\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 160, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/160\hfil Nonlinear elliptic BVP in unbounded domains]
{Existence of weak solutions for degenerate semilinear elliptic
equations in unbounded domains}

\author[V. Raghavendra, R. kar \hfil EJDE-2009/160\hfilneg]
{Venkataramanarao Raghavendra, Rasmita Kar}  % in alphabetical order

\address{Venkataramanarao Raghavendra \newline
Department of Mathematics and Statistics,
Indian Institute of Technology, Kanpur, India 208016}
\email{vrag@iitk.ac.in}

\address{Rasmita Kar \newline
Department of Mathematics and Statistics,
Indian Institute of Technology, Kanpur, India 208016}
\email{rasmita@iitk.ac.in}

\thanks{Submitted October 25, 2009. Published December 15, 2009.}
\subjclass[2000]{35J70, 35D30}
\keywords{Degenerate equations; weighted Sobolev space;
unbounded domain}

\begin{abstract}
 In this study, we prove the existence of a weak solution for
 the degenerate semilinear elliptic Dirichlet boundary-value problem
  \begin{gather*}
  Lu-\mu u g_{1} + h(u) g_{2}= f\quad \text{in }\Omega,\\
  u = 0\quad \text{on }\partial\Omega
 \end{gather*}
 in a suitable weighted Sobolev space.
 Here the domain $\Omega\subset\mathbb{R}^{n}$, $n\geq 3$, is
 not necessarily bounded, and  $h$ is a continuous bounded
 nonlinearity. The theory is also extended for $h$ continuous
 and unbounded.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

Let $\Omega\subset\mathbb{R}^n$, $n\geq 3$, be a domain (not necessarily
bounded) with boundary $\partial\Omega$. Let $L$ be an elliptic
operator in divergence form
\begin{equation*}
   Lu(x)=-\sum_{i,j=1}^nD_j(a_{ij}(x) D_iu(x))\quad
\text{with }D_j=\frac{\partial}{\partial x_j},
\end{equation*}
with coefficients $a_{ij}/\omega \in L^{\infty}(\Omega)$
which are symmetric and satisfy the degenerate ellipticity
condition
\begin{equation}\label{eq:a1}
  \lambda|\xi|^2\omega(x)\leq\sum_{i,j=1}^{n}a_{ij}(x)\xi_i\xi_j
  \leq\Lambda|\xi|^2\omega(x),\quad \text{a.e. }x\in\Omega,
\end{equation}
 for all $\xi\in\mathbb{R}^n$ and $\omega$ is an ${A_2}$-weight
$(\lambda>0,\Lambda>0)$. Let
$f/\omega \in L^{2}(\Omega,\omega)$ and $h$ be a real valued
continuous function defined on $\mathbb{R}$. Recently  Cavalheiro
\cite{sem} studied the BVP
\begin{equation}\label{eq:a2}
\begin{gathered}
  Lu-\mu ug_1+h(u)g_2=f\quad \text{in }\Omega, \\
  u=0\quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $g_1/\omega \in L^{\infty}(\Omega)$, $\mu>0$, $h$ is a bounded
continuous function and where $\Omega$ is bounded. In general, the
Sobolev spaces $W^{k,p}(\Omega)$ without weights occurs as spaces of
solutions for elliptic and parabolic PDEs. For degenerate problems
with various types of singularities in the coefficients it is
natural to look for solutions in weighted Sobolev spaces; for
example, see \cite{app,wei,chi,wie2,loc,fra}.

The treatment of problem \eqref{eq:a2} has not been effective since
the usual compactness arguments for bounded domains may not extend
to unbound domains. One natural approach is to approximate a
solution of \eqref{eq:a2} by a sequence of solutions in bounded
subdomains of $\Omega$. The present work is a generalization of the
work by  Cavalheiro \cite{sem}, for unbounded domain $\Omega$
such that,
$\Omega=\cup^\infty_{i=1}\Omega_{i}$,
$\Omega_i\subseteq \Omega_{i+1}$, for each $i\geq1$.
Section 2 deals with preliminaries
and some basic results.
Section 3 contains the existence of a
sequence of solutions $\{u_i\}$ of \eqref{eq:a2} in each bounded
subdomains $\Omega_i$ and a uniform bound for them. The main result
is about the extraction of a solution for \eqref{eq:a2} from
$\{u_i\}$. Finally section 4 deals with extension for a class of
continuous function $h$, not necessarily bounded.

\section{Preliminaries}

We need the following preliminaries for the ensuing study. Let
$\Omega\subset \mathbb{R}^{n}$, $n\geq3$ be an open connected set. Let
$\omega:\mathbb{R}^{n}\to\mathbb{R}^{+}$ be a locally integrable non
negative function with $0<\omega<\infty$ a.e. We say that $\omega$
belongs to the Muckenhoupt class $A_p$, $1 < p < \infty$, or that
$\omega$ is an $A_p$-weight, if there is a constant
$c = c_{p,\omega}$  such that
\begin{equation*}
\big(\frac{1}{|B|}\int_{B}\omega(x)dx\big)\big(\frac{1}{|B|}
\int_{B}{\omega}^{\frac{1}{1-p}}(x)dx\big)^{p-1}
\leq c,
\end{equation*}
for all balls $B$ in $\mathbb{R}^{n}$, where $|.|$ denotes the
$n$-dimensional Lebesgue measure in $\mathbb{R}^{n}$. We assume that
$\omega$ belongs to Muckenhoupt class $A_p$, $1<p<\infty$ (i.e.
$\omega$ is an $A_p$-weight). For more details on $A_p$-weight, we
refer the reader to \cite{gar,hei,pot2}. We shall denote by
$L^p(\Omega,\omega)$ $(1\leq p<\infty)$ the usual Banach space of
measurable real valued functions, $f$, defined in $\Omega$ for which
\begin{equation}
  \|f\|_{p,\Omega}=\Big(\int_{\Omega} |f(x)|^p\omega(x)
  dx\Big)^{1/p}<\infty
\end{equation}
For $p\geq1$ and $k$ a non-negative integer, the weighted Sobolev
space $W^{k,p}(\Omega,\omega)$ is defined by
\begin{equation*}
  W^{k,p}(\Omega,\omega):=\{u\in L^{p}(\Omega,\omega): D^\alpha u\in
  L^{p}(\Omega,\omega),1\leq|\alpha|\leq k\}
\end{equation*}
with the associated norm
\begin{equation}\label{eq:b1}
  \|u\|_{k,p,\Omega}=\|u\|_{p,\Omega}+\sum_{1\leq|\alpha|\leq
  k}\|D^\alpha u\|_{p,\Omega}.
\end{equation}
If $\omega\in A_p$ then $W^{k,p}(\Omega,\omega)$ is
the closure of $C^{\infty}(\overline\Omega)$ with respect to the
norm (\ref{eq:b1}) and the space $W_0^{k,p}(\Omega,\omega)$ is
defined as the closure of $C^{\infty}_0(\Omega)$ with respect to the
norm
\begin{equation*}
  \|u\|_{0,k,p,\Omega}=\sum_{1\leq|\alpha|\leq k}\|D^\alpha
  u\|_{p,\Omega}.
\end{equation*}
For details we refer the reader to \cite[Proposition 3.5]{chi}. We
also note that $W^{k,2}(\Omega,\omega)$ and
$W^{k,2}_0(\Omega,\omega)$, are Hilbert spaces. At each step, a
generic constant is denoted by $c$ or $k_0$ in order to avoid too
many suffices.
We need the following result.

\begin{lemma} \label{lem2.1}
Let $\Omega\subset\mathbb{R}^n$, $n\geq3$ be a bounded domain and
let $\omega$ be $A_2$ weight. Then
\begin{equation}\label{eq:b2}
  W_0^{1,2}(\Omega,\omega)\hookrightarrow\hookrightarrow
  L^2(\Omega,\omega)
\end{equation}
(i.e the inclusion is compact) and there exists $C_{\Omega}>0$ such that
\begin{equation}\label{eq:b3}
  \|u\|_{2,\Omega}\leq C_{\Omega}\|u\|_{0,1,2,\Omega},\quad
 \forall u\in   W^{1,2}_0(\Omega,\omega),
 \end{equation}
where $C_{\Omega}$ may be taken to depend only on $n,2$ and
the diameter of $\Omega$.
\end{lemma}

A proof of the above statement can be found in
\cite[Theorem 4.6]{fra}.


\begin{definition} \label{def2.2} \rm
Let $\Omega\subset\mathbb{R}^n$ be an open connected set.
We say that $u\in W^{1,2}_0(\Omega,\omega)$ is a called a weak
solution of \eqref{eq:a2} if
\begin{align*}
&\int_{\Omega} a_{ij}D_iu(x)D_j\phi(x)dx-\int_\Omega\mu
  u(x)g_1(x)\phi(x)dx+\int_\Omega h(u(x))g_2(x)\phi(x)dx\\
&=\int_\Omega   f(x)\phi(x)dx
\end{align*}
for every $\phi\in W^{1,2}_0(\Omega,\omega)$.
\end{definition}

In section 3, we use the following result.

\begin{theorem} \label{thm2.3}
Let $B,N : X\to X^\ast$ be operators on the real separable
reflexive Banach space $X$.
\begin{enumerate}
  \item the operator $B:X\to X^\ast$ is linear and continuous;
  \item the operator $N:X\to X^\ast$ is demicontinuous and bounded;
  \item $B+N$ is asymptotically linear;
  \item for each $T\in X^{\ast}$ and for each $t\in[0,1]$,
  the operator $A_t(u)=Bu+t(Nu-T)$ satisfies condition $(S)$ in $X$.
\end{enumerate}
If $Bu=0$ implies $u=0$, then for each $T\in X^{\ast}$, the equation
$Bu+Nu=T$ has a solution in $X$.
\end{theorem}

For a detailed proof of the above Theorem, we refer to \cite{hes}
 or to \cite[Theorem 29.C]{zed1}.


\begin{definition} \label{def2.4} \rm
Let $B :X\to X^*$ be an operator
on the real separable reflexive Banach space $X$. Then, $B$
satisfies condition (S) if
\begin{equation}\label{e:s}
     u_n\rightharpoonup u \text{ and }
     \lim_{n\to\infty}(Bu_n-Bu|u_n-u)=0, \text{ implies } u_n\to u,
\end{equation}
where $(f|x)$ denotes the value of linear functional $f$ at $x$.
\end{definition}

We need the following hypotheses for further study.
\begin{enumerate}
\item [(H1)] Let $h:\mathbb{R}\to\mathbb{R}$ be a continuous
 and bounded function;

\item [(H2)] $\omega\in A_2$;

\item [(H3)] Assume $g_1/\omega \in L^{\infty}(\Omega)$,
 $g_2/\omega \in  L^{2}(\Omega,\omega)$ and
 $f/\omega \in    L^{2}(\Omega,\omega)$.
\end{enumerate}

\begin{remark} \label{rmk2} \rm
If $u_k\in W^{1,2}_0(\Omega_k,\omega)$
is a solution of  (\ref{eq:b5}) (see below) on $\Omega_k$, then,
 for any $k\geq i$, $u_k$ is also a solution of (\ref{eq:b5}) on
$\Omega_i$, which has been used in Lemma \ref{lem2.4}.
\end{remark}

\begin{lemma} \label{lem2.4}
Assume {\rm (H1)-(H3)}. Let $\mu>$$0$ not be  an
eigenvalue of
\begin{gather*}
  Lu-\mu u(x)\omega(x)=0\quad \text{in }\Omega_i,\\
 u=0\quad \text{on   }\partial \Omega_i
\end{gather*}
for $i=1,2,3,\dots$ Then, the BVP
\begin{equation}\label{eq:b5}
\begin{gathered}
  Lu-\mu ug_1 + h(u)g_2=f\quad \text{in }\Omega_i,\\
 u=0\quad \text{on  }\partial \Omega_i
\end{gathered}
\end{equation}
has a solution $u=u_i\in W^{1,2}_0(\Omega_i,\omega)$.
In addition, if
\begin{equation}\label{eq:41}
  \lambda>\mu\;C_{\Omega_i}\big\|\frac{g_1}{\omega}\big\|_{\infty,\Omega},
\end{equation}
 then for $k\geq i$,
$\|u_k\|_{0,1,2,\Omega_i}\leq k_0$,   where $k_0$
is  independent of $k$.
\end{lemma}

\begin{proof}
We define the operators $B_1,B_2: W_0^{1,2}(\Omega_i,\omega)\times
W_0^{1,2}(\Omega_i,\omega)\to\mathbb{R}$ by
\begin{gather*}
  B_1(u,\phi)=\int_{\Omega_i}
  a_{ij}D_iu(x)D_j\phi(x)dx-\int_{\Omega_i}\mu u(x)g_1(x)\phi(x)dx\\
  B_2(u,\phi)=\int_{\Omega_i} h(u(x))g_2(x)\phi(x)dx.
\end{gather*}
Also define $T:W_0^{1,2}(\Omega_i,\omega)\to\mathbb{R}$ by
\begin{equation*}
  T(\phi)=\int_{\Omega_i}f(x)\phi(x)dx.
\end{equation*}
A function $u=u_i\in W_0^{1,2}(\Omega_i,\omega)$ is a solution
of (\ref{eq:b5}) if
\[
B_1(u,\phi)+B_2(u,\phi)=T(\phi),\quad
\forall\phi\in W_0^{1,2}(\Omega_i,\omega).
\]
Using the identification principle
\cite[Theorem 21.18]{zed2}, we have
$W_0^{1,2}(\Omega_i,\omega)=[W_0^{1,2}(\Omega_i,\omega)]^*$ and
$\langle u ,v \rangle=(u|v)$, where $\langle .,.\rangle$ denotes
 the inner product on a Hilbert
space. We define the operators
$B,N:W_0^{1,2}(\Omega_i,\omega)\to
W_0^{1,2}(\Omega_i,\omega)$ as
$$
(Bu|\phi)=B_1(u,\phi),\quad
(Nu|\phi)=B_2(u,\phi),\quad \text{for }
u,\phi\in W_0^{1,2}(\Omega_i,\omega).
$$
Then, problem (\ref{eq:b5}) is equivalent to operator equation
$Bu+Nu=T$, $u\in W_0^{1,2}(\Omega_i,\omega)$.
The proof of the existence for
(\ref{eq:b5}) is similar to that given in \cite{sem}. The proof of
the  latter part of the theorem (which is not in \cite{sem}) is
given below. Let $|h(t)|\leq A,t\in\mathbb{R}$.
Let $u_k\in W_0^{1,2}(\Omega_k,\omega)$ be the solutions
of (\ref{eq:b5}). Then,
from the hypotheses, with the help of Lemma \ref{lem2.1} and from
the Remark \ref{rmk2},  we note that, for $k\geq i$,
\begin{gather*}
  |B_1(u_k,u_k)|\leq (c+ C_{\Omega_i}|\mu|\|\frac{g_1}{\omega}
  \|_{\infty,\Omega_i})\|u_k\|_{0,1,2,\Omega_i}
  \|u_k\|_{0,1,2,\Omega_i}\\
  |B_2(u_k,u_k)|\leq A C_{\Omega_i}\|\frac{g_2}{\omega}\|_{2,\Omega_i}
  \|u_k\|_{0,1,2,\Omega_i}\\
  |T(u_k)|\leq C_{\Omega_i}\|\frac{f}{\omega}\|_{2,\Omega_i}
  \|u_k\|_{0,1,2,\Omega_i},
\end{gather*}
where $C_{\Omega_i}$ (is the constant of Lemma \ref{lem2.1}) and $A$ are
constants independent of $k$. Also, $B_1(.,.)$ is a regular
G{\aa}rding form \cite[p.364]{zed2}. In fact, we obtain, for
$k\geq i$
\begin{align*}
   B_1(u_k,u_k)
&\geq\lambda\int_{\Omega_i}|Du_k|^2\omega dx-\mu\big\|\frac{g_1}{\omega}
   \big\|_{\infty,\Omega_i}\int_{\Omega_i}{ u_{k}^2\omega} dx\\
&=\lambda \|u_k\|^2_{0,1,2,\Omega_i}-\mu\|\frac{g_1}{\omega}
  \|_{\infty,\Omega_i}\|u_k\|^2_{2,\Omega_i}
\end{align*}
Now, by Lemma \ref{lem2.1}, we have
\begin{equation*}
B_1(u_k,u_k)\geq\big(\lambda-C_{\Omega_i}\mu\|\frac{g_1}{\omega}
\|_{\infty,\Omega_i}\big) \|u_k\|^2_{0,1,2,\Omega_i}.
\end{equation*}
Since, $\lambda> C_{\Omega_i}\mu\|\frac{g_1}{\omega}
\|_{\infty,\Omega_i}$, we obtain
\begin{equation}\label{e:m1}
  \|u_k\|^2_{0,1,2,\Omega_i}\leq\big(\frac{1}
  {\lambda-C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega_i}}
  \big)B_1(u_k,u_k)
\end{equation}
Also, we note that
\begin{equation}\label{e:m2}
  |B_1(u_k,u_k)|\leq C_{\Omega_i}\big\{
 A\|\frac{g_2}{\omega}\|_{2,\Omega_i}+
  \|\frac{f}{\omega}\|_{2,\Omega_i}\big\}\|u_k\|_{0,1,2,\Omega_i}.
\end{equation}
By (\ref{e:m1}) and (\ref{e:m2}), we have
\begin{align*}
  \|u_k\|_{0,1,2,\Omega_i}
&\leq \frac{{C_{\Omega_i}}\big\{A\|\frac{g_2}{\omega}
  \|_{2,\Omega_i}+\|\frac{f}{\omega}\|_{2,\Omega_i}\}}
  {(\lambda-C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega_i})}\\
&\leq \frac{{C_{\Omega_i}}\big\{A\|\frac{g_2}{\omega}\|_{2,\Omega}
  +\|\frac{f}{\omega}\|_{2,\Omega}\}}
  {(\lambda-C_{\Omega_i}\mu \|\frac{g_1}{\omega}\|_{\infty,\Omega})}=k_0,
\end{align*}
where $k_0$ is independent of $k$. Hence,
\begin{equation}\label{e:uni}
  \|u_k\|_{0,1,2,\Omega_i}\leq
  k_0,\quad \forall k\geq i
\end{equation}
\end{proof}

\begin{corollary} \label{coro2.5}
Under the hypotheses of Lemma \ref{lem2.4}, let $M$ be any open bounded
domain in $\Omega$ such that $M\subseteq\Omega_i$, for some $i$. For
$k\geq i$, let $u_k$ be a solution of
\begin{gather*}%\label{eq:b8}
  Lu-\mu ug_1+h(u)g_2=f\quad \text{in }\Omega_k,\\
u=0\quad \text{on  }\partial\Omega_k
\end{gather*}
Then, there exists a constant $k_0>0$ such that
$\|u_k\|_{0,1,2,M}\leq k_0$, where $k_0$ is independent of $k$.
\end{corollary}

The proof of this result is similar to that of Lemma \ref{lem2.4}
and hence omitted.

\begin{remark} \label{rmk3}
Corollary \ref{coro2.5} is needed in the main result stated in $\S 3$.
Lemma \ref{lem2.4} is a ``modification" of the
result in \cite{sem}, which gives a uniform ${u_k}$, $k\geq i$ at the
cost of the restriction on $\mu$ as given by (\ref{eq:41}).
\end{remark}


\section{Main results}

In this section,  we dispense with the condition
(\ref{eq:41}) when $g_1$ does not change sign. We consider a BVP
\begin{equation}\label{eq:c1}
\begin{gathered}
  Lu-\mu ug_1+ h(u)g_2=f\quad \text{in }G,\\
u=0\quad \text{on }\partial G
\end{gathered}
\end{equation}
where $G\subset\mathbb{R}^n$ is an open bounded set, $n\geq 3$. The two
results are related to the cases when $g_1>0$ with $\mu<0$ and
$g_1<0$ with $\mu>0$. These results are similar to that found in
\cite{sem} but with suitable changes.

\begin{proposition} \label{prop3.1}
Let $G\subset\Omega$ be an open bounded set in $\mathbb{R}^n$,
$n\geq 3$. Suppose that {\rm (H1)--(H3)} hold. Let $g_1>0$
 and $\mu<0$, then the BVP
\begin{equation}\label{eq:c2}
\begin{gathered}
  Lu-\mu ug_1 + h(u)g_2=f\quad\text{in }G,\\
u=0\quad\text{on }\partial G
\end{gathered}
\end{equation}
has a solution $u\in W^{1,2}_0(G,\omega)$.
\end{proposition}

\begin{proof}
As in Lemma \ref{lem2.4}, the basic idea is to reduce the problem
\eqref{eq:c2} to  an operator equation $Bu+Nu=T$ with the help
of the Theorem \ref{thm2.3}. To do proceed, we define $B,N$, and $T$
with $\Omega_i$ replaced by $G$, as in Lemma \ref{lem2.4} and after a little
bit of computation, we have
\begin{gather*}
|B_1(u,\phi)|\leq (c+C_G|\mu|\|\frac{g_1}{\omega}\|_{\infty,G})
  \|u\|_{0,1,2,G}\|\phi\|_{0,1,2,G}\\
|B_2(u,\phi)| \leq   C_GA\|\frac{g_2}{\omega}\|_{2,G}\|\phi\|_{0,1,2,G}\\
|T(\phi)| \leq C_G\|\frac{f}{\omega}\|_{2,G}\|\phi\|_{0,1,2,G}
\end{gather*}
where $c$ (a generic constant), $A$ are constants depending on $n$,
$p$ and the constant $C_G$ comes from Lemma \ref{lem2.1}. With these
preliminaries, \eqref{eq:c2} is equivalent to
\[
Bu+Nu=T,\quad u\in W^{1,2}_0(G,\omega).
\]
 The compact embedding of
$W^{1,2}_0(G,\omega)\hookrightarrow\hookrightarrow L^{2}(G,\omega)$,
shows that $B_1(.,.)$ is a strict regular G{\aa}rding form. Also,
$\mu<0$ and $g_1>0$ yields
\begin{equation}
  B_1(u,u)=\int_G a_{ij}D_iu(x)D_ju(x)dx-\int_G\mu
  u^2(x)g_1(x)dx\geq \lambda\|u\|^2_{0,1,2,G}
\end{equation}
Next, we also show that $B+N$ is asymptotically linear and $N$
strongly continuous. The proof is similar to the one in \cite{sem}
and we omit the same for brevity. Since $\mu$ is not an eigenvalue
of
\begin{equation}\label{eq:c3}
\begin{gathered}
  Lu-\mu u(x)\omega(x)=0\quad\text{in }G,\\
u=0\quad\text{on }\partial G,
\end{gathered}
\end{equation}
$Bu=0$ implies $u=0$. By Theorem \ref{thm2.3}, $Bu+Nu=T$ has a solution
$u\in W^{1,2}_0(G,\omega)$ which equivalently shows the BVP \eqref{eq:c2}
has a solution $u\in W^{1,2}_0(G,\omega)$.
\end{proof}

We consider the boundary-value problem
\begin{equation}\label{eq:c8}
\begin{gathered}
  Lu-\mu ug_1+ h(u)g_2=f\quad\text{in }\Omega_i,\\
u=0\quad\text{on   }\partial \Omega_i
\end{gathered}
\end{equation}
where $\Omega_i\subseteq \mathbb{R}^n,~n\geq 3$ is an open bounded set,
for $i\geq 1$.

\begin{corollary} \label{coro3.2}
Let the hypotheses of Proposition \ref{prop3.1} hold for $\Omega_i$ in
place of $G$, for $i\geq1$. Then, there exists
$u_i\in W^{1,2}_0(\Omega_i,\omega)$ which satisfies \eqref{eq:c8} and in
addition, for $k\geq i$,
\begin{equation}
  \|u_k\|_{0,1,2,\Omega_i}\leq k_0,
\end{equation}
where $k_0$ is a constant independent of $k$.
\end{corollary}

The proof of the above corollary is similar to the later part of the
Lemma \ref{lem2.4} and hence omitted. With suitable changes in the proof of
Proposition \ref{prop3.1}, we arrive at the following result.

\begin{theorem} \label{thm3.3}
Let  the hypotheses of Proposition \ref{prop3.1} hold, except that
$g_1<0$ and $\mu>0$. Let  $\mu$ not be an eigenvalue of
\begin{equation}\label{eq:c3b}
\begin{gathered}
  Lu-\mu u(x)\omega(x)=0\quad\text{in }G,\\
u=0\quad\text{on }\partial G
\end{gathered}
\end{equation}
Then the \eqref{eq:c2} has a solution $u\in W^{1,2}_0(G,\omega)$.
\end{theorem}

\begin{corollary} \label{coro3.4}
Let the hypotheses of Proposition \ref{prop3.1} hold for $\Omega_i$ in
place of $G$, for $i\geq1$. Then, there exists $u_i\in
W^{1,2}_0(\Omega_i,\omega)$ which satisfies \eqref{eq:c8} and in
addition, for $k\geq i$,
\begin{equation*}
  \|u_k\|_{0,1,2,\Omega_i}\leq k_0,
\end{equation*}
where $k_0$ is a constant independent of $k$.
\end{corollary}

The proof of the above corollary is similar to Corollary \ref{coro3.2} and
hence omitted.

\begin{theorem} \label{thm3.5}
Let $\Omega = \cup^\infty_{i=1}{\Omega_i},\Omega_i\subseteq
\Omega_{i+1}$ be open bounded domains in $\Omega$. Let $\mu>0$
not be an eigenvalue of
\begin{equation}\label{eq:b9}
\begin{gathered}
  Lu-\mu u(x)\omega(x)=0\quad\text{in }\Omega_i,\\
 u=0\quad\text{on  }\partial \Omega_i
\end{gathered}
\end{equation}
for $i=1,2,3,\dots$ and in addition the condition
$\lambda>C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega}$ be
fulfilled. Under the hypotheses {\rm (H1)-(H3)}, \eqref{eq:a2} has a
weak solution $u \in W^{1,2}_{0}(\Omega,\omega)$.
\end{theorem}

\begin{proof}
A part of this proof follows from \cite{eag,nou1,nou2}.
Let $\{u_k\}$ be the sequence of solutions for
\eqref{eq:c8} in $W^{1,2}_{0}(\Omega_k,\omega),(k\geq 1)$. Let
$\tilde{u}_k(\text{ for }k\geq 1)$ denote the extension of $u_k$ by
zero outside $\Omega_k$, which we continue to denote it by $u_k$.
 From (\ref{e:uni}), we have
\[
\|u_k\|_{0,1,2,\Omega_l}\leq k_0, \text{ for }k\geq l.
\]
Then, $\{u_k\}$ has a subsequence $\{u_{k_m^1}\}$ which converges
weakly to $u^1$, as $m\to\infty$, in
$W^{1,2}_{0}(\Omega_1,\omega)$. Since $\{u_{k_m^1}\}$ is bounded in
$W^{1,2}_{0}(\Omega_2,\omega)$, it has a convergent subsequence
$\{u_{k_m^2}\}$ converging weakly to $u^2$ in
$W^{1,2}_{0}(\Omega_2,\omega)$. By induction, we have
$\{u_{k_m^{l-1}}\}$ has a subsequence $\{u_{k_m^l}\}$ which weakly
converges to $u^l$ in $W^{1,2}_{0}(\Omega_l,\omega)$, i.e in short,
we have $u_{k_m^l}\rightharpoonup u^l$ in
$W^{1,2}_{0}(\Omega_l,\omega),~l\geq 1$. Define $u:
\Omega\to\mathbb{R}$ by
$$
u(x):= {u}^l(x),\quad\text{for }x\in\Omega_l.
$$
(Here there is no confusion occurs since
${u}^l(x)={u}^m(x)$ for $x\in\Omega$ for any $m\geq l$).

Let $M$ be any fixed (but arbitrary) bounded domain such that
${M }\subseteq\Omega$. Then there exists an integer $l$ such that
$M\subseteq \Omega_l$. We note that, the diagonal sequence
$\{u_{k_m^m};m\geq l\}$
converges weakly to $u=u^l$ in $W^{1,2}_{0}(M,\omega)$,
 as $m\to\infty$.

What remains is to show that $u$ is the required weak solution. It
is sufficient to show that $u$ is a weak solution of \eqref{eq:a2}
for an arbitrary bounded domain $M$ in $\Omega$. Since
$u_{k_m^m}\rightharpoonup u^l$ in $W^{1,2}_{0}(M,\omega)$, we have
\[
\int_M\nabla(u_{k_m^m}-u).\nabla\phi \omega dx\to0,\quad
\text{as } m\to\infty,
\]
implies
\[
\int_M{D_i(u_{k_m^m}-u)}D_j\phi \omega
dx\to0,\quad\text{as } m\to\infty.
\]
 From (\ref{eq:a1}), for a constant $c$, we have
$|a_{ij}|\leq c \omega$.
\begin{equation}\label{e:d0}
\begin{aligned}
\int_M{a_{ij}D_i(u_{k_m^m}-u)}D_j\phi\,dx
&\leq\int_M|a_{ij}||D_i(u_{k_m^m}-u)||D_j\phi|dx \\
&\leq c\|D_i(u_{k_m^m}-u)\|_{2,M}\|D_j\phi\|_{2,M}\to 0,
\end{aligned}
\end{equation}
as $m\to\infty$. Also, by Lemma \ref{lem2.1}, $u_{k_m^m}$ $\to u$ in
$L^2(M,\omega)$. We have
\begin{align*}
\big|\int_M(u_{k_m^m}-u){g_1}\phi dx\big|
&\leq\int_M|(u_{k_m^m}-u)||{g_1}||\phi|dx\\
&\leq\int_M|(u_{k_m^m}-u)||\frac{g_1}{\omega}||\phi|\omega \,dx\\
&\leq \|\frac{g_1}{\omega}\|_{\infty,M}\|u_{k_m^m}-u\|_{2,M}
\|\phi\|_{2,M}.
\end{align*}
So we have now
\begin{equation}\label{e:d1}
  \mu\int_M{u_{k_m^m}}g_1\phi dx\to\mu\int_M{ug_1\phi}dx\,.
\end{equation}
A little computation shows that
\begin{equation}\label{e:d2}
  \int_Mh(u_k(x))\to\int_M h(u(x)),
\end{equation}
which follows from dominated convergence theorem, if needed through
a subsequence. Since $M$ is an arbitrary bounded domain in
$\Omega$, it follows from (\ref{e:d0}), (\ref{e:d1}) and
(\ref{e:d2}),
\begin{align*}
&\int_\Omega a_{ij}D_iu(x)D_j\phi(x)dx-\int_\Omega\mu
  u(x)\phi(x)g_1(x)dx+\int_\Omega h(u(x))\phi(x)g_2(x)\\
&=\int_\Omega
  f(x)\phi(x)dx
\end{align*}
which completes the proof of the theorem.
\end{proof}

\begin{theorem} \label{thm3.6}
Let $\Omega = \cup^\infty_{i=1}{\Omega_i}$,
$\Omega_i\subseteq \Omega_{i+1}$ be open bounded domains in $\Omega$.
Let $g_1>0$ and $\mu<0$.
Under hypotheses {\rm (H1)-(H3)}, \eqref{eq:a2} has a weak
solution $u \in W^{1,2}_{0}(\Omega,\omega)$.
\end{theorem}

The proof is similar to that of Theorem \ref{thm3.6} and hence omitted.
We remark that the above theorem is also true when
$g_1<0$ and $\mu>0$ is not an eigenvalue of (\ref{eq:b9}).
% \label{rmk4}

\section{Extensions}

In section 3, the nonlinearity $h$ is assumed to be continuous and
bounded. In this section, we extend these results for a class of
functions $h$ which are continuous only. Generalized H\"{o}lder's
inequality comes handy for establishing suitable estimates. Below,
we consider the problem
\begin{equation}\label{eq:d1}
\begin{gathered}
  Lu-\mu ug_1 + h(u)g_2=f\quad\text{in }\Omega,\\
  u=0\quad\text{on }\partial \Omega,
\end{gathered}
\end{equation}
where $\Omega\subseteq\mathbb{R}^n,n\geq 3$ is an open and connected
set and $h:\mathbb{R}\to\mathbb{R}$ be defined by
$h(t)=|t|^{\epsilon},0<\epsilon<1$. We establish the existence of
weak solution in a bounded domain $G$.

Again, we consider the cases $g_1<0$ and $g_1>0$ separately.
Although the proofs are similar to the ones in section 3, we
restrict ourselves to sketch the differences wherever needed. The
result of \cite{sem} is not applicable here since $h$ is not
bounded.
We collect the common hypotheses for convenience.
\begin{itemize}
\item[(H1')] Suppose that $h:\mathbb{R}\to\mathbb{R}$ defined by
  $h(t)=|t|^\epsilon,t\in \mathbb{R},0<\epsilon<1$;
\item[(H2')] $g_1/\omega \in L^{\infty}(\Omega)$,
  $g_2/\omega \in L^{\infty}(\Omega)$ and
  $f/\omega \in L^{2}(\Omega,\omega)$, where $\omega$ is an $A_2$
weight.
\end{itemize}

\begin{theorem} \label{thm4.1}
Let $G\subset\mathbb{R}^n$, $n\geq 3$ be any open bounded set. Let the
hypotheses {\rm (H1'), (H2')} hold. Let $g_1>0$ and $\mu<0$ then
the problem
\begin{equation}\label{eq:d7}
\begin{gathered}
  Lu-\mu ug_1+ h(u)g_2=f\quad\text{in }G,\\
u=0\quad\text{on }\partial G
\end{gathered}
\end{equation}
has a solution $u\in W^{1,2}_0(G,\omega)$.
\end{theorem}

\begin{proof}
We give only a sketch of the proof as it is similar to the proof of
Proposition \ref{prop3.1}.  From the hypotheses and by Lemma \ref{lem2.1} and for
$u\in W^{1,2}_0(G,\omega)$, we note that
\begin{equation}\label{eq:d4}
\begin{gathered}
  |B_1(u,\phi)|\leq \big(c+C_G|\mu|\|\frac{g_1}{\omega}
  \|_{\infty,G}\big)\|u\|_{0,1,2,G}\|\phi\|_{0,1,2,G},\\
 |T(\phi)|\leq
  C_G\|{\frac{f}{\omega}\|_{\infty,G}}\|\phi\|_{0,1,2,G},
\end{gathered}
\end{equation}
where $c$ is a generic constant and the constant $C_G$ comes from
Lemma \ref{lem2.1}. Again, by Lemma \ref{lem2.1} and generalized H\"{o}lder's
inequality \cite[p.67]{fun}, we have
\[
  |B_2(u,\phi)|\leq\int_G|h(u(x))||\phi(x)||\frac{g_2}{\omega}|\omega\,dx
 \leq \|u\|^\epsilon_{2,G}\|\phi\|_{2,G}\|\frac{g_2}{\omega}
  \|_{\frac{2}{1-\epsilon},G}. %\label{eq:d5}
\]
We also observe that $B_1$ satisfies condition (S) by a similar
argument as in \cite{sem} (also refer to
\cite[Proposition 27.12]{zed1}). We observe that
\[
  |(Nu|\phi)|=|B_2(u,\phi)|\leq
  C_G\|u\|^{\epsilon}_{0,1,2,G}\|\phi\|_{0,1,2,G}\|\frac{g_2}{\omega}
  \|_{\frac{2}{1-\epsilon},G}
\]
which implies
\[
\|Nu\|\leq   C_G\|u\|^{\epsilon}_{0,1,2,G}\|\frac{g_2}{\omega}
  \|_{\frac{2}{1-\epsilon},G}\leq c C_G\|u\|^{\epsilon}_{0,1,2,G},
\]
So
\begin{equation}\label{e:d6}
  \frac{\|Nu\|}{\|u\|_{0,1,2,G}}\leq
  \frac{c {C_G}\|u\|^{\epsilon}_{0,1,2,G}}{\|u\|_{0,1,2,G}}\to
  0\quad \text{as }\|u\|_{0,1,2,G}\to\infty.
\end{equation}
This shows that $B+N$ is asymptotically linear. Also,
$u\in L^2(\Omega,\omega)$  implies
$h(u)\in   L^{\frac{2}{\epsilon}}(\Omega,\omega)$
and define the Nemyckii operator
\begin{equation}\label{eq:h1}
   h_u:L^2(\Omega,\omega)\to
   L^{\frac{2}{\epsilon}}(\Omega,\omega)
\end{equation}
by $h_u(x)=h(u(x))$; we have $h_u$ is continuous
(by \cite[Theorem 2.1]{kra}).
Let $u_n\rightharpoonup u$ in $W^{1,2}_0(G,\omega)$, then
\begin{align*}
   |(Nu_n|\phi)-(Nu|\phi)|
&\leq\int_G{|h(u_n)-h(u)||\frac{g_2}{\omega}||\phi|\omega  dx}\\
&\leq C_G\|h(u_n)-h(u)\|_{\frac{2}{\epsilon},G}
   \|\frac{g_2}{\omega}\|_{\frac{2}{1-\epsilon},G}\|\phi\|_{0,1,2,G}.
\end{align*}
Hence we have
\begin{equation}\label{eq:d4n}
  \|Nu_n-Nu\|\to 0\quad\text{as  }n\to\infty
\end{equation}
By a similar argument as in \cite{sem}, the operator
$A_t(u)=Bu+t(Nu-T)$ satisfies condition (S). If $\mu<0$ is not an
eigenvalue of the linear problem
\begin{gather*}
   Lu-\mu u(x)\omega(x)=0\quad\text{in }G,\\
u=0\quad\text{on }\partial G
\end{gather*}
shows that the operator equation $Bu+Nu=T$ has a solution
$u\in W^{1,2}_0(G,\omega)$, which completes the proof.
\end{proof}

An immediate consequence is the following result.

\begin{corollary} \label{coro4.2}
Let $\Omega$ be any open set in $\mathbb{R}^n$ such that
$\Omega=\cup_{i=1}^{\infty}\Omega_i,\Omega_i\subseteq\Omega_{i+1},
\Omega_i$ is an open bounded subset of $\mathbb{R}^n$ for each
$i=1,2,3.$. Let the hypotheses of Theorem \ref{thm4.1} hold. Let
$g_1>0$ and $\mu<0$ then,
 the problem
\begin{equation}\label{eq:d2}
\begin{gathered}
  Lu-\mu ug_1 + h(u)g_2=f\quad\text{in }\Omega_i,\\
u=0\quad\text{on }\partial \Omega_i
\end{gathered}
\end{equation}
has a solution $u=u_i\in W^{1,2}_0(\Omega_i,\omega)$, for $i=1,2.$.
in addition $ \|u_k\|_{0,1,2,\Omega_i}\leq k_0$  for all $k\geq i$,
where $k_0$ is independent of $k$.
\end{corollary}

\begin{remark} \label{rmk5} \rm
 Theorem \ref{thm4.1} and Corollary \ref{coro4.2} hold if
$g_1<0$ and $\mu>0$ with the remaining intact. But when $\mu>0$ and
$g_1$ changes sign, we need additional conditions on $\mu$ and
$g_1$(stated below) to obtain a uniform bound $k_0$ for $\|u_k\|$,
$k=1,2$. where $k_0$ is independent of $k$. This uniform
boundedness is essential to establish the existence of solution when
$\Omega$ is not necessarily bounded. We state these results below in
Theorem \ref{thm4.3} and Corollary \ref{coro4.4} but we give a sketch of the proof. We
note that in (\ref{e:d6} the required asymptotic linearity of $B+N$ is
a consequence of $\epsilon$ lying between 0 and 1.
\end{remark}

\begin{theorem} \label{thm4.3}
Let $G$ be an open bounded set in $\mathbb{R}^n,n\geq 3$.
Let the hypotheses {\rm (H1'), (H2')} hold. Also,
let $\mu>0$ not be an eigenvalue of (\ref{eq:c3}). Then the BVP
\begin{equation}\label{eq:d8}
\begin{gathered}
  Lu-\mu ug_1+ h(u)g_2=f\quad\text{in }G,\\~
u=0\quad\text{on }\partial G
\end{gathered}
\end{equation}
has a solution $u\in W^{1,2}_0(G,\omega)$.
\end{theorem}

The proof is omitted since it is along
the same lines of the proof of Theorem \ref{thm4.1}.
 As a consequence of Theorem \ref{thm4.3}, we have the following result.

\begin{corollary} \label{coro4.4}
In addition to the hypotheses of Theorem \ref{thm4.3}, let
$\lambda>C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega}$.
 Then  \eqref{eq:d2} has a solution $u=u_i\in W^{1,2}_0(\Omega_i,\omega)$,
for $i=1,2\dots$ and in addition
\[
\|u_k\|_{0,1,2,\Omega_i}\leq k_0,\quad \text{for all }k\geq i.
\]
where $k_0$ is a constant independent of $k$.
\end{corollary}

\begin{proof}
The proof for existence of solutions $u=u_i\in W^{1,2}_0(\Omega_i,\omega)$
for \eqref{eq:d2} is similar to the proof of Theorem \ref{thm4.1}
and hence omitted. We note that on $\Omega_i$, for
$k\geq i$,
\begin{align*}
&(\lambda-C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega_i})
 \|u_k\|^2_{0,1,2,\Omega_i}\\
&\leq C_{\Omega_i}
  \big \{\|u_k\|^{\epsilon}_{0,1,2,\Omega_i}
  \|\frac{g_2}{\omega}\|_{\frac{2}{1-\epsilon},\Omega_i}
  +\|\frac{f}{\omega}\|_{2,\Omega_i}\}\|u_k\|_{0,1,2,\Omega_i},
\end{align*}
where $C_{\Omega_i}$ is independent of $k$. Since
$\lambda>C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega_i}$,
we obtain
\begin{equation}\label{e:ep}
   \|u_k\|_{0,1,2,\Omega_i}
  \leq \frac{{{C_{\Omega_i}}{\big(\|u_k\|^{\epsilon}_{0,1,2,\Omega_i}}
  \|\frac{g_2}{\omega}\|_{\frac{2}{1-\epsilon},\Omega_i}+
  \|\frac{f}{\omega}\|_{2,\Omega_i}\big)}}{(\lambda-C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega_i})}
\end{equation}

\textbf{Case 1:} If $\|u_k\|_{0,1,2,\Omega_i}\leq1$, then from
(\ref{e:ep}), we have
\begin{align*}
  \|u_k\|_{0,1,2,\Omega_i}
&\leq\frac{{C_{\Omega_i}}\big(\|\frac{g_2}{\omega}
  \|_{\frac{2}{1-\epsilon},\Omega_i}
 +\|\frac{f}{\omega}\|_{2,\Omega_i}\big)}{(\lambda-C_{\Omega_i}
 \mu\|\frac{g_1}{\omega}\|_{\infty,\Omega_i})}\\
&\leq\frac{{C_{\Omega_i}}\big(\|\frac{g_2}{\omega}\|_{\frac{2}{1-\epsilon},\Omega}
 +\|\frac{f}{\omega}\|_{2,\Omega}\big)}{(\lambda-C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega})}
=c^*,
\end{align*}
where $c^*$ is a constant independent of $k$.
Hence, we obtain
  $$
\|u_k\|_{0,1,2,\Omega_i}\leq c^{*},\text{ for all }k\geq i.
$$
\textbf{Case 2:} If $\|u_k\|_{0,1,2,\Omega_i}>1$, from (\ref{e:ep}),
we have
\begin{align*}
  \|u_k\|_{0,1,2,\Omega_i}
&\leq\frac{{C_{\Omega_i}}\big(\|u_k\|^{\epsilon}_{0,1,2,\Omega_i}
 \|\frac{g_2}{\omega}
  \|_{\frac{2}{1-\epsilon},\Omega_i}+\|\frac{f}{\omega}
 \|_{2,\Omega_i}\big)}
  {(\lambda-C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega_i})}\\
&\leq\frac{{C_{\Omega_i}}\big(\|\frac{g_2}{\omega}\|_{\frac{2}{1-\epsilon},
 \Omega}
+\|\frac{f}{\omega}\|_{2,\Omega}\big){\|u_k\|
 ^{\epsilon}_{0,1,2,\Omega_i}}}{(\lambda-C_{\Omega_i}\mu\|
\frac{g_1}{\omega}\|_{\infty,\Omega})}
\end{align*}
where $C_{\Omega_i}$ is independent of  $k$. This implies
$$
\|u_k\|^{1-\epsilon}_{0,1,2,\Omega_i}\leq c,0<\epsilon<1,\quad
\|u_k\|_{0,1,2,\Omega_i}\leq c^{\frac{1}{1-\epsilon}}=c',
$$
where $c$ and $c'$ are constants independent of $k$. Since
$\Omega_i\subseteq\Omega_{i+1},\forall ~i\geq1$, we have
\begin{equation*}
  \|u_k\|_{0,1,2,\Omega_i}\leq   c',\quad \text{for all }k\geq i.
\end{equation*}
Let $k_0=\max\{c^*,c'\}$. Hence, we have
\begin{equation}\label{e:u}
 \|u_k\|_{0,1,2,\Omega_i}\leq
  k_0,\quad \text{for all }k\geq i,
\end{equation}
where $k_0$ is independent of $k$.
\end{proof}

Now we state the main result of this section.

\begin{theorem} \label{thm4.5}
Let $\Omega = \cup^\infty_{i=1}{\Omega_i},\Omega_i\subseteq
\Omega_{i+1}$ be open bounded domains in $\Omega$. Let $\mu>0$
not be an eigenvalue of
\begin{equation}\label{eq:b91}
\begin{gathered}
  Lu-\mu u(x)\omega(x)=0\quad\text{in }\Omega_i,\\
 u=0\quad\text{on   }\partial \Omega_i
\end{gathered}
\end{equation}
for $i=1,2,3,\dots$ and in addition let
$\lambda>C_{\Omega_i}\mu\|\frac{g_1}{\omega}\|_{\infty,\Omega}$.
Under  hypotheses {\rm (H1'),  (H2')}, \eqref{eq:d1} has a weak
solution $u \in W^{1,2}_{0}(\Omega,\omega)$.
\end{theorem}

\begin{proof}
Let $\{u_k\}$ be the sequence of solutions for \eqref{eq:d2} in
$W^{1,2}_{0}(\Omega_k,\omega),(k\geq 1)$.
Let $\tilde{u}_k$ (for $k\geq 1$) denote the extension of $u_k$
by zero outside $\Omega_k$, which we continue to denote it by $u_k$.
 From (\ref{e:u}), we have
\[
\|u_k\|_{0,1,2,\Omega_l}\leq k_0, \quad \text{for }k\geq l.
\]
Then, $\{u_k\}$ has a subsequence $\{u_{k_m^1}\}$ which converges
weakly to $u^1$, as $m\to\infty$, in
$W^{1,2}_{0}(\Omega_1,\omega)$. Since $\{u_{k_m^1}\}$ is bounded in
$W^{1,2}_{0}(\Omega_2,\omega)$, it has a convergent subsequence
$\{u_{k_m^2}\}$ converging weakly to $u^2$ in
$W^{1,2}_{0}(\Omega_2,\omega)$. By induction, we have
$\{u_{k_m^{l-1}}\}$ has a subsequence $\{u_{k_m^l}\}$ which weakly
converges to $u^l$ in $W^{1,2}_{0}(\Omega_l,\omega)$, i.e in short,
we have $u_{k_m^l}\rightharpoonup u^l$ in
$W^{1,2}_{0}(\Omega_l,\omega),~l\geq 1$. Define $u:
\Omega\to\mathbb{R}$ by
$$
u(x):= {u}^l(x),\quad\text{for }x\in\Omega_l.
$$
(Here there is no confusion occurs since
${u}^l(x)={u}^m(x)$ for $x\in\Omega$ for any $m\geq l$).

Let $M$ be any fixed (but arbitrary) bounded domain such that
${M}\subseteq\Omega$. Then there exists an integer $l$ such that
$M\subseteq \Omega_l$. We note that, the diagonal sequence
$\{u_{k_m^m};m\geq l\}$
 weakly converges to $u=u^l$ in $W^{1,2}_{0}(M,\omega), \quad\text{as }
m\to\infty$.

What remains is to show that $u$ is the required weak solution. It
is sufficient to show that $u$ is a weak solution of \eqref{eq:d1}
for an arbitrary bounded domain $M$ in $\Omega$. Since
$u_{k_m^m}\rightharpoonup u^l$ in $W^{1,2}_{0}(M,\omega)$, we have
\begin{equation*}
\int_M\nabla(u_{k_m^m}-u).\nabla\phi \omega
dx\to 0,\quad\text{as } m\to\infty,
\end{equation*}
which implies
\[\int_M{D_i(u_{k_m^m}-u)}D_j\phi \omega
dx\to0,\quad\text{as } m\to\infty.
\]
 From (\ref{eq:a1}), for a constant $c$, we have $|a_{ij}|\leq c \omega$.
\begin{equation}\label{e:d61}
\begin{aligned}
\int_M{a_{ij}D_i(u_{k_m^m}-u)}D_j\phi dx
&\leq\int_M|a_{ij}||D_i(u_{k_m^m}-u)||D_j\phi|dx \\
&\leq c\|D_i(u_{k_m^m}-u)\|_{2,M}\|D_j\phi\|_{2,M}\to 0,
\quad\text{as }   m\to\infty.
\end{aligned}
\end{equation}
Also, by Lemma \ref{lem2.1}, $\{u_{k_m^m}\}\to u$ in $L^2(M,\omega)$.
We have
\begin{align*}
\big|\int_M(u_{k_m^m}-u){g_1}\phi\, dx\big|
&\leq\int_M|(u_{k_m^m}-u)||{g_1}||\phi|dx\leq
\int_M|(u_{k_m^m}-u)||\frac{g_1}{\omega}||\phi|\omega dx\\
&\leq \|\frac{g_1}{\omega}\|_{\infty,M}\|u_{k_m^m}-u\|_{2,M}\|\phi\|_{2,M}.
\end{align*}
So we have
\begin{equation}\label{e:d7}
  \mu\int_M{u_{k_m^m}}g_1\phi dx\to\mu\int_M{ug_1\phi}dx
\end{equation}
By (\ref{eq:h1}) and generalized H\"{o}lder's inequality, we obtain
\[
\int_M{|h(u_{k_m^m})-h(u)||\frac{g_2}{\omega}||\phi|\omega
   dx}\leq \|h(u_{k_m^m})-h(u)\|_{\frac{2}{\epsilon},M}
   \|\frac{g_2}{\omega}\|_{\frac{2}{1-\epsilon},M}\|\phi\|_{2,M}.
\]
Hence, we have
\begin{equation}\label{e:d8}
  \int_Mh(u_k(x))g_2\phi dx\to\int_M{h(u(x))g_2\phi}dx\,.
\end{equation}
 Since $M$ is an arbitrary bounded domain in
$\Omega$, it follows from (\ref{e:d61}), (\ref{e:d7}) and
(\ref{e:d8}), that
\begin{align*}
&\int_\Omega a_{ij}D_iu(x)D_j\phi(x)dx-\int_\Omega\mu
  u(x)\phi(x)g_1(x)dx+\int_\Omega h(u(x))\phi(x)g_2(x)\\
&=\int_\Omega   f(x)\phi(x)dx
\end{align*}
which completes the proof of the theorem.
\end{proof}

\begin{theorem} \label{thm4.6}
Let $\Omega = \cup^\infty_{i=1}{\Omega_i},\Omega_i\subseteq
\Omega_{i+1}$ be open bounded domains in $\Omega$. Let $g_1>0$ and
$\mu<0$. Under the hypotheses $(H'_1)$-$(H'_2)$, \eqref{eq:d1} has a
weak solution $u \in W^{1,2}_{0}(\Omega,\omega)$.
\end{theorem}

The proof is similar to Theorem \ref{thm4.5} and hence omitted.
 Above theorem is also true when, $g_1<0$ and $\mu>0$ is not an
 eigenvalue of  (\ref{eq:b91}).

\begin{remark} \label{rmk6} \rm
 The main results Theorem \ref{thm3.5} and
Theorem \ref{thm4.5} hold, if $h$ is continuous,
$|h(t)-h(s)|\leq c|t-s|^{\epsilon}$, $0<\epsilon<1$  and
$h(0)=0$.
\end{remark}

\subsection*{Acknowledgements}
We want to thank the anonymous referee for the constructive
comments and suggestions.

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