\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 110, pp. 1--4.\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 2003 Texas State University-San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2003/110\hfil
Existence of large solutions via sub and supersolutions]
{A remark on the existence of large solutions via sub and supersolutions}
\author[Jorge Garc\'{\i}a-Meli\'an\hfil EJDE--2003/110\hfilneg]
{Jorge Garc\'{\i}a-Meli\'an}
\address{Dpto. de An\'alisis Matem\'atico,
Universidad de La Laguna,
c. Astrof\'{\i}sico Francisco S\'anchez s/n, 38271 - La Laguna, Spain
\hfill\break
Centro de Modelamiento Matem\'atico,
Universidad de Chile, Blanco Encalada 2120, 7 piso - Santiago, Chile}
\email{jjgarmel@ull.es}
\date{}
\thanks{Submitted July 4, 2003. Published November 4, 2003.}
\thanks{Supported by grant BFM2001-3894 from FEDER and MCYT (Spain)
\hfill\break\indent and by Centro de Modelamiento Matem\'atico (Chile).}
\subjclass[2000]{35J60, 35J25}
\keywords{Boundary blow-up, sub and supersolutions}
\begin{abstract}
We study the boundary blow-up elliptic problem
$\Delta u=a(x) f(u)$ in a smooth bounded domain
$\Omega\subset \mathbb{R}^N$, with $u|_{\partial\Omega}=+\infty$.
Under suitable growth assumptions on $a$ near $\partial\Omega$
and on $f$ both at zero and at infinity, we prove the existence
of at least a positive solution. Our proof is based on the
method of sub and supersolutions, which permits on the one hand
oscillatory behaviour of $f(u)$ at infinity and on the other hand
positive weights $a(x)$ which are unbounded and/or oscillatory
near the boundary.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
Let $\Omega\subset \mathbb{R}^N$, $N\ge 2$, be a smooth bounded
domain. In this work we consider the boundary blow-up elliptic
problem
\begin{equation}\label{problema}
\begin{gathered}
\Delta u= a(x) f(u) \quad \hbox{in } \Omega\\
u=+\infty \quad \hbox{on } \partial\Omega\,,
\end{gathered}
\end{equation}
where $a$ is a H\"older continuous positive function defined in
$\Omega$ and $f$ is locally H\"older in $(0,+\infty)$. We are
interested in the existence of positive classical solutions to
(\ref{problema}), that is solutions $u\in C^2(\Omega)$ to $\Delta
u = a(x) f(u)$ such that $u(x)\to +\infty$ as
$d(x):=\mathop{\rm dist}(x,\partial\Omega)\to 0+$.
Boundary blow-up problems like (\ref{problema}) have received a
great deal of attention in the recent years. Without being
exhaustive with the references, let us quote \cite{B} (as the
starting point for these problems), \cite{K}, \cite{BM},
\cite{DL}, \cite{LM1}, \cite{LM2}, \cite{MV}, \cite{Z} and
\cite{GLS} (see also references therein).
In the reference situation $f(u)=u^p$ (see hypotheses (1.3)
below), existence and uniqueness of positive solutions to problem
(\ref{problema}) have been obtained before under different kinds
of assumptions on the weight $a(x)$. For instance in \cite{BM} and
\cite{MV} when $a$ is bounded and bounded away from zero,
\cite{GLS} when $a$ is bounded, but {\em is} zero on
$\partial\Omega$, with a prescribed behaviour or \cite{Z} and
\cite{GX} where $a$ goes to $+\infty$ also in a completely
determined way. Our existence result (Theorem \ref{primero})
covers all these situations, and also more general weights which
can behave as the three cases above in different parts of
$\partial \Omega$. The advantage of our approach is that all
possible cases are treated together with a very simple proof,
based on the case $a(x)\equiv 1$. Also, at the best of our
knowledge, this seems to be the first time where the method of sub
and supersolutions is used to prove existence of solutions to
boundary blow-up problems.
We start by quoting our hypotheses on $a$ and $f$. We will assume
that $a\in C^\nu (\Omega)$ for some $0<\nu<1$, $a>0$ in $\Omega$,
and that $f \in C^\nu(0,+\infty)$. In addition there exist
constants $C_1, C_2>0$ and $\gamma_2\ge \gamma_1 >-2$ such that
\begin{equation}
C_2 d(x)^{\gamma_2} \le a(x) \le C_1 d(x)^{\gamma_1}, \quad x\in\Omega.
\label{A}
\end{equation}
Note that $\gamma_1$ and $\gamma_2$ can have
different signs, and so $a$ is permitted to be bounded in some
parts of $\partial \Omega$, and to go to $+\infty$ or even
oscillate in some others (it can also go to zero).
For the nonlinearity $f$ we further assume that there exist
$p_1\ge p_2>1$ such that
\begin{equation}
f(u) \le C_1 u^{p_1} \quad u\in \mathbb{R}^+, \quad f(u) \ge C_2 u^{p_2}
\quad \hbox{for large } u \,. \label{F}
\end{equation}
Note that we can take the constants $C_1$ and $C_2$ to be the same as
in \eqref{A}. These assumptions allow $f(u)$ to be oscillating for large $u$,
i.e. $f$ does not need to be increasing at infinity. We remark that
$\gamma_2\ge \gamma_1$ and $p_1\ge p_2$ are nothing else but
compatibility conditions, and $\gamma_1>-2$ is necessary in order
to have positive solutions to (\ref{problema}) (compare with
\cite{GX} in the radial case and $f(u)=u^p$). We now state our
Theorem.
\begin{theorem} \label{primero}
Assume $a$ and $f$ verify
hypotheses \eqref{A} and \eqref{F}. Then problem (\ref{problema}) has at least
a positive solution $u$ which verifies
\begin{equation}\label{estimacion}
D_1 d(x)^{-\alpha_1} \le u(x) \le D_2 d(x)^{-\alpha_2} \quad
\hbox{in }\Omega,
\end{equation}
where $\alpha_i=(2+\gamma_i)/(p_i-1)$, and $D_1$, $D_2$ are
positive constants.
\end{theorem}
\begin{remark} \rm
(a) Note that $p_1\ge p_2$, $\gamma_1\le \gamma_2$
imply that $\alpha_1 \le \alpha_2$, and thus (\ref{estimacion})
makes sense.
\noindent b) In the light of the possible oscillatory behaviour of
$a$ and $f$, according to hypotheses (1.2) and (1.3), one would
expect that estimates (\ref{estimacion}) can not be improved in
general, even if $p_1=p_2$ and $\gamma_1=\gamma_2$
($\alpha_1=\alpha_2$), in contrast with the case when $a$ has a
prescribed asymptotics near $\partial\Omega$ and $f$ near
$+\infty$.
\noindent c) If $f(u)=u^p$, $p>1$, uniqueness of positive
solutions verifying (\ref{estimacion}) can be achieved for
positive weights $a$ satisfying (1.2) with $\gamma_1=\gamma_2$,
through an adaptation of the proof of Theorem 3.4 in \cite{Ki}.
\noindent d) The regularity assumptions on $a$ and $f$ can of
course be relaxed to continuity, obtaining weak solutions to
(\ref{problema}) in that case.
\noindent e) Theorem \ref{primero} can be adapted to nonlinearities with a
different type of growth, for instance exponential:
\begin{equation} \label{F'}
f(u) \le C_1 e^{p_1 u} \quad u\in \mathbb{R}^+, \quad
f(u) \ge C_2 e^{p_2 u} \quad \hbox{for large } u,
\end{equation}
and we obtain the existence of at least a classical solution $u$
such that $2\lambda_2 \log d + C \le u \le 2\lambda_1 \log d +C'$,
where $\lambda_i=(\gamma_i+2)/p_i$.
\end{remark}
\section{Results}
This section is devoted to the proof of Theorem \ref{primero}
and the method of sub and supersolutions. First we state and prove
an adaptation of the method of sub and supersolutions to problem
(\ref{problema}) (Lemma \ref{subsuper} below is indeed a
slight generalization of Lemma 4 in \cite{GLS}, which was not proved there),
then we introduce an auxiliary problem which will turn out to be very
important for our purposes, and we finally will proceed to the
proof of Theorem \ref{primero}.
A function $\underline{u} \in C^2(\Omega)$ is a (classical)
subsolution to problem (\ref{problema}) if $\underline{u}=+\infty$
on $\partial\Omega$ and $\Delta \underline{u} \ge
a(x)f(\underline{u})$ in $\Omega$. Similarly, $\bar{u}$ is a
supersolution if $\bar{u}=+\infty$ on $\partial\Omega$ and $\Delta
\bar{u} \le a(x)f(\bar{u})$ in $\Omega$. When $\underline{u}$ and
$\bar{u}$ are ordered we have the next result.
\begin{lemma} \label{subsuper}
Assume there exist a subsolution
$\underline{u}$ and a supersolution $\bar{u}$ to the problem
(\ref{problema}) such that $\underline{u} \le \bar{u}$. Then there
exists at least a classical solution $u$ such that
$\underline{u}\le u\le \bar{u}$.
\end{lemma}
\begin{proof} For $n\in \mathbb{N}$, we introduce the
domain $\Omega_n:=\{x\in \Omega: d(x)>1/n\}$, and consider the
problem
\begin{equation}\label{finito}
\begin{gathered}
\Delta u= a(x) f(u) \quad \hbox{in } \Omega_n\\
u=\underline{u} \quad \hbox{on } \partial\Omega_n\ .
\end{gathered}
\end{equation}
Since $\underline{u}$ is a subsolution and $\bar{u}$ a
supersolution, this problem has at least a positive classical
solution $u_n$ such that $\underline{u} \le u_n\le \bar{u}$. This
in particular gives local bounds for the sequence $\{u_n\}$ which
in turn leads to local bounds in $C^{2,\nu}$ (cf. \cite{GT}). Thus
for every $k\in \mathbb{N}$, we can select a subsequence $\{u_n^k\}$
which converges in $C^2(\overline{\Omega}_k)$. A diagonal
procedure gives a subsequence (denoted again by $\{u_n\}$) which
converges to a function $u$ in $C^2_{\rm loc}(\Omega)$. Passing to
the limit in (\ref{finito}) we see that $u$ is a classical
solution of the equation in (\ref{problema}), verifying
$\underline{u} \le u \le \bar{u}$. In particular, we deduce that
$u=+\infty$ on $\partial\Omega$. This proves the Lemma.
\end{proof}
As already remarked, a fundamental role in our approach is
played by the well-known blow-up problem:
\begin{gather*}
\Delta U= U^{r_i} \quad \hbox{in } \Omega\\
U=+\infty \quad \hbox{on } \partial\Omega\,,
\end{gather*}
where $r_i=1+2/{\alpha_i}>1$. This problem has a unique positive
solution $U_i$ such that
$C d(x)^{-\alpha_i} \le U_i(x) \le C' d(x)^{-\alpha_i}$, for some
positive constants $C$ and $C'$ (see \cite{BM}).
\begin{proof}[Proof of Theorem \ref{primero}]
The proof consists
in choosing adequate ordered sub and supersolutions in terms of
the functions $U_1$ and $U_2$ defined above. Indeed, we set
$\underline{u} =\lambda U_1$. Then $\underline{u}$ will be a
subsolution provided that
$$
\lambda U_1^{r_1} \ge a(x) f(\lambda U_1)\,.
$$
By hypothesis \eqref{F} on $f$, this is a consequence of
$\lambda \le (C_1\sup_\Omega a(x) U_1(x) ^{p_1-r_1})^{-\frac{1}{p_1-1}}$,
which holds for small $\lambda$ if the supremum is finite. But note
that $a(x) U_1^{p_1-r_1}\le C d(x)^{\gamma_1-\alpha_1(p_1-r_1)}=C$
in virtue of hypotheses \eqref{A}, and the claim follows.
In a similar way we can see that $\bar u=\Lambda U_2$ is a supersolution
for large $\Lambda$. Since $\alpha_1\le \alpha_2$, it also follows that
$\lambda U_1 \le \Lambda U_2$, and
Lemma \ref{subsuper} shows that there exists at least a positive
classical solution to (\ref{problema}), which in addition verifies
the estimates (\ref{estimacion}). This proves the Theorem.
\end{proof}
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\end{document}