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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 13, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/13\hfil Existence of solutions]
{Existence of solutions for  quasilinear
 parabolic equations at resonance}

\author[G. Jia, L.-N. Huang,  X.-J. Zhang \hfil EJDE-2013/13\hfilneg]
{Gao Jia, Xiao-Juan Zhang, Li-Na Huang}  % in alphabetical order

\address{Gao Jia\newline
College of Science,
University of Shanghai for Science and Technology,
Shanghai 200093, China}
\email{gaojia79@yahoo.com.cn, gaojia79@139.com}

\address{Xiao-Juan Zhang \newline
College of Science,
University of Shanghai for Science and Technology,
Shanghai 200093, China}
\email{ zhangxiaojuan0609@163.com}

\address{Li-Na Huang \newline
College of Science,
University of Shanghai for Science and Technology,
Shanghai 200093, China}
\email{ hln881229@163.com}

\thanks{Submitted March 29, 2012. Published January 14, 2013.}
\thanks{Supported by grant 11171220 from  the
National Natural Science Foundation of China}
\subjclass[2000]{35H30,  35K58,  65L60}
\keywords{Weighted Sobolev space; quasilinear parabolic equation; resonance}

\begin{abstract}
 In this article, we show the existence of nontrivial solutions
 for a class of quasilinear parabolic differential equations.
 To obtain the  solution in a weighted Sobolev space,
 we use the  Galerkin method, Brouwer's theorem, and a
 compact Sobolev-type embedding theorem  proved by Shapiro.
\end{abstract}

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

 \section{Introduction}

 Many results on the existence of solutions
of the quasilinear parabolic resonance problems have been
presented in \cite{b2,k2,l2,l3,s1} and their references cited therein.
Shapiro \cite{s1} considered a weak solution
of the following problem, in the Hilbert space
 $\widetilde{H}(\widetilde{\Omega},\Gamma)$,
\begin{equation} \label{e1.1}
 \begin{gathered}
\rho D_{t}u+\mathcal{Q}u=[\lambda_{j_0}u+f(x,u)+g(x,t,u)]\rho, \quad
  (x,t)\in\widetilde{\Omega},\\
 u\in\widetilde{H}(\widetilde{\Omega},\Gamma),
 \end{gathered}
\end{equation}
where
$$
\mathcal{Q}u=-\sum_{i=1}^{N} D_i\big[p_i^{1/2}(x)A_i(x,u,Du)\big]
+qB_0(x,u,Du)u.
$$
Kuo \cite{k2} also discussed the existence of a nontrivial solution
for a quasilinear parabolic equation in the Hilbert space
 $\widetilde{H}^{m}_0$:
\begin{equation}  \label{e1.2}
 \begin{gathered}
D_{t}u+\widetilde{\mathcal{Q}}u-\lambda_1u+f(x,t,u)=h(x,t), \quad
  (x,t)\in\widetilde{\Omega},\\
 u=0,\quad  (x,t)\in\partial\widetilde{\Omega},
 \end{gathered}
\end{equation}
where
$$
\widetilde{\mathcal{Q}}(u)(v)=\sum_{|\alpha|\leq m}
\int_{\widetilde{\Omega}}A_{\alpha}(x,\xi(u))D^{\alpha}v,
$$
and $ \lambda_1$ is the first eigenvalue of $ \widetilde{\mathcal{Q}}$.

Motivated by Shapiro \cite{s1}, in this paper, we show the existence of solutions
for the  quasilinear parabolic equation in the weighted Sobolev space
 $\widetilde{H}(\widetilde{\Omega},\Gamma)$:
\begin{equation} \label{e1.3}
 \begin{gathered}
\rho D_{t}u+\widetilde{\mathcal{M}}u=[\lambda_{j_0}u+f(x,u)+g(x,t,u)]\rho-G,
\quad (x,t)\in\widetilde{\Omega},\\
 u\in\widetilde{H}(\widetilde{\Omega},\Gamma),
 \end{gathered}
\end{equation}
where
\begin{equation}\label{e1.4}
\widetilde{\mathcal{M}}u=-\sum_{i,j=1}^{N} D_i[p_i^{1/2}(x)p_j^{1/2}(x)
\sigma_i^{1/2}(u)\sigma_j^{1/2}(u)b_{ij}(x)D_ju]+b_0(x)
\sigma_0(u)qu,
\end{equation}
and $\lambda_{j_0}$ is an eigenvalue of $\mathcal{L}$.

In fact, \eqref{e1.3} is one of the most useful sets of Navier-Stokes
equations which describe the motion of viscous fluid substances.
 They are widely used in the design of aircrafts and cars, the
study of blood flow and the design of power stations, etc.
Furthermore, coupled with Maxwell's equations, the Navier-Stokes
equations can also be used to model and study magnetohydrodynamics.

The method of this paper is based on the Galerkin method \cite{l1},
the generalized Brouwer's theorem \cite{k1} and a weighted compact Sobolev-type
embedding theorem \cite{s1} established by Shapiro.
The nonlinearity in \eqref{e1.3} satisfies the generalized
Landesman-Lazer type conditions \cite{l1}. Compared with the problem
\eqref{e1.1} in \cite{s1}, the operator $ \widetilde{\mathcal{M}}$
in \eqref{e1.3} has an extensive presentation format and wider applications.

Now, we give the assumptions and definitions which are needed
for the proof of Theorem \ref{thm1.1}.

Let $ \Omega\subset R^{N},N\geq 1 $, be an open (possibly unbounded)
set and let $\rho(x),p_i(x)\in C^{0}(\Omega)$ be positive functions
with the property that
\begin{equation}\label{e1.5}
\int_{\Omega} \rho(x)dx<\infty,\quad
\int_{\Omega} p_i(x)dx<\infty,\quad i=1,2,\dots,N.
\end{equation}
Let $q(x)\in C^{0}(\Omega)$ be a nonnegative function and
 $\Gamma\subset\partial\Omega$ be a fixed closed set.
Note that $ \Gamma $ may be an empty set and $ q(x) $ may be zero.
On the other hand, $ q(x) $ will satisfy:
There exists $K>0 $, such that
\begin{equation}\label{e1.6}
0\leq q(x)\leq K\rho(x),  \quad\text{for all } x \in\Omega.
\end{equation}
Here $\mathcal{A}$ is a set of real-valued functions defined as
$$
\mathcal{A}=\{u\in C^{0}( \bar{\Omega} \times R ) :
u ( x,t+2 \pi )=u ( x,t ), \text{ for all } (x,t) \in \bar{\Omega} \times R\}.
$$
Setting $ \widetilde{\Omega} = \Omega \times T$,
$T =(-\pi,\pi), p=(p_1,\dots,p_{N})$ and
$ D_i=\frac{\partial u }{\partial x_i},i=1,2,\dots,N$,
we consider the following pre-Hilbert spaces (see \cite{s1}):
$$
\widetilde{C}_{\rho}^{0}(\widetilde{\Omega})
=\big\{u\in C^{0}(\widetilde{\Omega}):
\int_{\widetilde{\Omega}}|u(x,t)|^2\rho(x) \,dx\,dt<\infty \big\},
$$
with the inner product
$$
 \langle u,v\rangle^{\sim}_{\rho}
=\int_{\widetilde{\Omega}}u(x,t)v(x,t)\rho(x) \,dx\,dt ,
$$
 and the space
\begin{align*}
 \widetilde{C}_{p,\rho}^{1}(\widetilde{\Omega},\Gamma)
=\Big\{&u\in \mathcal{A}\cap C^{1}(\Omega \times R) :
u(x,t)=0, \text{ for all } (x,t)\in \Gamma \times R; \\
&\int_{\widetilde{\Omega}}[\sum_{i=1}^{N}|D_iu|^2p_i+(u^2+|D_{t}u|^2)\rho]
 <\infty \Big\}
\end{align*}
with inner product
\begin{equation}\label{e1.7}
\langle u,v\rangle_{\widetilde{H}}
=\int_{\widetilde{\Omega}} \big[ \sum_{i=1}^{N}p_iD_iuD_iv+(uv+D_{t}uD_{t}v)
\rho \big] \,dx\,dt.
\end{equation}
Let $\widetilde{L}^2_{\rho}=L^2_{\rho}(\widetilde{\Omega})$
 denote the Hilbert space obtained from the completion of
$\widetilde{C}^{0}_{\rho}$ with the norm
$\|u\|_{\rho}=(\langle u,u\rangle_{\rho}^{\sim})^{1/2}$
by using Cauchy sequences, and
$ \widetilde{H}=\widetilde{H}(\widetilde{\Omega},\Gamma) $
denote the completion of the space $\widetilde{C}^{1}_{p,\rho}$ with the norm
$ \|u\|_{\widetilde{H}}=\langle u,u\rangle^{1/2}_{\widetilde{H}}$.
Similarly, we have $\widetilde{L}_{p_i}^2,(i=1,2,\cdot\cdot\cdot,N)$ and
$\widetilde{L}_{q}^2$.

It is assumed throughout this paper that $\sigma_i(u)(i=0,1,\dots,N)$ meets:
\begin{itemize}
\item[(S1)] $\sigma_i(u): \widetilde{H}\to \mathbb{R}$ is weakly sequentially
 continuous;
\item[(S2)]  there are $\eta_0,\eta_1>0$, such that
 $\eta_0 \leq \sigma_i(u) \leq \eta_1$, and
 $\sigma_i(u)$ is measurable, for $ u\in \widetilde{H}$.
\end{itemize}
The functions $a_{ij}(i,j=1,2,\dots,N)$ and $ a_0(x) $ satisfy
(also $b_{ij}(x)$ and $ b_0(x))$:
\begin{equation}\label{e1.8}
\begin{gathered}
 a_0(x),  a_{ij}(x)\in C^{0}(\Omega) \cap L^{\infty}(\Omega), \quad
   i,j=1,2,\dots,N;
\\
 a_{ij}(x)=a_{ji}(x),\quad \forall  x\in \Omega, \; i,j=1,2,\dots,N;
\\
 a_0(x)\geq \beta_0 >0\quad (b_0(x)\geq \beta_1>0), \quad \text{for } x\in \Omega;
\\
\text{there is a $c_0>0$  ($c_1>0$) for $x\in\Omega$,
 $\xi\in \mathbb{R}^{N}$, such  that }
\\
\sum_{i,j=1}^{N}a_{ij}(x)\xi_i\xi_j\geq c_0|\xi|^2\quad
(\sum_{i,j=1}^{N}b_{ij}(x)\xi_i\xi_j\geq c_1|\xi|^2).
 \end{gathered}
\end{equation}
The function $g(x,t,s)$ meets the following conditions:
\begin{itemize}
\item[(G1)] $g(x,t,s)$ satisfies the Caratheodory assumptions;

\item[(G2)] for any $\varepsilon>0$, there is  a
$g_{\varepsilon}(x,t)\in\widetilde{L}_{\rho}^2$, such that
$|g(x,t,s)|\leq\varepsilon|s|+g_{\varepsilon}(x,t)$, for a.e.
$(x,t)\in\widetilde{\Omega}$, and all $s\in \mathbb{R}$.
\end{itemize}

\begin{definition} \label{def1.1} \rm
For the quasilinear differential operator $\widetilde{\mathcal{M}}$,
the two-form is
\begin{equation}\label{e1.9}
\widetilde {\mathcal{M}}(u,v)
= \sum_{i,j = 1}^N \int_{\widetilde{\Omega}}
 \big[p_i^{\frac12}p_j^{\frac12}\sigma_i^{\frac12}(u)
\sigma_j^{\frac12}(u) b_{ij}D_j u D_iv \big]
+ \int_{\widetilde{\Omega}} q\sigma_0(u)b_0 uv ,
\end{equation}
for $u,v \in \widetilde{H} (\widetilde {\Omega} ,\Gamma )$.
\end{definition}

Defining
\begin{equation}\label{e1.10}
\mathcal{L}u =  - \sum_{i,j = 1}^N D_i
\big[ p_i^{\frac12}p_j^{\frac12} a_{ij} D_ju  \big] + a_0 q u,
 \end{equation}
for $ u \in H_{p,q,\rho } = H_{p,q,\rho }(\Omega ,\Gamma)$
(as described in \cite{s1}), and
\[
\widetilde{\mathcal{L}}u
=  - \sum_{i,j = 1}^N D_i\big[ p_i^{\frac12}p_j^{\frac12} a_{ij} D_ju  \big]
+ a_0 q u, \quad u \in \widetilde H( {\widetilde \Omega ,\Gamma }),
\]
then the two-form of $\mathcal{L}$ is
\begin{equation}\label{e1.11}
\mathcal{L}(u,v) = \sum_{i,j = 1}^N  \int_\Omega  p_i^{1/2} p_j^{1/2}
 a_{ij}(x) D_ju D_iv  + \int_\Omega  {{a_0}u} vq,\quad
  u,v \in H_{p,q,\rho }( \Omega,\Gamma),
\end{equation}
and the two-form of $ \widetilde{\mathcal{L}}$ is
\begin{equation}\label{e1.12}
\widetilde{\mathcal{L}}(u,v)
 = \sum_{i,j = 1}^N {\int_{\widetilde \Omega } {p_i^{1/2}p_j^{1/2}{a_{ij}}
(x){D_j}u} {D_i}v}
 + \int_{\widetilde \Omega } {{a_0}uv} q,\quad
  u,v \in \widetilde H( {\widetilde \Omega ,\Gamma } ).
\end{equation}

\begin{definition} \label{def1.2}\rm
We say that $\widetilde{\mathcal{M}}$ is $\# \widetilde H$-related to
$\widetilde{\mathcal{L}}$ if the following condition holds:
$$
\lim_{\|u\|_{\widetilde{H}}\to\infty}
\frac{[\widetilde{\mathcal{M}}(u,v) - \widetilde{\mathcal{L}}(u,v)]}
{\|u\|_{\widetilde{H}}}=0,\quad\text{uniformly for }  \| v \|_{\widetilde{H}}
\leq 1.
$$
\end{definition}

\begin{definition} \label{def1.3} \rm
The pair $(\Omega ,\Gamma )$ is a $ {V_L}({\Omega ,\Gamma })$ if
\begin{itemize}
\item[(VL1)] there is a complete orthonormal system
$ \{ \varphi _{n}\}_{n = 1}^{\infty} $ in
$L_{\rho}^2$.
 Also ${\varphi _n} \in {H_{p,q,\rho }} \cap {C^2}$ for all $n$;

\item[(VL2)]  there is a sequence of eigenvalues
$\{ {\lambda _n}\} _{n = 1}^\infty $ with
$0 < {\lambda _1} < {\lambda _2} \le {\lambda _3} \le  \dots
\le {\lambda _n} \to \infty $
such that $\mathcal{L}({\varphi _n},v)
= {\lambda _n}{\langle {{\varphi _n},v} \rangle _\rho }$ for all
$v \in {H_{p,q,\rho }}( {\Omega ,\Gamma })$.
 Also ${\varphi _1} > 0$ in $\Omega$.
\end{itemize}
\end{definition}

We set
\begin{equation}\label{e1.13}
\gamma  = (\lambda _{j_0 + j_1} - \lambda _{j_0} )/2,
\end{equation}
where $\lambda _{j_0}$ is an eigenvalue of $\mathcal{L}$
of multiplicity ${j_1}$.   So ${\lambda _{{j_0} + {j_1}}}$ is the next
eigenvalue strictly greater than ${\lambda _{j_0}}$.
Also, we set
\begin{equation}\label{e1.14}
\mathcal{F}^ \pm (x) = \limsup _{s \to  \pm \infty }f(x,s)/s,\quad
\mathcal{F}_ \pm (x) = \liminf_{s \to  \pm \infty } f(x,s)/s\,.
\end{equation}
For $f(x,s)$, we have:
\begin{itemize}
\item[(F1)] $f(x,s)$ satisfies the Caratheodory conditions;

\item[(F2)] $| {f(x,s) - \gamma s}| \le \gamma| s | + {f_0}(x)$  for  all
$s \in \mathbb{R}$, a.e. $ x \in \Omega $, where ${f_0} \in L_\rho ^2 $;

\item[(F3)]
\begin{equation}\label{e1.15}
\int_{\Omega  \cap ({v > 0})} {({{\lambda _{{j_0} + {j_1}}} - {\lambda _{j_0}}
- {\mathcal{F}^ + }})} {v^2}\rho
 + \int_{\Omega  \cap ({v < 0})} {({{\lambda _{{j_0} + {j_1}}}
- {\lambda _{j_0}} - {\mathcal{F}^ - }})} {v^2}\rho  > 0,
\end{equation}
for every nontrivial ${\lambda _{{j_0} + {j_1}}} $-eigenfunction $v$
of $\mathcal{L}$, and
\begin{equation}\label{e1.16}
\int_{\Omega  \cap ({w > 0})} {{\mathcal{F}_ + }} {w^2}\rho
+ \int_{\Omega  \cap ({w < 0})} {{\mathcal{F}_ - }} {w^2}\rho  > 0,
\end{equation}
for every nontrivial ${\lambda _{j_0}}$-eigenfunction $w$ of $\mathcal{L}$ .
\end{itemize}

\begin{remark} \label{rmk1.1} \rm
 If $\widetilde{\mathcal{M}}$, as defined by \eqref{e1.4},
satisfies {\rm (S1)--(S2)}, then
\begin{equation}\label{e1.17}
\widetilde M(v,{D_t}v) = 0,\quad\forall  v
\in \widetilde C_{p,\rho }^{1b} = \{v \in \widetilde C_{p,\rho }^1:
{D_t}v \in \widetilde C_{p,\rho }^1\}.
\end{equation}
\end{remark}

Now, we state the main result of this article.

\begin{theorem} \label{thm1.1}
Let $\Omega  \subset {R^N}({N \ge 1})$,
$ T = ( - \pi ,\pi )$, $ \widetilde \Omega  = \Omega  \times T$,
$ p = ({p_1}, \dots ,{p_N})$, $\rho $  and
${p_i}(i = 1, \dots ,N)$   be positive functions in ${C^0}(\Omega )$
satisfying \eqref{e1.5}, $q \in {C^0}(\Omega )$
be a nonnegative function satisfying \eqref{e1.6}, and
$\Gamma  \subset \partial \Omega $  be a closed set.
Let $\mathcal{L}$  and $\widetilde{\mathcal{M}} $ be given by \eqref{e1.9}
 and \eqref{e1.4}  satisfying \eqref{e1.8}, {\rm (S1), (S2)} respectively
and $\mathcal{L}$  satisfies the conditions of $ {V_L}({\Omega ,\Gamma })$.
If ${\lambda _{j_0}}$ is an eigenvalue of $\mathcal{L}$  of multiplicity
$j_1$, $\widetilde{\mathcal{M}} $  is $\# \widetilde H $- related to
 $\widetilde{\mathcal{L}} $,  and {\rm (F1)--(F3), (G1)--(G2)}
 hold, then problem \eqref{e1.3} has at least one weak solution; i.e.,
there exits ${u^ *} \in \widetilde H$ such that
\begin{equation}\label{e1.18}
 \langle D_{t}u^{*},v \rangle _{\rho} ^{\sim} + \widetilde{\mathcal{M}}( u^{*},v)
= \lambda _{j_0}\langle u^ {* },v \rangle _{\rho} ^{\sim}
+ \langle f( x,u^{ * } ) + g( x,t,u^ {* } ),v \rangle _{\rho} ^{\sim} - G(v)\,.
\end{equation}
\end{theorem}

The rest of this article is arranged as follows.
In section 2, we will give some preliminary lemmas;
In section 3, we will prove the main results on the quasilinear parabolic
differential  equations.


 \section{Preliminary lemmas}

In this section, we introduce some lemmas, and
concepts which will be used later.
If both \eqref{e1.8} and the conditions of $ {V_L}({\Omega ,\Gamma })$
hold, we have 
\begin{equation}\label{e2.1}
\{\widetilde{\varphi}_{jk}^c\}_{j = 1,k = 0}^{\infty,\infty}
\cup \{  \widetilde{\varphi}_{jk}^{s}\}_{j = 1,k = 1}^{\infty,\infty}
\text{is a CONS for } \widetilde L_\rho ^2,
\end{equation}
where
\begin{equation}\label{e2.2}
\begin{gathered}
\widetilde \varphi _{jk}^c(x,t) =
\begin{cases}
\varphi _j (x)/\sqrt {2\pi } &  k = 0,\; j = 1,2, \dots ,\\
 \varphi _j(x) \cos (kt)/\sqrt{\pi} & j,k = 1,2, \dots ,
\end{cases}
\\
\widetilde \varphi _{jk}^s(x,t)
= \varphi _j(x)\sin(kt) /\sqrt{\pi} \quad  j,k = 1,2, \dots .
\end{gathered}
\end{equation}
Obviously, both $\widetilde{\varphi}_{jk}^c$  and
 $\widetilde{\varphi}_{jk}^{s}$  are in
 $\widetilde H({\widetilde \Omega ,\Gamma })$.
 Define
\begin{equation}\label{e2.3}
\mathcal{L}_1(u,v) = \widetilde{\mathcal{L}}(u,v)
+ \langle u,v \rangle_{\rho}^{\sim},\quad  \forall u,v \in \widetilde{H}.
\end{equation}
It is clear that $\mathcal{L}_1(u,v)$ is an inner product on
$\widetilde{H}$ and from \eqref{e1.6}-\eqref{e1.8}, \eqref{e1.12} and
\eqref{e2.3}, there are
${K_1},{K_2} > 0$  such that
\begin{equation}\label{e2.4}
K_1\| v \|_{\widetilde{ H}}^2 \le \mathcal{L}_1(v,v)
 + \| D_{t}v \|_{\rho}^2 \le K_2\| v \|_{\widetilde{ H}}^2,\quad
\forall  v \in \widetilde{H}.
\end{equation}
For $v \in \widetilde L_\rho ^2$, setting
\begin{equation}\label{e2.5}
\widehat v^{c}(j,k) = \langle v,\widetilde {\varphi}_{jk}^{c}
\rangle _{\rho} ^{\sim},\quad
\widehat v^{s}(j,k) = \langle v,\widetilde {\varphi}_{jk}^{s}
 \rangle _{\rho} ^{\sim},
\end{equation}
and from (VL2), \eqref{e1.12} and \eqref{e2.3}, we see that for
$ v \in \widetilde H$,
\begin{equation}\label{e2.6}
\mathcal{L}_1(v,\widetilde{ \varphi} _{jk}^{s})
 = (\lambda _j + 1)\widehat v^{s}(j,k),   \quad
{\mathcal{L}_1}(v,\widetilde{ \varphi} _{jk}^{c})
 = (\lambda _j + 1)\widehat v^{c}(j,k).
\end{equation}

\begin{lemma} \label{lem2.1}
 If $\{\widetilde{\varphi}_{jk}^c\}_{j = 1,k = 0}^{\infty,\infty}
\cup \{  \widetilde{\varphi}_{jk}^{s}\}_{j = 1,k = 1}^{\infty,\infty}$
 is a {\rm CONS} for  $L_\rho ^2(\widetilde \Omega )$ defined by \eqref{e2.2},
setting
\begin{equation}\label{e2.7}
{\tau _n}(v) = \sum_{j = 1}^n {{{\widehat v}^c}(j,0)} \widetilde \varphi _{j0}^c
 + \sum_{j = 1}^n \sum_{k = 1}^n
[ {{{\widehat v}^c}(j,k)\widetilde \varphi _{jk}^c
+ {{\widehat v}^s}(j,k)\widetilde \varphi _{jk}^s} ] ,
\end{equation}
we have
\begin{equation}\label{e2.8}
\lim_ { n  \to \infty }\| {{\tau _n}(v) - v} \|_{\widetilde {H}} = 0,\quad
\text{for all }  v \in \widetilde{H}.
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2.2}
{\rm (i)} If $v \in \widetilde H$, then
\begin{equation}\label{e2.9}
\begin{aligned}
&\mathcal{L}_1(v,v) + \| {{D_t}v} \|_\rho ^2 \\
&= \sum_{j = 1}^\infty  {{{| {{{\widehat v}^c}(j,0)}|}^2}({{\lambda _j} + 1})}
+ \sum_{j = 1}^\infty  {\sum_{k = 1}^\infty
 {[ {{{| {{{\widehat v}^c}(j,k)} |}^2} + {{| {{{\widehat v}^s}(j,k)} |}^2}} ]} }
 ({{\lambda _j} + 1 + {k^2}}).
\end{aligned}
\end{equation}
{\rm (ii)} If $v \in L_{\rho} ^2(\widetilde {\Omega} )$  and
$\mathcal{L}_1(v,v) + \| D_{t}v\|_{\rho} ^2< \infty $,
then $ v \in \widetilde{H}$.
\end{lemma}

\begin{lemma} \label{lem2.3}
 Let $ \widetilde \Omega ,\rho ,p,q$, and $\mathcal{L}$  be as in the
hypothesis of Theorem \ref{thm1.1} and assume that $( {\Omega ,\Gamma } )$
is a ${V_L}({\Omega ,\Gamma })$. Then $\widetilde H$
is compactly imbedded in $L_\rho ^2(\widetilde \Omega )$.
\end{lemma}

The proofs of Lemmas \ref{lem2.1}--\ref{lem2.3},  can be found in \cite{s1}.
We define
\begin{equation}\label{e2.10}
{S_n} =\Big\{ v \in \widetilde H:
v = \sum_{j = 1}^n {\eta _{j0}^c} \widetilde \varphi _{j0}^c
+ \sum_{j = 1}^n {\sum_{k = 1}^n {\eta _{jk}^c\widetilde \varphi _{jk}^c
 + \eta _{jk}^s\widetilde \varphi _{jk}^s} },\;
\eta _{jk}^c,\eta _{jk}^s \in \mathbb{R} \Big\}.
\end{equation}

\begin{remark} \label{rmk2.1}
 If ${u_n} \in {S_n}$, then $\widetilde{\mathcal{M}}({u_n},{D_t}{u_n}) = 0$.
\end{remark}

\section{Proof of main results}

In this section, we will give the proof of Theorem \ref{thm1.1}.
To do this, we divide the proof into three parts.
In part 1, we construct approximation solutions in a finite dimension
space ${S_n}$. That is,

\begin{lemma} \label{lem3.1}
Assume that all the conditions in the hypothesis of Theorem \ref{thm1.1}
 hold except for {\rm (F3)}. Let $S_{n}$  be the subspace of
$\widetilde H$  defined by \eqref{e2.10}. Taking ${n_0} = {j_0} + {j_1} $,
then for $n \ge {n_0}$, there is a ${u_n} \in {S_n}$ with the property that
\begin{equation}\label{e3.1}
\begin{aligned}
&\langle D_{t}u_{n},v \rangle _{\rho} ^{\sim}
+ \widetilde{\mathcal{M}}( u_{n},v )\\
& = ( \lambda _{j_0} + \gamma n^{ - 1} )\langle u_{n},v\rangle _{\rho }^{\sim}
+ (1 - n^{ - 1})\langle f( {x,u_{n} )
+ g( x,t,u_{n}),v} \rangle _{\rho} ^{\sim} - G( v ),\quad
\forall v\in S_{n}.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
First observe that from \eqref{e2.10},
\begin{equation}  \label{e3.2}
 \begin{gathered}
(1)\quad v\in S_{n}\Rightarrow D_{t}v\in S_{n}, \\
(2)\quad \langle D_{t}(\alpha \widetilde{\varphi}_{jk}^c
+\beta \widetilde{\varphi}_{jk}^s), \alpha \widetilde{\varphi}_{jk}^c
+\beta \widetilde{\varphi}_{jk}^s\rangle_{\rho}^{\sim}=0, \quad
\text{for } j,k\geq1,\alpha,\beta\in \mathbb{R}.
 \end{gathered}
\end{equation}
Let $\{ {{\psi _i}} \}_{i = 1}^{2{n^2} + n}$  be an enumeration of
$ \{\widetilde{\varphi}_{jk}^c\}_{j = 1,k = 0}^{n,n}
\cup \{  \widetilde{\varphi}_{jk}^{s}\}_{j = 1,k = 1}^{n,n}$, and set
\begin{equation}\label{e3.3}
 {n^*} = ({j_0} + {j_1} - 1)(2n + 1).
\end{equation}
So $\{ {{\psi _i}} \}_{i = 1}^{{n^*}}$  is an enumeration of
$\{\widetilde{\varphi}_{jk}^c\}_{j = 1,k = 0}^{{j_0}
+ {j_1} - 1,n} \cup \{  \widetilde{\varphi}_{jk}^{s}\}_{j = 1,
k = 1}^{{j_0} + {j_1} - 1,n}$, where $n \ge {n_0}$.

For $\alpha  = ({{\alpha _1}, \dots ,{\alpha _{2{n^2} + n}}})$, setting
\begin{equation}\label{e3.4}
u = \sum_{i = 1}^{2{n^2} + n} {{\alpha _i}} {\psi _i}, \quad
\widetilde u = \sum_{i = 1}^{2{n^2} + n} {{\delta _i}{\alpha _i}} {\psi _i},
\end{equation}
where
\begin{equation}\label{e3.5}
\delta_i= \begin{cases} -1, & 1\leq i\leq n^{*},\\
1, & n^{*}+1\leq i\leq 2{n^2} + n,
 \end{cases}
\end{equation}
we define
\begin{equation}\label{e3.6}
\begin{aligned}
 F_i(\alpha )
&= \langle D_{t}u,\delta _i\psi _i \rangle _{\rho} ^{\sim}
+ \widetilde{\mathcal{M}}(u,\delta _i\psi _i) - (\lambda _{j_0}
+ \gamma n^{ - 1})\langle u,\delta _i\psi _i \rangle _{\rho} ^{\sim}\\
&\quad - (1 - n^{ - 1})\langle f(x,u)
 + g(x,t,u),\delta _i \psi _i\rangle _{\rho} ^{\sim} + G(\delta _i\psi_i ).
\end{aligned}
\end{equation}
 It is clear from the orthogonality that
$\langle D_{t}u,\widetilde {u} \rangle _{\rho} ^{\sim} = 0$.
From \eqref{e3.4} and \eqref{e3.6}, we obtain
\begin{equation}\label{e3.7}
\begin{aligned}
\sum_{i = 1}^{2{n^2} + n} F_i(\alpha ) \alpha _i
& = \widetilde{\mathcal{M}}( u,\widetilde {u} ) - ( \lambda _{j_0}
 + \gamma n^{ - 1} )\langle u,\widetilde{ u} \rangle _{\rho} ^{\sim}\\
&\quad  - (1 - n^{ - 1})\langle f(x,u) + g(x,t,u),\widetilde {u}
 \rangle _{\rho} ^{\sim} + G(\widetilde {u} ).
\end{aligned}
\end{equation}
From \eqref{e3.3}--\eqref{e3.5}, we have
\begin{equation}\label{e3.8}
\begin{gathered}
\widetilde u = \sum_{j = 1}^n {{\delta _j}{{\widehat u}^c}(j,0)}
 \widetilde \varphi _{j0}^c + \sum_{j = 1}^n {\sum_{k = 1}^n
 {{\delta _j}[{{\widehat u}^c}(j,k)\widetilde \varphi _{jk}^c
 + {{\widehat u}^s}(j,k)\widetilde \varphi _{jk}^s]} } ,\\
u = \sum_{j = 1}^n {{{\widehat u}^c}(j,0)} \widetilde \varphi _{j0}^c
 + \sum_{j = 1}^n {\sum_{k = 1}^n [{{{\widehat u}^c}(j,k)
 \widetilde \varphi _{jk}^c + {{\widehat u}^s}(j,k)\widetilde \varphi _{jk}^s} }] ,\\
{\delta _j} =  \begin{cases}
 - 1, &1\le j \le {j_0} + {j_1} - 1,\\
1,    & j_0 + j_1 \le j \le n.
\end{cases}
\end{gathered}
\end{equation}
Consequently, we obtain
\[
\widetilde L(u,\widetilde u)
= \sum_{j = 1}^n {{\delta _j}{\lambda _j}{{| {{{\widehat u}^c}(j,0)}
|}^2}}
+ \sum_{j = 1}^n {\sum_{k = 1}^n {{\lambda _j}{\delta _j}
[ {{{| {{{\widehat u}^c}(j,k)} |}^2}
+ {{| {{{\widehat u}^s}(j,k)} |}^2}} ]} } .
\]
Adding and subtracting
$ - \gamma  \langle u,\widetilde {u} \rangle _{\rho} ^{\sim}
+ \widetilde{\mathcal{L}}(u,\widetilde{ u})$
to the right-hand side of \eqref{e3.7}, we see that
\begin{equation}\label{e3.9}
\begin{aligned}
\sum_{i = 1}^{2{n^2} + n} {{F_i}(\alpha )} {\alpha _i}
&= \sum_{j = 1}^n \delta _j(\lambda _j - \lambda _{j_0}
 - \gamma){{| {{{\widehat u}^c}(j,0)} |}^2}\\
&\quad + \sum_{j = 1}^n \sum_{k = 1}^n \delta _j(\lambda _j - \lambda _{j_0}
 - \gamma)[ | \widehat u^c (j,k) |^2 + | \widehat u^s (j,k)|^2 ]\\
&\quad   - (1 - n^{ - 1})\langle f(x,u)
 - \gamma u,\widetilde {u} \rangle _{\rho} ^{\sim}
 - (1 - n^{ - 1})\langle g(x,t,u),\widetilde{ u}  \rangle _{\rho} ^{\sim}\\
&\quad  + G(\widetilde{ u}) + \widetilde{\mathcal{M}}(u,\widetilde{ u})
 -\widetilde{\mathcal{L}}(u,\widetilde {u}).
\end{aligned}
\end{equation}
By \eqref{e3.8}, it is obvious that
${\delta _j}({{\lambda _j} - {\lambda _{j_0}} - \gamma }) \ge \gamma $,
for $j = 1, \dots ,n$. From (F2), \eqref{e2.1} and \eqref{e3.9}
there exists is a $K > 0$, such that
\begin{equation}\label{e3.10}
\begin{aligned}
\sum_{i = 1}^{2n^2 + n}F_i(\alpha ) \alpha _i
&\ge \gamma n^{ - 1}\| u \|_{\rho} ^2
- ( 1 - n^{ - 1})\langle g(x,t,u),\widetilde{ u} \rangle _{\rho} ^{\sim} \\
&\quad - K\| u \|_{\rho } + G( \widetilde {u} )
 + \widetilde{\mathcal{M}}(u,\widetilde{ u} )
 - \widetilde{\mathcal{L}}(u,\widetilde{ u} ).
\end{aligned}
\end{equation}
Now from (G2), it follows that
\begin{equation}\label{e3.11}
\lim_ {\| u\|_{\rho} \to \infty }|\langle g(x,t,u),\widetilde{u}
\rangle^{\sim}_{\rho}|\big/\|u\|^2_{\rho}=0.
\end{equation}
For a fixed $n$ , it follows from \eqref{e2.4}, \eqref{e2.9},
and \eqref{e3.8} that there is a $K>0$, such that
\begin{equation}\label{e3.12}
\| u \|_{\widetilde {H}} \le K\| u \|_{\rho },  \quad  u \in S_{n},
\end{equation}
and since $\widetilde{\mathcal{M}}$   is $\# \widetilde H$- related
to $\widetilde{\mathcal{L}}$,
\begin{equation}\label{e3.13}
\lim_ {\| u\|_{\rho} \to \infty }|\widetilde{\mathcal{M}}(u,\widetilde{u})
-\widetilde{\mathcal{L}}(u,\widetilde{u})|\big/\|u\|^2_{\rho}=0,\quad
u \in S_{n}.
\end{equation}
From $\| u \|_{\rho} ^{2 }= | \alpha  |^2$  and
$G \in ( \widetilde{ H})'$, we conclude from \eqref{e3.10}--\eqref{e3.13}
that there is an $s_0>0$  such that
\begin{equation}\label{e3.14}
\sum_{i = 1}^{2{n^2} + n} F_i(\alpha )\alpha _i \ge
\frac{\gamma |\alpha|^2}{2n},\quad\text{for }|\alpha| > {s_0}.
\end{equation}
From the generalized Brouwer's theorem  \cite{k1}, there exists
${\alpha ^ * } = (\alpha _1^ * , \dots ,\alpha _{2{n^2} + n}^ * )$
satisfying ${F_i}({{\alpha ^ * }}) = 0$. Thus, setting
${u_n} = \sum_{i = 1}^{2{n^2} + n} {\alpha _i^ * } {\psi _i}$,
and from \eqref{e3.6}, we have
\begin{align*}
&\langle D_{t}u_{n},\psi _i\rangle _{\rho} ^{\sim}
+ \widetilde{\mathcal{M}}( u_{n},\psi _i ) \\
&= ( \lambda _{j_0} + \gamma n^{ - 1} )\langle u_{n},\psi _i
\rangle _{\rho} ^{\sim} +( 1 - n^{ - 1} )\langle f(x,u_{n})
+ g(x,t,u_{n}),\psi _i \rangle _{\rho} ^{\sim} - G( \psi _i ),
\end{align*}
for $i = 1, \dots ,2{n^2} + n$. The proof of Lemma \ref{lem3.1} is completed by
the definition of ${S_n}$.
\end{proof}

\begin{lemma} \label{lem3.2}
Assume that the conditions in Lemma \ref{lem3.1} hold.
If (F3) holds, then the sequence $\{u_{n} \}$
obtained in Lemma \ref{lem3.1} is uniformly bounded in $\widetilde{H}$
with respect to the norm
$\|u_{n}\|_{\widetilde{H}}=\langle u_{n},u_{n}\rangle_{\widetilde{H}}^{1/2}$.
\end{lemma}

\begin{proof}
 We assume that $\lambda_{j_0+j_1}$ is an eigenvalue of $\mathcal{L}$
of multiplicity $j_2$. By Lemma \ref{lem3.1}, for $u_{n} \in S_{n}$, we have
\begin{equation} \label{e3.15}
\begin{aligned}
&\langle D_{t}u_{n},v \rangle_{\rho}^{\sim}+\widetilde{\mathcal{M}}(u_{n},v)\\
&= (\lambda_{j_0}+\gamma n^{-1})\langle u_{n},v \rangle_{\rho}^{\sim}
+(1-n^{-1})\langle f(x,u_{n})
+g(x,t,u_{n}),v\rangle_{\rho}^{\sim}-G(v),
\end{aligned}
\end{equation}
 for all $v \in S_{n}$, $n\ge n_1=j_0+j_1+j_2$.
We claim that there is a constant $K$ such that
\begin{equation} \label{e3.16}
\|u_{n}\|_{\widetilde{H}} \leq K,\quad \text{for all } n \geq n_1.
\end{equation}
 Suppose that \eqref{e3.16} fails. For ease of notation and without
loss of generality, we assume
\begin{equation} \label{e3.17}
\lim_{n \to \infty}\|u_{n}\|_{\widetilde{H}} =\infty.  \hskip 4cm
\end{equation}
Taking $v=u_{n}$ in \eqref{e3.15}, from \eqref{e3.11}, \eqref{e3.2}(1),
(F2), (G2), $G \in (\widetilde{H})'$ and Schwarz's inequality, there
exists a $K>0$ such that
\begin{equation} \label{e3.18}
\widetilde{\mathcal{M}}(u_{n},u_{n}) \leq K\|u_{n}\|_{\rho}^2+K\|u_{n}\|_{\rho},\quad\text{for } n\geq n_1.
\end{equation}
We observe from \eqref{e3.2}(1) and Remark \ref{rmk2.1} that
\begin{equation} \label{e3.19}
\widetilde{\mathcal{M}}(u_{n},D_{t}u_{n})=0,\quad\text{for } n \geq n_1.
\end{equation}
Thus, replacing $v$ by $D_{t}u_{n}$ in \eqref{e3.15}, from (F2), (G2)
 and $G \in (\widetilde{H})'$,  there is a $K >0$ such that
\begin{equation} \label{e3.20}
\|D_{t}u_{n}\|_{\rho} \leq K\|u_{n}\|_{\rho}+K.
\end{equation}

From \eqref{e1.8} and (S2), there is  a $K >0$ such that
\begin{equation} \label{e3.21}
K \Big(\sum_{i=1}^{N}\int_{\widetilde{\Omega}}p_i|D_iu_{n}|^2
+\int_{\widetilde{\Omega}}qu_{n}^2\Big)
\leq \widetilde{\mathcal{M}}(u_{n},u_{n}).
\end{equation}
Therefore, from \eqref{e1.7}, \eqref{e3.18}, \eqref{e3.20} and \eqref{e3.21},
it is easy to obtain from \eqref{e3.17} that
\begin{equation} \label{e3.22}
\lim_{n\to\infty}\|u_{n}\|_{\rho}=\infty,
\end{equation}
and there exist $n_2>0$ and $K>0$ such that
\begin{equation} \label{e3.23}
\|u_{n}\|_{\widetilde{H}} \leq K\|u_{n}\|_{\rho},\quad  \text{for } n \geq n_2.
\end{equation}
Set
\begin{equation} \label{e3.24}
u_{n}=u_{n1}+u_{n2}+u_{n3}+u_{n4},
\end{equation}
 where
\begin{gather*}
u_{n1}=\sum_{j=1}^{j_0-1}\widehat{u}_{n}^{c}(j,0)\widetilde{\varphi}_{j0}^{c}
 + \sum _{j=1}^{j_0-1}
\sum _{k=1}^{n}[\widehat{u}_{n}^{c}(j,k)\widetilde{\varphi}_{jk}^{c}
 + \widehat{u}_{n}^{s}(j,k)\widetilde{\varphi}_{jk}^{s}],
\\
u_{n2}=\sum_{j=j_0}^{j_0+j_1-1}\widehat{u}_{n}^{c}(j,0)
 \widetilde{\varphi}_{j0}^{c} + \sum _{j=j_0}^{j_0+j_1-1}
\sum _{k=1}^{n}[\widehat{u}_{n}^{c}(j,k)\widetilde{\varphi}_{jk}^{c}
 + \widehat{u}_{n}^{s}(j,k)\widetilde{\varphi}_{jk}^{s}],
\\
u_{n3}=\sum_{j=j_0+j_1}^{j_0+j_1+j_2-1}\widehat{u}_{n}^{c}(j,0)
 \widetilde{\varphi}_{j0}^{c} + \sum_{j=j_0+j_1}^{j_0+j_1+j_2-1}
\sum _{k=1}^{n}[\widehat{u}_{n}^{c}(j,k)\widetilde{\varphi}_{jk}^{c}
 + \widehat{u}_{n}^{s}(j,k)\widetilde{\varphi}_{jk}^{s}],
\\
u_{n4}=\sum_{j=j_0+j_1+j_2}^{n}\widehat{u}_{n}^{c}(j,0)
 \widetilde{\varphi}_{j0}^{c} + \sum _{j=j_0+j_1+j_2}^{n}
\sum _{k=1}^{n}[\widehat{u}_{n}^{c}(j,k)\widetilde{\varphi}_{jk}^{c}
+ \widehat{u}_{n}^{s}(j,k)\widetilde{\varphi}_{jk}^{s}].
\end{gather*}


\noindent\textbf{Step 1:} We claim that
\begin{equation} \label{e3.25}
\begin{gathered}
(1)\quad  \lim_{n\to\infty}(\|u_{n1}\|_{\rho}^2 
+ \|u_{n4}\|_{\rho}^2)/\|u_{n}\|_{\rho}^2=0,\\
(2)\quad
\lim_{n\to\infty}(\|u_{n4}\|_{\widetilde{H}}) /\|u_{n}\|_{\rho}=0.
\end{gathered}
\end{equation}
Defining
\begin{equation} \label{e3.26}
\widetilde{u}_{n}=-u_{n1}-u_{n2}+u_{n3}+u_{n4},
\end{equation}
and from \eqref{e3.2}(2), we have
\begin{equation} \label{e3.27}
\langle D_{t}u_{n},\widetilde{u}_{n} \rangle _{\rho}^{\sim}=0.
\end{equation}
As a result, from \eqref{e3.15} with $v=\widetilde{u}_{n}$, we obtain
\begin{equation} \label{e3.28}
\begin{aligned}
&\widetilde{\mathcal{L}}(u_{n},\widetilde{u}_{n})-(\lambda_{j_0}
 + \gamma)\langle u_{n},\widetilde{u}_{n} \rangle _{\rho}^{\sim}\\
&=(1-n^{-1})\langle f(x,u_{n})-\gamma u_{n},\widetilde{u}_{n}
\rangle _{\rho}^{\sim}
+\langle g(x,t,u_{n}),\widetilde{u}_{n} \rangle _{\rho}^{\sim}
 - G(\widetilde{u}_{n})\\
&\quad + \widetilde{\mathcal{L}}(u_{n},\widetilde{u}_{n})
 -\widetilde{\mathcal{M}}(u_{n},\widetilde{u}_{n}).
\end{aligned}
\end{equation}
Set
\begin{gather*}
I=\widetilde{\mathcal{L}}(u_{n},\widetilde{u}_{n})
 - (\lambda_{j_0} + \gamma)\langle u_{n},\widetilde{u}_{n}
 \rangle _{\rho}^{\sim}, \\
\begin{aligned}
II&=(1-n^{-1})\langle f(x,u_{n})-\gamma u_{n},\widetilde{u}_{n}
 \rangle _{\rho}^{\sim}
+\langle g(x,t,u_{n}),\widetilde{u}_{n} \rangle _{\rho}^{\sim}\\
&\quad - G(\widetilde{u}_{n}) + \widetilde{\mathcal{L}}(u_{n},\widetilde{u}_{n})
 -\widetilde{\mathcal{M}}(u_{n},\widetilde{u}_{n}).
\end{aligned}
\end{gather*}
Now from \eqref{e3.8} and $\lambda_{j_0} + \gamma=\lambda_{j_0+j_1} - \gamma$,
we see that
\begin{equation} \label{e3.29}
I  \ge \gamma \| u_{n} \|_{\rho} ^2 +I^{*},
\end{equation}
where
\begin{align*}
I^{*} &= \sum_{j = 1}^{j_0 - 1} (\lambda _{j_0}
-\lambda _j) | \widehat{u}_{n}^{c}(j,0)|^2
+ \sum_{j = 1}^{j_0 - 1} (\lambda _{j_0}
-\lambda _j) \sum_{k = 1}^{n}[ |\widehat{u}_{n}^{c}(j,k)| ^2
 + |\widehat{u}_{n}^{s}(j,k)| ^2]
\\
&\quad +\sum_{j =j_0 +j_1 +j_2}^{n}(\lambda _j
 - \lambda _{j_0 + j_1})|\widehat{u}_{n}^{c}(j,0)|^2
+ \sum_{j =j_0 +j_1 +j_2}^{n}(\lambda _j
 - \lambda _{j_0 + j_1})\sum_{k = 1}^{n}[ |\widehat{u}_{n}^{c}(j,k)| ^2\\
&\quad  + |\widehat{u}_{n}^{s}(j,k)| ^2].
\end{align*}
Hence, from \eqref{e3.24}, we see that
\begin{equation} \label{e3.30}
I \ge \gamma\|u_{n}\|_{\rho}^2 + (\lambda _{j_0}
- \lambda _{j_0-1})\|u_{n1}\|_{\rho}^2 + (\lambda _{j_0 + j_1+ j_2}
- \lambda _{j_0 + j_1})\|u_{n4}\|_{\rho}^2.
\end{equation}
On the other hand, for any $\varepsilon  > 0$, from  (F2), (G2)
and $G \in (\widetilde{H})'$  we see that
\begin{equation} \label{e3.31}
\begin{aligned}
II &\le [(\gamma + \varepsilon )\|u_{n}\|_{\rho}
 + \|f_0\|_{\rho}+ \|g_{\varepsilon}\|_{\rho}]\|\widetilde{u}_{n}\|_{\rho}\\
&\quad  + K_0\|\widetilde{u}_{n}\|_{\widetilde{H}}
 + |\widetilde{\mathcal{L}}(u_{n},\widetilde{u}_{n})
 -\widetilde{\mathcal{M}}(u_{n},\widetilde{u}_{n})|,
\end{aligned}
\end{equation}
and $\widetilde{\mathcal{M}}$ being  $\# \widetilde{H}$-related
to $\widetilde{\mathcal{L}}$, we have
\begin{equation} \label{e3.32}
\lim_{n\to\infty}|\widetilde{\mathcal{L}}(u_{n},\widetilde{u}_{n})
-\widetilde{\mathcal{M}}(u_{n},\widetilde{u}_{n})|/\|u_{n}\|_{\rho}^2=0.
\end{equation}
Therefore, from \eqref{e3.28} and \eqref{e3.32}, we see on dividing
both \eqref{e3.30} and \eqref{e3.31} by $\|u_{n}\|_{\rho}^2$ and passing
to the limit as $n\to\infty$, that
$$
\lim_{n\to\infty} [(\lambda _{j_0} - \lambda _{j_0-1})\|u_{n1}\|_{\rho}^2
+ (\lambda _{j_0 + j_1+ j_2}
 - \lambda _{j_0 + j_1})\|u_{n4}\|_{\rho}^2]/\|u_{n}\|_{\rho}^2=0.
$$
Since $\lambda _{j_0} - \lambda _{j_0-1}>0$ and
$\lambda _{j_0 + j_1+ j_2} - \lambda _{j_0 + j_1}>0$, we see that
claim (1) in \eqref{e3.25} is true.

Set $\delta= 1 - \lambda _{j_0 + j_1}/ \lambda _{j_0 + j_1+ j_2}$.
It implies
\begin{equation} \label{e3.33}
\lambda _j - \lambda _{j_0+j_1} \ge \delta \lambda _j,\quad\text{for }
j \ge j_0 + j_1+ j_2.
\end{equation}
Then,  \eqref{e3.24} and \eqref{e3.33} give
\begin{equation} \label{e3.34}
I^{*} \ge \delta \widetilde{\mathcal{L}}(u_{n4},u_{n4}).
\end{equation}
Hence, from \eqref{e3.28}-\eqref{e3.30}, \eqref{e3.34}, we obtain
\begin{align*}
\gamma \|u_{n}\|_{\rho}^2 + \delta \widetilde{\mathcal{L}}(u_{n4},u_{n4})
&\le  |\widetilde{\mathcal{L}}(u_{n},\widetilde{u}_{n})
 -\widetilde{\mathcal{M}}(u_{n},\widetilde{u}_{n})| 
 + [(\gamma + \varepsilon )\|u_{n}\|_{\rho} + \|f_0\|_{\rho}\\
&\quad + \|g_{\varepsilon}\|_{\rho}]\|\widetilde{u}_{n}\|_{\rho}
 + K_0\|\widetilde{u}_{n}\|_{\widetilde{H}} ,
\end{align*}
and on dividing by $\|u_{n}\|_{\rho}^2$ on both sides and letting
$ n\to\infty$, we see that
\begin{equation} \label{e3.35}
\lim_{n\to\infty} \widetilde{\mathcal{L}}(u_{n4},u_{n4}) /\|u_{n}\|_{\rho}^2 =0.
\end{equation}
Since
\begin{equation} \label{e3.36}
\langle f(x,v),D_{t}v \rangle _{\rho}^{\sim}
= \int_{\widetilde{\Omega}}f(x,v(x,t))D_{t}v(x,t)\rho \,dt\,dx=0,
\end{equation}
for $v \in \widetilde{C}^{1b}_{p,\rho}$, we conclude from \eqref{e3.36},
(F2) and the definition of $S_{n}$ that
$$
\langle f(x,u_{n}), D_{t}u_{n} \rangle _{\rho}^{\sim}=0,\quad
\text{for } u_{n} \in S_{n},\; n \ge n_2.
$$
Hence, replacing $v$ by $D_{t}u_{n}$ in \eqref{e3.15} and from
Remark \ref{rmk1.1}, Schwarz's inequality and $G \in (\widetilde{H})'$, we obtain
$\|D_{t}u_{n}\|_{\rho} \le \|g(x,t,u_{n})\|_{\rho} + K$ and
\begin{equation} \label{e3.37}
\lim_{n\to\infty} \|D_{t}u_{n}\|_{\rho}^2 /\|u_{n}\|_{\rho}^2 =0.
\end{equation}
From claim \eqref{e3.25}(1), \eqref{e2.4}, \eqref{e3.35}, \eqref{e3.37} and
$\|D_{t}u_{n4}\|_{\rho}^2 \le \|D_{t}u_{n}\|_{\rho}^2$,
claim \eqref{e3.25}(2) is true.
\medskip

\noindent\textbf{Step 2:}  We show that $W(x) = W_{(2)}(x) + W_{(3)}(x)$.
Setting
\begin{equation} \label{e3.38}
W_{n}(x)= u_{n}/\|u_{n}\|_{\rho},\quad
W_{ni}(x)= u_{ni}/\|u_{n}\|_{\rho},\quad \text{for } i=1, \dots ,4,
\end{equation}
 from \eqref{e3.23}  there is a $K$ such that
\begin{equation} \label{e3.39}
 \|W_{n}\|_{\widetilde{H}} \le K \quad\text{and} \quad
\|W_{ni}\|_{\widetilde{H}} \le K,
\end{equation}
for $i=1, \dots ,4$, and $n \ge n_2$. From \eqref{e3.39} and
Lemma \ref{lem3.1}, we obtain that there is a $W \in \widetilde{H}$ such that
\begin{equation}  \label{e3.40}
\begin{gathered}
(1)\quad \lim_{n\to\infty} \|W_{n}-W\|_{\rho}=0,\\
(2)\quad \lim_{n\to\infty} W_{n}(x,t)=W(x,t),\quad \text{a.e. in } \overline{\Omega},\\
(3)\quad  \lim_{n\to\infty} \langle W_{n},v \rangle_{\widetilde{H}}
=\langle W,v \rangle_{\widetilde{H}},\quad\text{for } v \in \widetilde{H}.
\end{gathered}
\end{equation}
Since $\widetilde{\mathcal{M}}$ is $\# \widetilde{H}$-related
to $\widetilde{\mathcal{L}}$, we obtain from \eqref{e3.39} that
\begin{equation} \label{e3.41}
\lim_{n\to\infty}|\widetilde{\mathcal{L}}(u_{n},W_{ni}(x))
-\widetilde{\mathcal{M}}(u_{n},W_{ni}(x))|/\|u_{n}\|_{\rho}=0,
\text{for } i=1, \dots ,4.
\end{equation}
We observe from \eqref{e3.25} that $\lim_{n\to\infty}\|W_{n4}(x)\|_{\rho}=0$.
 Hence, if $n\to\infty$, then
$$
\langle W_{n},\widetilde{\varphi}_{jk}^{c} \rangle_{\rho}^{\sim}
 = \langle W_{n4},\varphi_{jk}^{c} \rangle_{\rho}^{\sim}\to 0,\quad
\text{for } j\ge j_0 + j_1+ j_2,
$$
and from \eqref{e3.40}(3), we obtain $\widehat{W}^{c}(j,k)=0$, for
$j\ge j_0 + j_1+ j_2$ and  all $  k$. In similar way, we have
$\widehat{W}^{s}(j,k)=0$, for $j\ge j_0 + j_1+ j_2$ and  all
 $ k$. Also, we observe from \eqref{e3.25} that
$\lim_{n\to\infty}\|W_{n1}(x)\|_{\rho}=0$. So we obtain
$\widehat{W}^{c}(j,k)=0$ and $\widehat{W}^{s}(j,k)=0$ for
$1 \le j \le j_0-1$ and all $ k$. Therefore, we have
\begin{gather} \label{e3.42}
\widehat{W}^{c}(j,k)=0 \quad \text{and} \quad
\widehat{W}^{s}(j,k)=0,\quad  \text{for $j\ge j_0 + j_1+ j_2$ and  all } k,
\\
\widehat{W}^{c}(j,k)=0 \quad \text{and} \quad
\widehat{W}^{s}(j,k)=0,\quad \text{for $1\le j\le j_0 -1$ and  all } k.
\label{e3.43}
\end{gather}
Next, for $k \ge 1$ and $j_0\le j\le j_0+ j_1+ j_2-1$,
 from \eqref{e2.2} and \eqref{e3.37} , we have
$$
k \widehat{W}^{c}(j,k)
=-\lim_{n\to\infty} \int_{\widetilde{\Omega}}D_{t}W_{n}(x,t)
\widetilde{\varphi}_{jk}^{s}(x,t)\rho(x)\,dx\,dt = 0.
$$
A similar situation prevails for $k \widehat{W}^{s}(j,k)$.
So we have
$$
\widehat{W}^{c}(j,k) = 0 \quad\text{and}\quad
 \widehat{W}^{s}(j,k)=0,
$$
for $k \ge 1$  and $j_0 \le j \le j_0 + j_1+ j_2-1$.
Hence, from \eqref{e3.42}, \eqref{e3.43} and the above formula,
 we see that $W(x,t)$ is a function unrelated to $t$; i.e.,
\begin{equation}  \label{e3.44}
\begin{gathered}
W(x,t)\equiv W(x),\\
W(x)= W_{(2)}(x) + W_{(3)}(x),\\
W_{(2)}(x)=\sum_{j=j_0}^{j_0 + j_1-1}\widehat{W}^{c}(j,0)
 \widetilde{\varphi}_{j0}^{c},\\
W_{(3)}(x)=\sum_{j=j_0+ j_1}^{j_0 + j_1+ j_2-1}
 \widehat{W}^{c}(j,0) \widetilde{\varphi}_{j0}^{c}.
\end{gathered}
\end{equation}

\noindent\textbf{Step 3:}
 We show that $\langle f^{*}, W_{(2)} \rangle _{\rho}^{\sim}= 0$ and
$\langle f^{*}, W_{(3)} \rangle _{\rho}^{\sim}
= (\lambda _{j_0 + j_1}- \lambda _{j_0})\| W_{(3)} \|_{\rho}^2$.
From \eqref{e3.38} and orthogonality we observe that
$$
\| W_{n}-W \|_{\rho}^2= \| W_{n1} \|_{\rho}^2 + \| W_{n2}
-  W_{(2)}\|_{\rho}^2 + \| W_{n3}-  W_{(3)}\|_{\rho}^2
+ \| W_{n4} \|_{\rho}^2.
$$
From \eqref{e3.40}(1), we conclude that
\begin{equation} \label{e3.45}
\lim_{n\to\infty}\| W_{n2}-  W_{(2)}\|_{\rho}^2=0 \quad \text{and}\quad
\lim_{n\to\infty} \| W_{n3}-  W_{(3)}\|_{\rho}^2=0.
\end{equation}
Putting $W_{ni}$ in place of $v$ in \eqref{e3.15}, we obtain
\begin{equation} \label{e3.46}
\begin{aligned}
&\widetilde{\mathcal{L}}(u_{n},W_{ni}) \\
&= (\lambda_{j_0}+\gamma n^{-1})\langle u_{n},W_{ni}\rangle_{\rho}^{\sim}
 + (1-n^{-1})\langle f(x,u_{n})+g(x,t,u_{n}),W_{ni} \rangle_{\rho}^{\sim}\\
&\quad -G(W_{ni})+\widetilde{\mathcal{L}}(u_{n},W_{ni})
 - \widetilde{\mathcal{M}}(u_{n},W_{ni}), \quad i=1,2.
\end{aligned}
\end{equation}
Dividing by $\|u_{n}\|_{\rho}^2$ on both sides of \eqref{e3.46}
and letting $n\to\infty$,  from \eqref{e3.24}, \eqref{e3.38},
\eqref{e3.40}, \eqref{e3.41}, \eqref{e3.45}, Schwarz's inequality,
$G \in (\widetilde{H})'$ and (G2)  we obtain
\begin{equation} \label{e3.47}
\lim_{n\to\infty} \langle f(x,u_{n}),W_{n3}
\rangle_{\rho}^{\sim}/\|u_{n}\|_{\rho}
= (\lambda _{j_0 + j_1}- \lambda _{j_0})\| W_{(3)} \|_{\rho}^2.
\end{equation}
In a similar way, from \eqref{e3.45}, we have
\begin{equation} \label{e3.48}
\lim_{n\to\infty} \langle f(x,u_{n}),W_{n2}
\rangle_{\rho}^{\sim}/\|u_{n}\|_{\rho}=0.
\end{equation}
Next, from (F2) and \eqref{e3.22} that there are $K$ and $n_3$ such that
\begin{equation} \label{e3.49}
 \| f(x,u_{n})\|_{\rho}/ \|u_{n}\|_{\rho} \le K, \quad\text{for  } n \ge n_3,
\end{equation}
where $n_3 \ge n_2$. Using the Banach-Saks theorem and other facts about
Hilbert spaces (see \cite[p. 181]{b1}), we obtain that there exits
 $ f^{*}(x,t) \in \widetilde{L}_{\rho}^2$ such that
\begin{equation} \label{e3.50}
\begin{gathered}
(1)\quad \lim_{n\to\infty} \langle \frac{f(x,u_{n})}{\|u_{n}\|_{\rho}},
 v \rangle_{\rho}^{\sim} = \langle f^{*} , v \rangle_{\rho}^{\sim},\quad
\forall  v \in \widetilde{L}_{\rho}^2;
\\
(2)\quad \lim_{n\to\infty} \| \frac{1}{n} \sum_{k=n_2}^{n_2+n}
 \frac{f(x,u_{k})}{\|u_{k}\|_{\rho}} -f^{*} \|_{\rho}=0;
\\
(3)\quad \text{there is $\{n_j\} \subset \{ n \}$, such that}\\
\lim_{j\to\infty} \frac{1}{n_j}\sum_{k=n_2}^{n_2+n_j}
\frac{f(x,u_{k})}{\|u_{k}\|_{\rho}}=f^{*}(x,t), \quad\text{a.e. in }
 \widetilde{\Omega}.
\end{gathered}
\end{equation}
From \eqref{e3.45}, \eqref{e3.47}, \eqref{e3.49} and \eqref{e3.50}(2) we obtain
\begin{equation} \label{e3.51}
\langle f^{*}, W_{(3)} \rangle _{\rho}^{\sim}
= (\lambda _{j_0 + j_1}- \lambda _{j_0})\| W_{(3)} \|_{\rho}^2.
\end{equation}
In a similar manner, from \eqref{e3.48} we obtain
\begin{equation} \label{e3.52}
\langle f^{*}, W_{(2)} \rangle _{\rho}^{\sim}= 0. \hskip 5cm
\end{equation}

\noindent\textbf{Step 4:} We show that
$\langle f^{*}, W_{(2)} \rangle _{\rho}^{\sim} >0$ and
 $\langle f^{*}, W_{(3)} \rangle _{\rho}^{\sim}
< (\lambda _{j_0 + j_1}- \lambda _{j_0})\| W_{(3)} \|_{\rho}^2$
under  assumption \eqref{e3.17}.

From \eqref{e3.38}, (F2), \eqref{e3.40}(2) and \eqref{e3.50}(3) it follows that
\begin{equation} \label{e3.53}
f^{*}(x,t)=0,\quad\text{a.e. in } \widetilde{\Omega}_0= \Omega_0 \times T,
\end{equation}
where $\Omega_0= \{ x \in \Omega :  W(x)=0 \}$. From (F2), for
$s \neq 0,\ \ ${\rm a.e.} $ x\in \Omega$, we have
$$
-\frac{f_0(x)}{|s|} \le \frac{f(x,s)}{s} \le 2\gamma + \frac{f_0(x)}{|s|},
$$
and for a.e. $ x \in \Omega$, from \eqref{e1.14} we see that
\begin{equation} \label{e3.54}
0 \le \mathcal{F}_{+}(x) \le \mathcal{F}^{+}(x)\le 2\gamma \quad\text{and}\quad
0 \le \mathcal{F}_{-}(x) \le \mathcal{F}^{-}(x)\le 2\gamma.
\end{equation}
Setting
\begin{equation} \label{e3.55}
\widetilde{\Omega}_{+}=\Omega _{+} \times T \quad\text{and}\quad
\widetilde{\Omega}_{-}=\Omega _{-} \times T,
\end{equation}
where $\Omega_{+}= \{ x \in \Omega :  W(x)>0 \}$ and
$\Omega_{-}= \{ x \in \Omega :  W(x)<0 \}$,
let $(x_0,t_0) \in \widetilde{\Omega}_{+}$ be such that $f^{*}(x_0,t_0)$
is finite, \eqref{e3.40}(2) and \eqref{e3.50}(3) hold, and $x_0$
be a value such that \eqref{e3.54} holds. Then given $\varepsilon > 0$,
we see that  there is an $s^{*}>0$ such that
$ f(x_0,s) \le \mathcal{F}^{+}(x_0)s + \varepsilon s$ for
$s \ge s^{*}$. Since $u_{n}(x_0,t_0)=\|u_{n}\|_{\rho}W_{n}(x_0,t_0)$,
from \eqref{e3.22}, \eqref{e3.40}(2) and \eqref{e3.50}(3), we obtain
 $f^{*}(x_0,t_0) \le \mathcal{F}^{+}(x_0)W(x_0)$.
Similarly, we have $\mathcal{F}_{+}(x_0)W(x_0) \le f^{*}(x_0,t_0)$.
Also, we can prevail for $\widetilde{\Omega}_{-}$.
 Hence, we conclude that
\begin{equation} \label{e3.56}
\begin{gathered}
 (1)\quad \mathcal{F}_{+}(x)W(x) \le f^{*}(x,t) \le \mathcal{F}^{+}(x)W(x),
\quad\text{a.e. in } \widetilde{\Omega}_{+};\\
 (2)\quad \mathcal{F}^{-}(x)W(x) \le f^{*}(x,t) \le \mathcal{F}_{-}(x)W(x),
\quad\text{a.e. in }  \widetilde{\Omega}_{-}.
\end{gathered}
\end{equation}
Since $\widetilde{\Omega}=\widetilde{\Omega}_0 \cup \widetilde{\Omega}_{+}
\cup \widetilde{\Omega}_{-}$, we define
\[
f^{**}(x,t)=
\begin{cases}
0,&  (x,t) \in \widetilde{\Omega}_0,\\
f^{*}(x,t)/W(x), & (x,t) \in \widetilde{\Omega}_{+} \cap \widetilde{\Omega}_{-}.
\end{cases}
\]
From \eqref{e3.53}--\eqref{e3.56}, we have
\begin{gather} \label{e3.57}
f^{*}(x,t)=f^{**}(x,t)W(x),\quad\text{a.e. in }  \widetilde{\Omega}, \\
 \label{e3.58}
0 \le f^{**}(x,t) \le 2\gamma,\quad\text{a.e. in }  \widetilde{\Omega}.
\end{gather}
Furthermore,  from \eqref{e3.44}, \eqref{e3.51}, \eqref{e3.52},
\eqref{e3.57} and \eqref{e3.58} we see that
$$
\langle (\lambda _{j_0 + j_1}-\lambda _{j_0}-f^{**})W_{(3)},
W_{(3)} \rangle _{\rho}^{\sim} + \langle f^{**}W_{(2)},
W_{(2)} \rangle _{\rho}^{\sim}= 0,
$$
and
\begin{equation} \label{e3.59}
\begin{gathered}
 \langle (\lambda _{j_0 + j_1}-\lambda _{j_0}-f^{**})W_{(3)},
 W_{(3)} \rangle _{\rho}^{\sim}=0,\\
 \langle f^{**}W_{(2)}, W_{(2)} \rangle _{\rho}^{\sim}= 0.
\end{gathered}
\end{equation}
Setting
\begin{equation} \label{e3.60}
\widetilde{\Omega}_2=\Omega _2 \times T \quad\text{and}\quad
\widetilde{\Omega}_3=\Omega _3 \times T,
\end{equation}
where $\Omega_i= \{ x \in \Omega :  W_{(i)}(x)\ne 0 \}$, $i=2,3$,
from \eqref{e3.58} and \eqref{e3.59}, we see that
$f^{**}(x,t)=\lambda _{j_0 + j_1}-\lambda _{j_0}$, a.e. in $\widetilde{\Omega}_3$.
Then, $\widetilde{\Omega}_2 \cap \widetilde{\Omega}_3$ is a set of
 Lebesgue measure zero. Also, both $W_{(2)}$ and $W_{(3)}$ are continuous
functions in $\Omega$ by \eqref{e3.44}. Therefore, both $\Omega_2$ and
$\Omega_3$ are open sets, and we see that $\Omega_2$ and $\Omega_3$
are disjoint sets. Since $W=W_{(2)}+W_{(3)}$, we find
\begin{equation} \label{e3.61}
W=W_{(2)}\text{ on } \Omega_2,\quad
W=W_{(3)}\text{ on } \Omega_3.
\end{equation}
Defining
\begin{equation} \label{e3.62}
\widetilde{\Omega}_{i+}=\Omega _{i+} \times T \quad \text{and}\quad
\widetilde{\Omega}_{i-}=\Omega _{i-} \times T,
\end{equation}
where $\Omega_{i+}= \{ x \in \Omega :  W_{(i)}(x)> 0 \}$,
$\Omega_{i-}= \{ x \in \Omega :  W_{(i)}(x)< 0 \}$, $i=2,3$,
from \eqref{e3.44} we see that $W_{(2)}(x)$ is a $\lambda _{j_0}$-eigenfuction
for $\mathcal{L}$. If $W_{(2)}(x)$ is nontrivial, then from \eqref{e1.16}
we  have
\begin{equation} \label{e3.63}
0 < \int_{\widetilde{\Omega}_{2+}}\mathcal{F}_{+}(x)W(x)W_{(2)}(x)\rho(x)\,dx\,dt
 + \int_{\widetilde{\Omega}_{2-}}\mathcal{F}_{-}(x)W(x)W_{(2)}(x)\rho(x)\,dx\,dt.
\end{equation}
If $(x,t)\in \widetilde{\Omega}_{2+} $, from \eqref{e3.61} and \eqref{e3.62},
 then $(x,t)\in \widetilde{\Omega}_{+} $.
So, from \eqref{e3.56} and \eqref{e3.63} we obtain
$$
0 < \int_{\widetilde{\Omega}_{2+}}f^{*}(x,t)W_{(2)}(x)\rho(x)\,dx\,dt
+ \int_{\widetilde{\Omega}_{2-}}f^{*}(x,t)W_{(2)}(x)\rho(x)\,dx\,dt;
$$
i.e., $\langle f^{*},W_{(2)}(x) \rangle _{\rho}^{\sim}>0$.
It is a contradiction of \eqref{e3.52}. As a result, we conclude that
 $ W_{(2)}(x)$ is indeed trivial; i.e., $ W_{(2)}(x)=0$, for all
$ x \in \Omega$. Hence, from \eqref{e3.44} we obtain
\begin{equation}\label{e3.64}
W(x)=W_{(3)}(x),\quad \text{for all } x \in \Omega.
\end{equation}
Since $W_{(3)}(x)$ is a $\lambda _{j_0+j_1}$-eigenfuction for
$\mathcal{L}$, so is $W$  from \eqref{e3.64}. If $W$ is a nontrivial function,
then from \eqref{e1.15} and \eqref{e3.55} we obtain
\begin{equation} \label{e3.65}
2 \gamma\|W\|_{\rho}^2 >
 \int_{\widetilde{\Omega}_{+}}\mathcal{F}^{+}(x)W^2(x)\rho(x)\,dx\,dt
+ \int_{\widetilde{\Omega}_{-}}\mathcal{F}^{-}(x)W^2(x)\rho(x)\,dx\,dt.
\end{equation}
Therefore, we obtain from \eqref{e3.56}(1)(2) and \eqref{e3.65} that
\begin{equation} \label{e3.66}
2 \gamma\|W\|_{\rho}^2
>  \int_{\widetilde{\Omega}_{+}}f^{*}(x,t)W(x)\rho(x)\,dx\,dt
+ \int_{\widetilde{\Omega}_{-}}f^{*}(x,t)W(x)\rho(x)\,dx\,dt;
\end{equation}
i.e., $\langle f^{*},W \rangle _{\rho}^{\sim}
<(\lambda _{j_0 + j_1}-\lambda _{j_0})\|W\|_{\rho}^2$.
It is a direct contradiction of \eqref{e3.51}.
 So we conclude that $W(x)=0$,  for all $x \in \Omega$.
Next, from \eqref{e3.40}(1), we obtain $\lim_{n\to\infty}\|W_{n}\|_{\rho}=0$.
However, from \eqref{e3.38} we see that $\lim_{n\to\infty}\|W_{n}\|_{\rho}=1$.
Obviously, it is a contradiction, and \eqref{e3.16} is indeed true.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
Since $\widetilde{H}$ is a separable Hilbert space, from \eqref{e3.16},
 (S1) and Lemma \ref{lem3.2}, there is a subsequence
(still denoted by $\{u_{n}\}_{n=n_3}^{\infty}$ and a function
 $u^{*} \in \widetilde{H}$) such that
\[
 \lim_{n\to\infty}\|u_{n}-u^{\ast}\|_{\rho}=0;
\]
there exists $W^{*}(x,t) \in \widetilde{L}_{\rho}^2$, such that
$|u_{n}(x,t)|\leq W^{*}(x,t)$, a.e. $(x,t)\in \widetilde{\Omega}$,
 $ n\ge n_3$;
\begin{equation} \label{e3.67}
\begin{gathered}
(1)\quad \lim_{n\to\infty}u_{n}(x,t)=u^{\ast}(x,t),\quad\text{a.e. }
(x,t)\in \widetilde{\Omega};
\\
(2)\quad  \lim_{n\to\infty}\langle D_iu_{n},v\rangle_{p_i}^{\sim}
=\langle D_iu^{\ast},v\rangle_{p_i}^{\sim},\quad \forall
  v\in \widetilde{L}_{p_i}^2,\; i=1,\dots,N;
\\
(3)\quad  \lim_{n\to\infty}\langle D_{t}u_{n},v\rangle_{\rho}^{\sim}
 =\langle D_{t}u^{\ast},v\rangle_{\rho}^{\sim},\quad \forall
  v\in \widetilde{L}_{\rho}^2;\\
(4)\quad \lim_{n\to\infty}\sigma_i(u_{n})=\sigma_i(u^{\ast}),\quad i=0,1,\dots,N.
\end{gathered}
\end{equation}

Let $v \in \widetilde{H}$ and $\tau_{J}(v)$ be defined by \eqref{e2.7}.
 Then $\tau_{J}(v)\in S_{J}$($J \ge n_3$) and from \eqref{e3.15},
for $n \ge J$, we have that
\begin{equation} \label{e3.68}
\begin{aligned}
&\langle D_{t}u_{n},\tau_{J}(v)
 \rangle_{\rho}^{\sim}+\widetilde{\mathcal{M}}(u_{n},\tau_{J}(v))\\
&= (\lambda_{j_0}+\gamma n^{-1})\langle u_{n},\tau_{J}(v)
 \rangle_{\rho}^{\sim}
+(1-n^{-1})\langle f(x,u_{n})+g(x,t,u_{n}),\tau_{J}(v)
\rangle_{\rho}^{\sim}\\
&\quad -G(\tau_{J}(v)).
\end{aligned}
\end{equation}
We conclude from \eqref{e1.4} and \eqref{e3.67} that
\begin{equation}\label{e3.69}
\lim_{n\to\infty}\widetilde{\mathcal{M}}(u_{n},\tau_{J}(v))
=\widetilde{\mathcal{M}}(u^{*},\tau_{J}(v)).
\end{equation}
Next, from (F2), (G2), \eqref{e3.67}(2) and the Lebesgue dominated
convergence theorem we obtain
\begin{equation}\label{e3.70}
\lim_{n\to\infty}\langle f(x,u_{n})+g(x,t,u_{n}),
\tau_{J}(v)\rangle_{\rho}^{\sim}= \langle f(x,u^{*})
+g(x,t,u^{*}),\tau_{J}(v)\rangle_{\rho}^{\sim}.
\end{equation}
From \eqref{e3.69}, \eqref{e3.70}, \eqref{e3.67}(1)(3), and \eqref{e3.68},
 we obtain
\begin{equation} \label{e3.71}
\begin{aligned}
&\langle D_{t}u^ {* },\tau _{J}( v ) \rangle _{\rho} ^{\sim}
+ \widetilde{\mathcal{M}}( u^{ * },\tau _{J}( v )) \\
&= \lambda _{j_0}\langle u^{ * },\tau _{J}( v ) \rangle _{\rho} ^{\sim}
 + \langle f(x,u^{ * }) + g(x,t,u^ {* }),\tau _{J}( v)\rangle _{\rho}
^{\sim}- G( \tau _{J}( v ) ).
\end{aligned}
\end{equation}
Passing to the limit as $J\to\infty$  on both sides of \eqref{e3.71},
we obtain
$$
\langle D_{t}u^{* },v \rangle _{\rho} ^{\sim}
 + \widetilde{\mathcal{M}}( u^ {* },v)
 = \lambda _{j_0}\langle u^{ * },v \rangle _{\rho} ^{\sim}
+ \langle f(x,u^{ * }) + g(x,t,u^ {* }),v\rangle _{\rho} ^{\sim} - G( v ),
$$
for all $v \in \widetilde H$, and the proof of Theorem \ref{thm1.1} is complete.
\end{proof}

\section{An example }

We give two functions to establish existence results for a function $f(x,s)$
that satisfies  (F1), (F2) and a function  $g(x,t,s)$ that
satisfies  (G1), (G2). Set
$\Omega  = \{ {x = ({{x_1},{x_2}}):x_1^2 + x_2^2 < 1} \}$  and
\[
{f_*}(x) = {({x_1^2 + x_2^2})^{ - \rho }},\quad 0 < \rho  < 1/4, \quad
x \in \Omega .
\]
 Also, $\gamma  > 0$  is given, and set
\[
f(x,s) = \begin{cases}
 - {s^2}{f_*}(x) + \gamma s,& 0 \le s < 1,\\
 - \sqrt s {f_*}(x) + \gamma s, &1 \le s <  + \infty ,
\end{cases}
\]
for $x\in\Omega$ and $0\leq s \leq+\infty $.
For  $- \infty  < s < 0$, we set  $f(x,s) =  - f(x, - s)$.
 Clearly, $f(x,s)$  meets (F1), (F2).
For $g(x,t,s)$ , set
 $\Omega  = \{ {x = ({{x_1},{x_2}}):x_1^2 + x_2^2 < 1} \}$,
$T = ( - \pi ,\pi )$,  and
\[
{g_0}({x,t}) = |t|{({x_1^2 + x_2^2})^{ - \rho }},\quad
0 < \rho  < 1/4, \quad ({x,t}) \in \widetilde \Omega .
\]
Also, we set
\[
g(x,t,s) = \begin{cases}
 - {s^2}{g_0}({x,t}), &0 \le s < 1,\\
 - \sqrt s {g_0}({x,t}), &1 \le s <  + \infty ,
\end{cases}
\]
for $({x,t}) \in \widetilde \Omega $  and $0 \le s <  + \infty $.
For $- \infty  < s < 0$, we set $g(x,t,s)=-g(x,t,-s)$. Clearly, $g(x,t,s)$
satisfies (G1)--(G2).

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\end{document}