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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 134, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/134\hfil Bounds and compactness for solutions]
{Bounds and compactness for solutions of second-order
elliptic equations}

\author[C. C. Aranda\hfil EJDE-2012/134\hfilneg]
{Carlos C. Aranda} 

\address{Carlos Cesar Aranda \newline
 Blue Angel Navire Research laboratory\\
 Rue Eddy 113 Gatineau QC, Canada}
\email{carloscesar.aranda@gmail.com}

\dedicatory{Dedicated to my mother Gregoria Ynes Zalazar}

\thanks{Submitted July 25, 2012. Published August 17, 2012.}
\subjclass[2000]{35J25, 35J60, 35J75}
\keywords{Elliptic equations; compact imbedding; Schauder approach}

\begin{abstract}
 In this article, we establish some connections between Sobolev 
 spaces   and nonlinear singular elliptic problems, to obtain bounds
 and compactness results for solutions of second-order elliptic equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction and results}

The main purpose of this paper is to develop some connections between 
Sobolev spaces and nonlinear singular elliptic problems to obtain bounds 
and compactness results for solutions of second-order elliptic equations, 
where the structure of the imbedding is nonlinear. 
The theory of singular nonlinear elliptic problems is fairly well developed. 
(See for example \cite{ag,dgr,hm,fm,gr,gs,gst} for a survey and bibliography.)
In \cite{lu} it is stated that
\begin{quote}
During the past half century, linear second order elliptic equations 
on bounded regions have been studied, if not exhaustively, at least with 
reasonable completeness and the fundamental questions concerning them have 
received rather simple solutions. In the works of Giraud and  Schauder
in the thirties, it was shown that the basic boundary value problems are 
solvable for such equations under the assumption of sufficient smoothness of 
the coefficients and of the boundary of the region.
Then, there were interpreted from the standpoint of functional analysis.
This approach was initiated by the article \cite{fie} of  Friedrichs in 1934 
on semibounded extensions of symmetric elliptic operators. 
This  article and further studies of Friedrichs,  Mikhlin, Vishik, and  others 
during the late forties showed that the solution of the classical boundary-value
problems for elliptic equations (we are only speaking of second order equations)
was equivalent to solving equations of the form $x+Ax=f$, for a completely 
continuous operator $A$ in certain Hilbert spaces constructed 
from the quadratic form  of the principal symmetric part of a differential operator.
\end{quote}
For a review of the state of the art on this topic, see \cite{b,br,dl,k,r,wyw}.

\begin{theorem}[{\cite[Theorem 7.26]{gt}}] \label{thm1}
 Let $\Omega$ be a $C^{0,1}$ domain in $\mathbb{R}^N$. Then
\begin{itemize}
\item[(i)] if $kp<N$, the space $W^{k,p}(\Omega)$ is continuously imbedded in 
$L^{p^*}(\Omega)$, $p^*=\frac{Np}{N-kp}$, and compactly imbedded in $L^q(\Omega)$ 
for any $q<p^*$.
\item[(ii)] if $0\leq m<k-\frac{N}{p}<m+1$, the space $W^{k,p}$ is continuously 
imbedded in $C^{m,\alpha}(\overline{\Omega})$, $\alpha=k-\frac{N}{p}-m$, 
and compactly imbedded in $C^{m,\beta}(\overline{\Omega})$ for any $\beta<\alpha$
\end{itemize}
\end{theorem}

\begin{theorem}[{\cite[Theorem 6.6]{gt}}] \label{thm2}
Let $\Omega$ be a $C^{2,\alpha}$ domain in $\mathbb{R}^N$ and let
 $u\in C^{2,\alpha}(\overline{\Omega})$ be a solutions of the equation
\begin{equation} \label{schauder}
\mathcal{L}u\equiv \sum_{i,j=1}^Na_{ij}u_{x_i,x_j}+\sum_{i=1}^Nb_iu_{x_i}+cu=f
\end{equation}
where $f\in C^\alpha(\overline{\Omega})$ and the coefficients of $\mathcal{L}$
satisfy, for positive constants $\lambda$, $\Lambda$, 
\begin{gather*}
\sum_{i,j=1}^Na_{ij}\xi_i\xi_j\geq\lambda|\xi|^2 \quad \text{for all }
x\in\Omega, \; \xi\in\mathbb{R}^N,
\\
|a_{i,j}|_{0,\alpha;\Omega}, \; |b_i|_{0,\alpha;\Omega}, \;
 |c|_{0,\alpha;\Omega}\leq\Lambda\,.
\end{gather*}
Let $\varphi\in C^{2,\alpha}(\overline{\Omega})$, and suppose $u=\varphi$
 on $\partial\Omega$. Then
\begin{equation}
|u|_{2,\alpha;\Omega}\leq C\{ |u|_{0,\Omega}+|\varphi|_{2,\alpha ;\Omega}
+|f|_{0,\alpha;\Omega}\}
\end{equation}
where $C=C(n,\alpha,\lambda,\Lambda)$.
\end{theorem}

Our main concern is related to a quotation from \cite{lu}:
\begin{quote}
We pose the following question: To what classes $L^s(\Omega)$ must the 
functions $a_i,  b_i,  c,  f_i$ and $f$ belong in order that all 
generalized solutions $u(x)$ of the equations
\begin{equation}
Lu\equiv\sum_{i=1}^n\frac{\partial}{\partial x_i}[a_{ij}(x)u_{x_j}+a_i(x)u]
+\sum_{i=1}^nb_i(x)u_{x_i}+a(x)u=f-\sum_{i=1}^n\frac{\partial f_i}{\partial x_i}
\end{equation}
in $W^{1,2}(\Omega)$ be bounded functions? To ascertain the necessary conditions,
we again take the function $u=\log |\log r|$ and regard it in the sphere $r\leq R<1$
as a solution of any one of the following equations
\begin{gather*}
\Delta u=F(r), \quad
\Delta u-\frac{\partial}{\partial x_i}\Big( \frac{x_i}{r^2\log r}\Big)=0,\\
\Delta u -\frac{F(r)}{\log|\log r|}u=0, \quad
\Delta u-\frac{\partial}{\partial x_i}
\Big( \frac{x_i u}{r^2\log r\log|\log r|}\Big)=0
\end{gather*}
where $F(r)$ has the same meaning as above. It is easy to see that in these
functions $f,a\in L^{\frac{N}{2}}(K_R)$ and $f_i,a_i\in L^N(K_R)$.
Therefore these last conditions does not ensures boundedness of the generalized
solutions. Therefore the requirements
$\| a_i, b_i, f_i \|_{L^q(\Omega)}\leq\mu<\infty$
 $\| a, f\|_{L^{\frac{q}{2}}(\Omega)}\leq\mu<\infty $; $q>N$  are necessary.
\end{quote}

We introduce now the equation
\begin{equation} \label{delilah}
-\mathcal{L}u=g(u),\text{ in }\Omega, \quad  u=0\text{ on }\Omega,
\end{equation}
where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq 3$,
$ c\leq 0$ in $\Omega$ and $g:(0,\infty) \to(0,\infty)$ is non-increasing
locally H\"{o}lder continuous function  singular
at the origin. It is well known that problem \ref{delilah} in the case
$\mathcal{L}=\Delta$ and $g(u)=u^{-\gamma}$ has a unique classical bounded
solution $u$ for all $\gamma >0$. This  solution $u$ belongs to the Sobolev
space $H^1_0(\Omega)$ if and only if $0<\gamma<3$. Moreover $\gamma>1$
implies $C_0\varphi_1^{-\frac{2\gamma s}{1+\gamma}}\leq u^{-s\gamma}
\leq C_1\varphi_1^{-\frac{2\gamma s}{1+\gamma}}$ where $\varphi_1$ is
the principal eigenfunction of the laplacian operator
($-\Delta\varphi_1=\lambda_1\varphi$, in $\Omega$, $\varphi_1=0$ on
 $\partial\Omega$)  \cite{ag,gs}.
Therefore $u^{-\gamma}$ not belong to any $L^s(\Omega)$, $s\geq 1$ for
$\gamma>1$ because   $\int_\Omega\varphi_1^{-r}dx<\infty$ for $r\geq 0$
if and only if $0\leq r<1$.
Nevertheless, we have an unexpected nonlinear compact imbedding:

\begin{theorem}[Aranda-Godoy \cite{ag}]\label{H}
Let $P$ be the positive cone in $L^\infty(\Omega)$.
Let $S_{\epsilon}:P\to P$ be the solution operator
for the problem
\begin{equation} \label{singular}
-\Delta u  =  g(u)+w  \text{ in }\Omega, \quad
        u  =  \epsilon   \text{ on }\partial\Omega,
\end{equation}
gives $S_{\epsilon}(w)=u$ where $\epsilon\geq 0$.
Then $S_{\epsilon}:P\to P$ is a continuous,
non decreasing and compact map with
$S_{\epsilon_0}(w)\leq S_{\epsilon_1}(w)$ for $\epsilon_0<\epsilon_1$.
\end{theorem}

The derivations of our results are very elementary using a Schauder approach.

We set
\[\mathcal{C}^{\alpha,g,+}_{\rm loc}(\Omega)
=\{f\in C^\alpha_{\rm loc}(\Omega)|0\leq f\leq g(u) 
\text{ where } u\text{ solves }\ref{delilah}\}
\]
Our main result follows.

\begin{theorem}\label{t1}
Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$, $N\geq 3$.
 Then the equation
\begin{equation}\label{samson}
-\mathcal{L}v=f \text{ in }\Omega, \quad v=0\text{ on }\partial\Omega,
\end{equation}
where $f\in\mathcal{C}^{\alpha,g,+}_{\rm loc}(\Omega)$ has a unique solution 
$v\in C^\alpha_{\rm loc}(\Omega)\cap C^0(\overline{\Omega})\cap C^2(\Omega)$ 
with $0\leq v\leq u$ in $\Omega$ and $u$ solves equation \ref{delilah}.
\end{theorem}

Our imbedding theorem is as follows.

\begin{theorem}\label{t2} 
Let $\mathcal{P}$ be the cone of positive functions in $C^0(\overline{\Omega})$.
Let $\mathcal{S}:\mathcal{C}^{\alpha,g,+}_{\rm loc}(\Omega)\to\mathcal{P}$
the solution operator of problem \ref{samson} gives $\mathcal{S}(f)=v$. 
Then $\mathcal{S}$ is continuous and compact. Moreover $g_m\leq g_{m+j}$ implies 
$\mathcal{C}^{\alpha,g_m,+}_{\rm loc}(\Omega)\subset
\mathcal{C}^{\alpha,g_{m+j},+}_{\rm loc}(\Omega)$.
\end{theorem}

 Finally our last result is the infinite tower property.

\begin{theorem}\label{t3}
Let us consider the equation
\begin{equation}
-\Delta u_{m}  =  g_m(u_m ) \text{ in }B_R(0),\quad
u_{\epsilon,m}  =  \epsilon   \text{ on }\partial B_R(0),
\end{equation}
where $g_m:(0,\infty)\to(0,\infty)$ is non increasing
locally H\"{o}lder continuous function
singular at the origin with the properties $g_m(s)=g(s)$ for all $s\geq 1$ 
and $\lim_{m\to\infty}g_m(s)=\infty$ for all $s\in(0,1)$, $m=1,\dots,\infty$. 
Then there exists $\delta>0$ and $u_\infty$ such that $\lim_{m\to\infty}u_m=u_\infty$ 
where $-\Delta u_\infty=\lim_{m\to\infty}g_m(u_m)=\infty$ on the annulus 
$A(R-\delta,R)$. Therefore the tower
\[
 \mathcal{C}^{\alpha,g_1,+}_{\rm loc}(\Omega)\subset\cdots
\subset\mathcal{C}^{\alpha,g_m,+}_{\rm loc}(\Omega)\subset\cdots
\subset\mathcal{C}^{\alpha,g_{m+j},+}_{\rm loc}(\Omega)\subset\cdots
 \]
 actually goes to infinite on the annulus $A(R-\delta,R)$.
\end{theorem}

\section{Auxiliary results}

Let us consider the problem
\begin{equation}\label{perdido estoy}
-\mathcal{L} u_m =  g_m(u_m ) \text{ in }\Omega,\quad
u_m    =  0  \text{ on }\partial\Omega,
\end{equation}
where  $g_{m+j}\geq g_m$, are non-increasing  locally H\"{o}lder continuous 
functions on $(0,\infty)$ and singular at zero.

\begin{lemma}\label{fulgor}
Let $u_m$ be a solution of  \eqref{perdido estoy}. 
Then $u_{m+j}\geq u_m$.
\end{lemma}

\begin{proof}
Suppose that there exists $x_0\in\Omega$ such that $u_m(x_0)>u_{m+j}(x_0)$.
 We define $\Omega_{\nu}=\{x\in\Omega |\nu+u_m(x)>u_{m+j}(x)\}$.
Then $\Omega_{\nu}\neq\emptyset$ for all $\nu\geq 0$.
Moreover  $g_m(u_m(x)+\nu)\leq g_{m+j}(u_m(x)+\nu)<g_{m+j}(u_{m+j}(x))$ 
for all $x\in\Omega_{\nu}$.
Setting
\[
\Omega_{\tau}=\{x\in\Omega |u_m(x)>\tau +u_{m+j}(x)\},
\]
we deduce that $\Omega_{\tau}\neq\emptyset$ and 
$\Omega_{\tau}\subset\Omega_{\nu}$ for $\tau$ small enough.
 Moreover, $g_{m}(u_m(x))\leq g_{m+j}(u_m(x))\leq g_{m+j}(u_{m+j}(x))$ 
for all $x\in\Omega_{\tau}$. Therefore
\[
-\mathcal{L} u_m \leq  -\mathcal{L} (u_{m+j}+\tau)
 \text{ in }\Omega_\tau, \quad
        u_m  = u_{m+j}+\tau   \text{ on }\partial\Omega_\tau.
\]
and we obtain $u_m\leq u_{m+j}+\tau$ in $\Omega_\tau$ 
\cite[Theorem 3.3]{gt}, a contradiction.
\end{proof}

\begin{lemma}\label{m29}
Let $u_m$ be a solution of  \eqref{perdido estoy}. Then
 $g_{m+j}(u_{m+j}(x))\geq g_m(u_m(x))$.
\end{lemma}

\begin{proof}
Suppose that there exists $x_0\in\Omega$ such that
 $g_m(u_m(x_0))>g_{m+j}(u_{m+j}(x_0))$. Then there exists
 $\hat{\Omega}\subset\Omega$ such that
\[
 -\mathcal{L} u_m \geq  -\mathcal{L} u_{m+j}
  \text{ in }\hat{\Omega}, \quad
         u_m  = u_{m+j}   \text{ on }\partial\hat{\Omega}.
\]
We infer that  $u_m\geq u_{m+j}$ in $\hat{\Omega}$ 
\cite[Theorem 3.3]{gt}. 
Therefore, $g_m(u_m(x))\leq g_m(u_{m+j}(x))\leq g_{m+j}(u_{m+j}(x))$ for all
 $x\in\hat{\Omega}$. A contradiction.
\end{proof}

\begin{proof}[Proof of Theorem \ref{t1}]
For any $f\in\mathcal{C}^{\alpha,g,+}_{\rm loc}(\Omega)$, we have 
$f_k=\min(k,f)\in C^\alpha(\overline{\Omega})$. Therefore there exist a 
unique solution $v_k\in C^{2,\alpha}(\overline{\Omega})$ of the problem
\begin{equation}
-\mathcal{L}v_k=f_k\text{ in }\Omega, \quad
 v_k=0\text{ on }\partial\Omega
\end{equation}
Using \cite[Corollary 6.3]{gt}, we obtain
\begin{align*}
d|Dv_k|_{0;\Omega'}+d^2|D^2v_k|_{0;\Omega'}
+d^{2+\alpha}[D^2v_k]_{\alpha ;\Omega'}
\leq C(|v_k|_{0;\Omega''}+|f_k|_{0,\alpha;\Omega''})
\end{align*}
where $\Omega'\subset\Omega''\subset\Omega$,
$d=\text{dist}(\Omega',\partial\Omega'')$  and $C$ is independent of $k$.
 Moreover $v_k\leq u$, it follows that $v_k\to v$ in
$C^2_{\rm loc}(\Omega)\cap C^0(\overline{\Omega})$ were $v$ solves
 equation \ref{samson}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{t2}]
This theorem is a direct consequence of the proof of Theorem \ref{t1} and
 Lemmas \ref{fulgor} and \ref{m29} .
\end{proof}

\begin{proof}[Proof of Theorem \ref{t3}] 
This theorem is a direct consequence of \cite{a}.
\end{proof}

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\end{document}