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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 43, pp. 1--11.\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-2012/43\hfil Weak solutions]
{Weak solutions for degenerate semilinear elliptic BVPs in
unbounded domains}

\author[R. Kar\hfil EJDE-2012/43\hfilneg]
{Rasmita Kar} 

\address{Rasmita Kar \newline
Department of Mathematics and Statistics, 
Indian Institute of Technology, Kanpur, 208016 India}
\email{rasmi07@gmail.com}

\thanks{Submitted September 17, 2011. Published March 20, 2012.}
\subjclass[2000]{46E35, 35J61}
\keywords{Semilinear elliptic boundary value problem; unbounded domain;
\hfill\break\indent pseudomonotone operator}

\begin{abstract}
 In this article, we prove the existence of a weak solution for the
 degenerate semilinear elliptic Dirichlet boundary-value problem
 \begin{gather*}
  Lu(x)+\sum_{i=1}^n g(x)h(u(x))D_iu(x)=f(x)\quad \text{in }\Omega,\\
  u=0\quad \text{on }\partial\Omega,
 \end{gather*}
 in a suitable weighted Sobolev space. Here
 $\Omega\subset\mathbb{R}^n$, $1\leq n\leq3,$ is not necessarily bounded.
\end{abstract}

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

\section{Introduction}

For $1\leq n\leq3$, let $\Omega\subset\mathbb{R}^n$, be a domain (not
necessarily bounded) with boundary $\partial\Omega$. We assume
$\Omega =\cup^\infty_{i=1} \Omega_{i}$,
$\overline{\Omega}_i\subseteq \Omega_{i+1}\subseteq
\overline{\Omega}_{i+1}\subset\Omega$, each $\Omega_i\subset \mathbb{R}^n$ is a bounded
domain with boundary $\partial\Omega_i$. 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))
\text{,}\quad D_j=\frac{\partial}{\partial x_j},
\end{equation*}
where the coefficients $a_{ij}$ are measurable, real valued functions, the matrix $\mathcal{A}=(a_{ij})$ is
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 weight function
$(\lambda>0,\Lambda>0)$. When $\omega =1$ in \eqref{eq:a1}, the condition \eqref{eq:a1} reduces to the usual ellipticity condition.
 However, such an ellipticity condition may not hold if $ a_{ij}$ are functions vanishing at some point $x\in \overline{ \Omega}$ leading to the degeneracy of the ellipticity condition. Let $f\in L^{2}(\Omega)$. In this paper, we study the existence of weak solutions to the degenerate semilinear elliptic BVP
\begin{equation}\label{eq:a2}
\begin{gathered}
 Lu(x)+ \sum_{i=1}^ng(x)h(u(x))D_iu(x)=f(x)\quad \text{in }\Omega,\\
 u=0\quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $g /\sqrt{\omega} \in L^{\infty}(\Omega)$ and $h$ is bounded and Lipschitz continuous. 
The tools used are pseudomonotone operators as introduced by of
Br\'ezis \cite{brz}, the compact embedding theorem in weighted
Sobolev spaces in a domain of $\mathbb{R}^n,n\leq3$ and a well-known technique used for unbounded domain as in Noussair and Swanson\cite{nou1}. Where as the restriction on dimension of the domain has yields us a required compactness condition.
The study is inspired by a non-degenerate problem in bounded domain given in the book by Zeidler \cite{zed1}.\par
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; see \cite{app,wei,chi,wie2,loc,fra}.
Elliptic
BVPs in unbounded domains present specific
difficulties, primarily due to lack of compactness.  Another
difficulty in the study of the elliptic BVPs is due to the non-availability of the Poincare-inequality in the Sobolev spaces
$W_{0}^{1,p}(\Omega)$ for a general unbounded domain say $\Omega$. %Some of the techniques used for problems in unbounded domains by many authors are as follows.
One of the classical technique employed  is extracting a solution on
unbounded domain $\Omega$ by solutions on bounded subdomains of $\Omega$ under the assumption the suitable upper and lower solutions exist.
The related literature are found in Noussair and Swanson\cite{nou1} and Cac \cite{pac}. Secondly, the use of Sobolev spaces of highly
symmetric functions, which admit compact embeddings, as in Berestycki and
Lions \cite{bere3, bere4}. Thirdly, the use of weighted-norm Sobolev
spaces which admit compact embeddings, as in Benci \cite{benci}, Bongers, Heinz and Kiipper \cite{bong}.

 In \cite{melv}, Berger and Schechter  have shown that a substitute for such embedding results can be obtained when $\Omega$ is unbounded, by introducing appropriate weighted $L^p$ norms.
These results are then applied by them to establish an existence theorem for the Dirichlet problem for quasilinear elliptic equations in an unbounded domain.
A few references for nonlinear boundary value
problems in unbounded domains with aid of
pseudomonotone operators %is studied in an article by Benci and
are found in \cite{brow,pavm,edm,landes}. The equation \eqref{eq:a2} considered in the present study is not a subclass of the equations studied in \cite{brow,pavm,edm,landes}.  The compactness condition for weighted Sobolev spaces has been assumed in \cite{pavm}, and it is shown how the assumption of monotonicity can be
weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated
or singular elliptic differential operators.

Section 2 deals with preliminaries. Section 3 deals with the
existence of a solution \eqref{eq:a2} in an arbitrary bounded domain say $G$.
In section 4, we obtain a uniform
bound for the solutions $\{u_i\}$ of
\eqref{eq:a2} in each bounded subdomains $\Omega_i$ and finally, extraction of a solution for \eqref{eq:a2} from the sequence $\{u_i\}$ has been shown.

\section{Preliminaries}

Let
$\Omega\subset \mathbb{R}^{n}$, $1\leq n\leq3$ be an open connected
set. Let $\omega:\mathbb{R}^{n}\to\mathbb{R}^{+}$ be a weight
function(\emph{i.e.}\ locally integrable non negative function with
$0<\omega<\infty$ a.e) in $\Omega$ satisfying the conditions
\begin{equation}
\omega\in L^1_{\rm loc}(\Omega),\quad
{\omega}^{-1/(p-1)}\in L^1_{\rm loc}(\Omega),\quad 1 < p < \infty.\label{n}
\end{equation}
We denote by $L^p(\Omega)$ $(1\leq p<\infty)$ the usual Banach space
of measurable real valued functions, $u$, defined in $\Omega$ for
which
\begin{equation}
  \|u\|_{p,\Omega}=\Big(\int_{\Omega} |u|^p  dx\Big)^{1/p}<\infty.
\end{equation}
For $p\geq1$, the weighted Sobolev space $W^{1,p}(\Omega,\omega)$ is defined by
\begin{equation*}
  W^{1,p}(\Omega,\omega):=\{u\in L^{p}(\Omega): D_{j} u\in
  L^{p}(\Omega,\omega),j=1,2\dots,n\}
\end{equation*}
with the associated norm
\begin{equation}\label{n1}
  \|u\|_{1,p,\Omega}=\Big(\int_\Omega |u|^{p}dx+\int_\Omega|D u|^{p}\omega\,  dx\Big)^{1/p},
\end{equation}
 where  $Du=(D_1u,\dots,D_nu)$. The space $W_0^{1,p}(\Omega,\omega)$ is defined as
the closure of $C^{\infty}_0(\Omega)$ with respect to the norm
\eqref{n1}. We also note that $W^{1,2}(\Omega,\omega)$ and
$W^{1,2}_0(\Omega,\omega)$, are Hilbert spaces.

\begin{proposition}\label{prop1}
 For abounded domain $\Omega\subset \mathbb{R}^{n}$, we have the compact embedding
\begin{equation}\label{c1}
    W_0^{1,p}(\Omega,\omega)\hookrightarrow\hookrightarrow L^{p+\eta}(\Omega)
\quad\text{for }0\leq\eta<p_s^*-p
\end{equation}
provided
\begin{equation}\label{n2}
\omega^{-s}\in L^1(\Omega)\quad\text{and}\quad
s\in\big(\frac{n}{p},\infty)\cap\big[\frac{1}{p-1},\infty),
\end{equation}
where
\begin{equation}\label{ps::1}
p_s=\frac{ps}{s+1} \quad\text{and}\quad p_s^*=\frac{np_s}{n-p_s}.
\end{equation}
\end{proposition}

For more details, we refer \cite{pav}.
It follows from the weighted Friedrichs inequality \cite[p.27]{pav} the norm
\begin{equation}\label{n3}
  \|u\|_{0,1,p,\Omega}=\Big(\int_\Omega|D u|^{p}\omega dx\Big)^{1/p}.
\end{equation}
on the space $W_0^{1,p}(\Omega,\omega)$($\Omega$ bounded) is equivalent to the norm
$\|u\|_{1,p,\Omega}$ defined by \eqref{n1} provided \eqref{n2} holds.
Hereafter, we assume the weight function $\omega$ satisfies conditions
\eqref{n} and \eqref{n2}.
We note in the following remark that the Proposition \ref{prop1} restricts
the dimension $n$ given the weight $\omega$ and the exponent $p$.

\begin{remark} \rm
Let $\Omega\subset \mathbb{R}^{n}$ be a bounded domain.
From \eqref{ps::1}, we note that
$$
2_s^*=\frac{2ns}{n(s+1)-2s}.
$$
Let
\begin{equation*}
\omega^{-s}\in L^1(\Omega)\quad \text{and}\quad
 s\in\big(\frac{n}{p},\infty\big)\cap\big[\frac{1}{p-1},\infty\big).
\end{equation*}
For $\eta=2$, from \eqref{c1}, we have
\begin{equation*}
    W_0^{1,2}(\Omega,\omega)\hookrightarrow\hookrightarrow L^{4}(\Omega) \quad
\text{for }0\leq 2<2_s^*-2.
\end{equation*}
Then
\begin{equation}\label{emb4}
2_s^*-2>2\Rightarrow\frac{2ns}{n(s+1)-2s}>4.
\end{equation}
Now, the inequality \eqref{emb4} holds, when $n\leq3$.
\end{remark}

\begin{example} \rm
Let $\Omega=\{x\in\mathbb{R}^n,n\leq3:|x|<1\}$ and $p=2$.
Then $\omega(x)=|x|^{\eta}$, $0<\eta<1$ is an admissible weight function.
\end{example}

For more details on weighted Sobolev spaces, we refer \cite{pav,gar,hei,pot2}.
At each step, a generic constant is denoted
by $c$ or ${\beta}_0$ in order to avoid too many suffices.

\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 weak
solution of \eqref{eq:a2} if
\begin{align*}
\int_{\Omega} \sum_{i,j=1}^na_{ij}D_iu(x)D_j\phi(x)dx+\int_\Omega\sum_{i=1}^n
g(x)h(u(x))D_iu\phi(x)dx=\int_\Omega f(x)\phi(x)dx
\end{align*}
for every $\phi\in W^{1,2}_0(\Omega,\omega)$.
\end{definition}

\begin{definition}[Pseudomonotone operators]\rm
 Let $A:X\to X^*$ be an operator on the real reflexive Banach space $X$.
The operator $A$ is called pseudomonotone if
 $u_n\rightharpoonup u$ as $n\to\infty$ and
\begin{equation*}
    \limsup_{n\to\infty}\langle Au_n,u_n-u\rangle\leq 0
\end{equation*}
implies
\begin{equation*}
    \langle Au,u-w\rangle\leq\liminf_{n\to\infty}\langle Au_n,u_n-w\rangle\quad
\text{for all }w\in X.
\end{equation*}
\end{definition}

We consider the operator equation
\begin{equation}\label{ps1}
    Au=b,\quad u\in X.
\end{equation}
In section 3, we use the following result.

\begin{proposition}[Br\'ezis(1968)] \label{pr1}
 Assume that the operator $A:X\to X^*$ is pseudomonotone, bounded and coercive
on the real,separable reflexive Banach space $X$.
Then, for each $b\in X^*$, the equation \eqref{ps1} has a solution.
\end{proposition}

For a proof of the above Theorem, we refer the reader to \cite[Theorem 27.A]{zed2}.

\section{Bounded domain}

Let $G$ be a bounded domain in $\mathbb{R}^n$ with $1\leq n\leq3$. We
consider the degenerate semilinear elliptic BVP
\begin{equation}\label{eq:b1}
\begin{gathered}
  Lu(x)+ \sum_{i=1}^n g(x)h(u(x))D_iu(x)=f(x)\quad\text{in }G,\\
  u(x)=0\quad \text{on }\partial G.
\end{gathered}
\end{equation}
 We need the following hypotheses for further study.
\begin{itemize}
\item [(H1)] Assume $g/\sqrt{\omega}\in L^{\infty}(G)$ and $f \in L^{2}(G)$.
\item [(H2)] Let $h:\mathbb{R}\to\mathbb{R}$ is a bounded
($|h(t)|\leq \mu, \forall t\in\mathbb{R}$, $\mu>0$), and Lipschitz continuous with Lipschitz
constant $A>0$ (e.g., $h(t)=\sin(t),\,\forall t\in \mathbb{R}$).
\end{itemize}
We define the functionals $B_1,B_2: W_0^{1,2}(G,\omega)\times
W_0^{1,2}(G,\omega)\to\mathbb{R}$ by
\begin{gather*}
 B_1(u,\phi)=\int_{G}
  \sum_{i,j=1}^na_{ij}(x)D_iu(x)D_j\phi(x)dx\\
  B_2(u,\phi)=r(u,u,\phi), \quad
r(u,v,\phi):=\int_{G}\sum_{i=1}^n g(x)h(u(x))D_iv(x)\phi(x)dx.
\end{gather*}
Also, define the functional $T:W_0^{1,2}(G,\omega)\to\mathbb{R}$ by
\begin{equation*}
  T(\phi)=\int_{G}f(x)\phi(x)dx.
\end{equation*}
A function $u\in W_0^{1,2}(G,\omega)$ is a weak solution
of \eqref{eq:b1} if
\begin{equation}\label{m1}
B_1(u,\phi)+B_2(u,\phi)=T(\phi),\quad\text{for all }\phi\in W_0^{1,2}(G,\omega).
\end{equation}

\begin{theorem}\label{t1}
Assume {\rm (H1)} and {\rm (H2)}. In addition, let the condition
$$
\mu C_{G}\|g/\sqrt{\omega}\|_{\infty,G}<\lambda,
$$
where $C_G$ is a constant (depending on $G$) arising out of weighted Fredrichs inequality.
Then the BVP \eqref{eq:b1} has a weak solution.
\end{theorem}

\begin{proof}
 First we write the BVP \eqref{eq:b1} as operator equation
\begin{equation}\label{bnt}
u\in W_0^{1,2}(\Omega,\omega): Bu+Nu=T\quad\text{in }
[W_0^{1,2}(\Omega,\omega)]^*,
\end{equation}
where $T\in [W_0^{1,2}(\Omega,\omega)]^*,B :W_0^{1,2}(\Omega,\omega)\to
[W_0^{1,2}(\Omega,\omega)]^*$
is linear, uniformly monotone and continuous,
$N :W_0^{1,2}(\Omega,\omega)\to [W_0^{1,2}(\Omega,\omega)]^*$
is strongly continuous and $B+N$ is coercive. Further we put Propositions \ref{pr1}
to this operator equation. The realization of this idea is split into 5 steps
for convenience.

\emph{Step 1:}
Since $|a_{ij}(x)|\leq c \omega (x)$, we have by H\"older's inequality
\begin{align*}
 B_1(u,v)&=\int_{G}
 \sum_{i,j=1}^na_{ij}(x)D_iu(x)D_jv(x)dx\\
 &\leq c \int_{G} \sum_{i,j=1}^n  |D_iu(x)||D_jv(x)|\omega dx\\
 &\leq c \|u\|_{0,1,2,G}\|v\|_{0,1,2,G},\quad
 \text{for all }u,v\in W_0^{1,2}(G,\omega).
\end{align*}
We define the operator
$B: W_0^{1,2}(G,\omega)\to
[W_0^{1,2}(G,\omega)]^*$ as
$$
(Bu|\phi)=B_1(u,\phi),\quad  \text{for }u,\phi\in W_0^{1,2}(G,\omega).
$$
Hence, the operator $B$ is well defined, linear, and continuous.
It follows from \eqref{eq:a1} that
\begin{align*}
\hspace{-0.3em} \big(Bu-Bv|u-v\big)
 &=B_1(u-v,u-v) \\&=\int_{G}
 \sum_{i,j=1}^na_{ij}D_i(u-v)D_j(u-v)dx \\& \geq \lambda\int_{G}|D(u-v)|^2w dx  \\
 &= \lambda\|u-v\|_{0,1,2,G}^2 \text{ for all }u,v\in X.
\end{align*}
Consequently, $B$ is uniformly monotone(and hence coercive).
For more details on monotone operators, we refer\cite{zed1}.

\emph{Step 2:} By (H1) and (H2), it follows from H\"{o}lder's inequality,
\begin{align*}
&\big|\int_G g(x)~h(u(x))D_iu(x)v(x)dx\big|\\
& \leq\int_G |g/\sqrt{\omega}||h(u(x))||D_iu(x)\sqrt{\omega}||v(x)|dx \\
&\leq  \mu\,\|g /\sqrt{\omega}\|_{\infty,G}\int_G |D_iu\,\omega^{1/2}||v| dx\\
&\leq \mu\,\|g / \sqrt{\omega}\|_{\infty,G}\Big(\int_G |D_iu|^2wdx\Big)^{1/2}
\Big(\int_G|v|^2dx\Big)^{1/2}
\end{align*}
and hence, by the weighted Friedrichs inequality \cite[p.27]{pav},
\begin{equation*}
|B_2(u,v)|\leq C_G\|u\|_{0,1,2,G}\|v\|_{0,1,2,G}\quad\text{for all }
u,v\in W_0^{1,2}(G,\omega).
\end{equation*}
 where $C_G>0$ is a constant(depending on domain $G$). Since $B_2(u,.)$
is linear and bounded, there exists an operator
$N:W_0^{1,2}(G,\omega)\to [W_0^{1,2}(G,\omega)]^*$ such that
 \[
(Nu|v)=B_2(u,v)\quad\text{for all  }u,v\in W_0^{1,2}(G,\omega).
\]
Then, problem \eqref{eq:b1} is equivalent to operator equation
$$
Bu+Nu=T,~u\in W_0^{1,2}(G,\omega).
$$

\emph{Step 3:} (I) By \eqref{c1}, the embedding
$W_0^{1,2}(G,\omega)\hookrightarrow\hookrightarrow L^4(G)$ is compact.
(II) Let $u_n\rightharpoonup u$ in $W_0^{1,2}(G,\omega)$ as $n\to\infty$.
Then, the sequence $\{u_n\}$ is bounded in $W_0^{1,2}(G,\omega)$. By (I),
$ u_n\to u$  in $L^4(G)$ as $n\to\infty$.
We claim that
\begin{equation*}
Nu_n\to Nu\quad\text{in }[W_0^{1,2}(G,\omega)]^*\quad\text{as }n\to\infty.
\end{equation*}
or
\begin{equation*}
\|Nu_n-Nu\|_{[W_0^{1,2}(G,\omega)]^*}=\sup_{\|v\|_{0,1,2,G}\leq 1}|(Nu_n-Nu|v)|\to0
\quad\text{as }n\to\infty.
\end{equation*}
Otherwise, there exists an $\epsilon_0>0$ and a sequence $\{v_n'\}$, which we
denote briefly by $\{v_n\}$, such that $\|v_n\|_{0,1,2,G}\leq 1$
for all $n$, with
\[
 (Nu_n-Nu|v_n)\geq\epsilon_0\quad\text{for all }n\label{s1}.
\]
Passing to a subsequence, if necessary, we assume that
$v_n\rightharpoonup v$ in $W_0^{1,2}(G,\omega)$ and it follows
that $v_n\to v$  in $L^4(G)$ as $n\to\infty$.
We note that
\begin{equation}
\begin{aligned}
&h(u_n)(D_iu_n)v_n -h(u)(D_iu)v_n\\
&= (h(u_n)- h(u))(D_iu_n)v_n+h(u)(D_iu_n)(v_n-v)\\
&\quad+h(u)(D_iu_n-D_iu)v+h(u)(D_iu)(v-v_n).
\end{aligned}\label{b4}
\end{equation}
Since $h$ is Lipschitz, we have
\begin{equation*}
|h(u_n)- h(u)|\leq A|u_n- u|,
\end{equation*}
by (I) and by the generalized H\"{o}lder's inequality, we obtain
\begin{equation}
\begin{aligned}
&\big|\int_G g(x)(h(u_n)- h(u))(D_iu_n)v_ndx\big|\\
&\leq \|g / \sqrt{\omega}\|_{\infty,G}\int_G|h(u_n)- h(u)||D_iu_n\omega^{1/2}||v_n|dx\\
&\leq A\|g/ \sqrt{\omega}\|_{\infty,G}\Big(\int_G{| u_n- u|^{4}dx}\Big)^{1/4}
  \Big(\int_G|D_iu_n|^{2}\omega dx\Big)^{1/2}
 \Big(\int_G|v_n|^{4}dx\Big)^{1/4}\\
&\leq C_G\| u_n- u\|_{4,G}\| u_n\|_{0,1,2,G} \| v_n\|_{0,1,2,G},
\end{aligned} \label{h1}
\end{equation}
where $C_G$ is a constant (depending on $G$) arising out of weighted Fredrichs inequality.
We have $u_n\to u$ and $v_n\to v$  in $L^4(G)$  as  $n\to\infty$; i.e.,
 $\| u_n- u\|_{4,G}\to0$ and $\| v_n- v\|_{4,G}\to0$.
Moreover, the sequences $\{u_n\}$ and $\{v_n\}$ are bounded in
$W_0^{1,2}(G,\omega)$. Again, we have $u_n\rightharpoonup u
\quad\text{in }W_0^{1,2}(G,\omega)$ and
\begin{align*}
|r(u,w,v)|&=\big|\int_{G}\sum_{i=1}^n g(x)h(u(x))D_iw(x)v(x)dx\big|\\
&\leq C_G\|w\|_{0,1,2,G}\|v\|_{0,1,2,G}\quad\text{for all
}u,v,w\in W_0^{1,2}(G,\omega),
\end{align*}
due to the H\"{o}lder's inequality, and hence, the linear functional
$w\mapsto r(u,w,v)$ is continuous on $W_0^{1,2}(G,\omega)$.
Finally, we have
\begin{equation}\label{r1}
r(u,u_n-u,v)\to0\quad\text{as }n\to\infty.
\end{equation}
By \eqref{b4},\eqref{r1} and by similar arguments as in\eqref{h1}, we have
\begin{equation}
\begin{aligned}
|(Nu_n-Nu|v_n)|
&=\sum_{i=1}^n \big|\int_{G}g(x)\{h(u_n)(D_iu_n)v_n-h(u)(D_iu)v_n\}dx\big|\\
&\leq\sum_{i=1}^n \int_{G}|g / \sqrt{\omega}||h(u_n)(D_iu_n)v_n-h(u)(D_iu)v_n
 |\omega^\frac{1}{2}dx  \\
&\leq \|g / \sqrt{\omega}\|_{\infty,G}C_G\{A\|u_n-u\|_{4}
 \| u_n\|_{0,1,2,G}\| v_n\|_{0,1,2,G}\\
&\quad +\mu\| v_n- v\|_{4,G}\| u_n\|_{0,1,2,G}+\mu\| v_n- v\|_{4,G}\|u\|_{0,1,2,G}\}\\
&\quad +|r(u,u_n-u,v)|\to0 \quad\text{as } n\to\infty.
\end{aligned} \label{b5}
\end{equation}
 Relation \eqref{b5} contradicts \eqref{s1} and hence,
 $N$ is strongly continuous.

\emph{Step 4:} For all $u\in W_0^{1,2}(G,\omega)$,
\begin{align*}
|B_2(u,u)|
&\leq \big|\int_G\sum_{i=1}^ng~{h(u)(D_iu)udx\big|}\\
&\leq \mu\|g/\sqrt{\omega}\|_{\infty,G} \int_G\sum_{i=1}^n|D_iu\,\omega^{1/2}||u|dx\\
&\leq \mu\|g/\sqrt{\omega}\|_{\infty,G}\sum_{i=1}^n
 \Big(\int_G|D_iu|^2\omega dx\Big)^{1/2}\Big(\int_G|u|^{2}dx\Big)^{1/2}\\
&\leq \mu C_G\|g/\sqrt{\omega}\|_{\infty,G}\|u\|_{0,1,2,G}^2,
\end{align*}
where $C_G$ is a constant(depending on $G$) arising out of weighted Fredrichs inequality.
By \eqref{eq:a1}, there exists a constant $\lambda>0$ such that
\begin{equation*}
B_1(u,u)\geq \lambda\|u\|_{0,1,2,G}^2\quad\text{for all }u\in W_0^{1,2}(G,\omega).
\end{equation*}
This implies
\begin{align*}
\big(Bu+Nu|u\big)&=B_1(u,u)+B_2(u,u)\\
&\geq(\lambda-\mu C_{G}\|g/\sqrt{\omega}\|_{\infty})\|u\|_{0,1,2,G}^2\quad
\text{for all }u\in W_0^{1,2}(G,\omega);
\end{align*}
i.e., $B+N$ is coercive if $\mu C_{G}\|g/\sqrt{\omega}\|_{\infty}<\lambda$.

\emph{Step 5:} Since $B$ is uniformly monotone and continuous, $N$ is strongly
continuous and $B+N$ is coercive, by \cite[Proposition 26.16, p.576]{zed1},
 we note that the operator $B+N$ is pseudomonotone. Also, we have $B+N$ is
continuous, and bounded. Now, for
$\mu C_{G}\|g/\sqrt{\omega}\|_{\infty}<\lambda$, by Proposition \ref{pr1}, problem
\eqref{eq:b1} has a weak $\text{solution in }W_0^{1,2}(G,\omega)$.
\end{proof}

\section{Unbounded domain}

Let $\Omega$ be a domain (not necessarily bounded) in $\mathbb{R}^n$ with
$1\leq n\leq3$.
We consider the degenerate semilinear elliptic BVP
\begin{equation}\label{u1}
\begin{gathered}
  Lu(x)+\sum_{i=1}^n  g(x)h(u(x))D_iu(x)=f(x)\quad\text{in }\Omega, \\
  u(x)=0\quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}

\begin{itemize}
\item [(H1')] Assume $g/\sqrt{{\omega}}\in L^{\infty}(\Omega)$
and $f \in L^{2}(\Omega)$.
\end{itemize}

\begin{lemma} \label{lem2.4}
Assume {\rm (H1')} and {\rm (H2)}.
If $\mu C_{\Omega_l}\|g/\sqrt{\omega}\|_{\infty,\Omega}<\lambda$, then the
BVP
\begin{equation}\label{eq:b5}
\begin{gathered}
  Lu+ \sum_{i=1}^n g~h(u)D_iu=f\quad \text{in }\Omega_l,\\
 u=0\quad \text{on  }\partial \Omega_l
\end{gathered}
\end{equation}
has a weak solution $u=u_l\in W^{1,2}_0(\Omega_l,\omega)$ for
$l=1,2,3,\dots$. In addition, for $k\geq l$,
$\|u_k\|_{0,1,2,\Omega_l}\leq {\beta}_0$,   where ${\beta}_0$ is independent of
$k$.
\end{lemma}

\begin{proof}
We use arguments similar to those in Theorem \ref{t1},
Let $u_k\in W_0^{1,2}(\Omega_k,\omega)$ be the solutions of \eqref{eq:b5} in
each bounded subdomains $\Omega_k$. Also $B_1,B_2$ and $T$ are defined in
a similar way as in section-3. Then, from the hypotheses and
relation \eqref{c1}, we note that
for $k\geq l$,
\begin{gather*}
  |B_1(u_k,u_k)|\leq c   \|u_k\|_{0,1,2,\Omega_l}^2\\
|B_2(u_k,u_k)| \leq \mu C_{\Omega_l}\|{\frac{g}{\sqrt{\omega}}}\|_{\infty,\Omega_l}
\|u_k\|_{0,1,2,\Omega_l}^2\\
|T(u_k)|\leq C_{\Omega_{l}}\|f\|_{2,\Omega_l} \|u_k\|_{0,1,2,\Omega_l},
\end{gather*}
where $C_{\Omega_l}$ (is the constant depending on the domain $\Omega_l$)
independent of $k$. Also, we have for $k\geq l$
\begin{align*}
B_1(u_k,u_k) \geq\lambda\int_{\Omega_l}|Du_k|^2\omega dx
=\lambda \|u_k\|^2_{0,1,2,\Omega_l}.
\end{align*}
We obtain
\begin{equation}\label{e:m1}
  \|u_k\|^2_{0,1,2,\Omega_l}\leq\frac{1}  {\lambda}  B_1(u_k,u_k)
\end{equation}
Also, we note that
\begin{align*}
     (Bu_k+Nu_k|u_k)
&=B_1(u_k,u_k)+B_2(u_k,u_k)\\
&\geq (\lambda-\mu C_{\Omega_l}\|\frac{g}{\sqrt{\omega}}\|_{\infty,\Omega_l})
 \|u_k\|^2_{0,1,2,\Omega_l}
\end{align*}
As, $T(u_k)= B_1(u_k,u_k)+B_2(u_k,u_k)$, we have
\begin{equation}\label{e:m2}
(\lambda-\mu C_{\Omega_l}\|\frac{g}{\sqrt{\omega}}\|_{\infty,\Omega_l})
\|u_k\|^2_{0,1,2,\Omega_l}
 \leq C_{\Omega_l}   \|{f}\|_{2,\Omega_l}\|u_k\|_{0,1,2,\Omega_l}.
\end{equation}
Since $\lambda> \mu C_{\Omega_l}\|\frac{g}{\sqrt{\omega}}\|_{\infty,\Omega_l}$, By
\eqref{e:m1} and \eqref{e:m2}, we have
\begin{align*}
  \|u_k\|_{0,1,2,\Omega_l}
&\leq \frac{{C_{\Omega_l}}\|{f}\|_{2,\Omega_l}}
  {(\lambda-\mu  C_{\Omega_l}\|\frac{g}{\sqrt{\omega}}\|_{\infty,\Omega_l})}\\
&\leq \frac{{C_{\Omega_l}}\|{f}\|_{2,\Omega}}
  {(\lambda-\mu  C_{\Omega_l}\|\frac{g}{\sqrt{\omega}}\|_{\infty,\Omega})}={\beta}_0,
\end{align*}
where ${\beta}_0$ is independent of $k$. Hence,
\begin{equation}\label{e:uni}
  \|u_k\|_{0,1,2,\Omega_l}\leq   {\beta}_0,\quad \text{for all }k\geq l
\end{equation}
\end{proof}

\begin{theorem}\label{thm3.5}
Let $\Omega = \cup^\infty_{l=1}{\Omega_l}$,
$\Omega_l\subseteq\overline{ \Omega}_{l}\subseteq\Omega_{l+1}\subseteq
\overline{\Omega}_{l+1}$ be bounded domains in $\Omega$, for
$l\geq1$ and let the condition
$\mu  C_{\Omega_l}\|\frac{g}{\sqrt{\omega}}\|_{\infty,\Omega}<\lambda$
be fulfilled. Under the hypotheses {\rm (H1')} and \rm{(H2)}, \eqref{u1}
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 of
\eqref{eq:b5} 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 \eqref{e:uni}, we have
\[
\|u_k\|_{0,1,2,\Omega_l}\leq {\beta}_0, \quad \text{for }k\geq l.
\]
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  since
${u}^l(x)={u}^m(x)$, $x\in\Omega$, 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 $m\to\infty$.

We still need to show that $u$ is the required weak solution. It is
sufficient to show that $u$ is a weak solution of \eqref{u1} 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
\[
\int_M D(u_{k_m^m}-u).D\phi \,\omega dx\to0,\quad \text{as } m\to\infty,
\]
implies
\[
\int_M{D_i(u_{k_m^m}-u)}D_j\phi \omega dx\to0,\quad\text{as } m\to\infty.
\]
 From \eqref{eq:a1}, for a constant $c$, we have $|a_{ij}|\leq c \omega$. 
We observe that
\begin{equation}\label{e:d0}
\int_M{\sum_{i,j=1}^na_{ij}D_i(u_{k_m^m}-u)}D_j\phi\,dx
\leq c\sum_{i,j=1}^n \int_MD_i(u_{k_m^m}-u)D_j\phi w\,dx \to 0,
\end{equation}
as $m\to\infty$. Also, by \eqref{c1}, $u_{k_m^m}$ $\to u$ in $L^4(M)$. We
have, by the generalized  H\"{o}lder's inequality
\begin{equation} \label{d1} 
\begin{aligned}
&\big|\int_M{g}(h(u_{k_m^m})-h(u))D_i(u_{k_m^m}-u)\phi dx\big|\\
&\leq A\int_M|{{g}}/{\sqrt{\omega}}||(u_{k_m^m}-u)||D_i(u_{k_m^m}-u)
 \sqrt{\omega}||\phi|dx \\
&\leq A\|\frac{{g}}{\sqrt{\omega}}\|_{\infty,M}\int_M|(u_{k_m^m}-u)||D_i(u_{k_m^m}-u)
 \sqrt{\omega}||\phi|dx\\
&\leq A\|\frac{{g}}{\sqrt{\omega}}\|_{\infty,M}
\Big(\int_M|(u_{k_m^m}-u)|^{4}dx\Big)^{1/4}
\Big(\int_M|D_i(u_{k_m^m}-u)|^{2}{\omega}dx\Big)^{1/2}\\
&\quad\times\Big(\int_M|\phi|^4dx\Big)^{1/4}\\
&\leq AC_M\|\frac{g}{\sqrt{\omega}}\|_{\infty,M}\|u_{k_m^m}-u\|_{4,M}
\|u_{k_m^m}-u\|_{0,1,2,M} \|\phi\|_{2,M}\to0,
\end{aligned}
\end{equation}
as $m\to\infty$.
Since $M$ is an arbitrary bounded domain in
$\Omega$, it follows from \eqref{e:d0} and
\eqref{d1},
\begin{align*}
&\int_\Omega \sum_{i,j=1}^na_{ij}(x)D_iu(x)D_j\phi(x)dx
+\int_\Omega \sum_{i=1}^ng(x)h(u(x))D_iu(x) \phi(x)dx \\
&=\int_\Omega f(x)\phi(x)dx
\end{align*}
for every $\phi\in W^{1,2}_0(\Omega,\omega)$,
which completes the proof.
\end{proof}

\begin{remark} \rm
The above results still hold if $h$ is a bounded and
continuous (not necessarily Lipschitz). We have to slightly modify
the argument used in the inequalities \eqref{h1} and \eqref{d1} and
the rest of the proof remains same. For a bounded domain $G$ and
bounded function $h$, if $u\in L^2(G)$, we have $h(u)\in L^{4}(G)$.
Define the Nemytskii operator
 %\label{eq:h1}
$h_u:L^2(G)\to   L^{4}(G)$ 
by $h_u(x)=h(u(x))$; we have $h_u$ is continuous  \cite[Theorem
2.1]{kra}. Let $u_n\rightharpoonup u$ in $W^{1,2}_0(G,\omega)$,
then
\begin{align*}
&\big|\int_G g(x)(h(u_n)- h(u))(D_iu_n)v_ndx\big|\\  
&\leq \|g / \sqrt{\omega}\|_{\infty,G}\int_G|h(u_n)-
h(u)||D_iu_n\omega^{1/2}||v_n|dx\\  
&\leq C_G\|g / \sqrt{\omega}\|_{\infty,G}\|h(u_n)- h(u)\|_{4,G}
  \|u_n\|_{0,1,2,G}\|v_n\|_{0,1,2,G}\to0, \quad\text{as } m\to\infty.
\end{align*}
Similar argument can be use to prove the inequality \eqref{d1} in
section 4.
\end{remark}

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\end{document}