ is a positive constant, $-2<\sigma$ and
$B_R=\{x\in \mathbb{R}^N: \|x\|<R\}$ with $N\geq2$.

 We see that  $\overline u =M-(2+\sigma)\ln (R^2-r^2)$ is an
explosive supersolution to  \eqref{e3.2} and $\underline{u}
=m-(2+\sigma)\ln(R^2-r^2)$ is an explosive subsolution to
\eqref{e3.2}. Where
$$
C_0e^M=2N(2+\sigma)R^2 \quad {\rm and}\quad C_0e^m=4(2+\sigma)R^2.
$$
Thus  \eqref{e3.2} has at least one solution $u$ satisfying
$$
m \leq u(x)+(2+\sigma)\ln (R^2-r^2)\leq M .
$$
In  particular, if $N=2$, $M=m$, then $u =m-
(2+\sigma)\ln(R^2-r^2)$ is an exact solution to  \eqref{e3.2}.
\end{example}

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\end{document}