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

\AtBeginDocument{{\noindent\small
2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2005, pp. 53--71.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{53}

\begin{document}

\title[\hfilneg EJDE/Conf/14 \hfil Higher order nonlinear degenerate problems]
{Higher order nonlinear degenerate elliptic problems with weak monotonicity}

\author[Y. Akdim, E. Azroul, M. Rhoudaf \hfil EJDE/Conf/14 \hfilneg]
{Youssef Akdim, Elhoussine Azroul, Mohamed Rhoudaf}  % in alphabetical order

\address{Youssef Akdim \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B. P. 1796 Atlas  F\`es, Maroc}
\email{akdimyoussef@yahoo.fr}

\address{E. Azroul \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B. P. 1796 Atlas  F\`es, Maroc}
\email{azroul\_elhoussine@yahoo.fr}

\address{Mohamed Rhoudaf \newline
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B. P. 1796 Atlas  F\`es, Maroc}
\email{rhoudaf\_mohamed@yahoo.fr}


\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{35J40, 35J70}
\keywords{Weighted Sobolev spaces; pseudo-monotonicity; \hfill\break\indent
nonlinear degenerate elliptic operators; boundary value problems}

\begin{abstract}
 We prove the existence of solutions for nonlinear degenerate elliptic
 boundary-value  problems of higher order. Solutions are obtained
 using pseudo-monotonicity theory in a suitable weighted
 Sobolev space.
\end{abstract}

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


\section{Introduction and statement of results}

Let $\Omega$ be an  open subset of $\mathbb{R}^N$ with finite
measure and let $m\geq 1$ be an integer and $p>1$ be a real
number. We will consider the degenerated partial differential
operators
\begin{equation}
Au(x)=A^mu(x)+A^{m-1}u(x),\label{e1.1} \end{equation}
on $\Omega$  where
\begin{equation}
A^mu(x)=\sum_{|\alpha|=m}(-1)^{|\alpha|}D^\alpha A_\alpha
(x,u,\dots ,\nabla^mu)\label{e1.2}
\end{equation}
is the top order part of the
degenerated quasilinear operator $A$. and where
\begin{equation}
A^{m-1}u(x)=\sum_{|\alpha|\leq m-1}(-1)^{|\alpha|}D^\alpha
A_\alpha (x,u,\dots ,\nabla^mu)\label{e1.3}
\end{equation}
is the lower order part of $A$. The coefficients
$\{A_\alpha(x,\eta,\zeta)$, $|\alpha|\leq m\}$ are real valued  functions
defined on $\Omega\times\mathbb{R}^{N_{m-1}}\times\mathbb{R}^{N_m} $
(with $N_{m-1}=\mathop{\rm card}\{\alpha\in \mathbb{N}^N$,
$|\alpha|\leq m-1\}$ and
$N_m=\mathop{\rm card}\{\alpha\in \mathbb{N}^N,\ \ |\alpha|= m \})$
which satisfy
suitable regularity and growth assumptions (see section 2).
Let $V$ be a subspace such that
\begin{equation}
W_0^{m,p}(\Omega,w)\subseteq
V\subseteq W^{m,p}(\Omega,w),\label{e1.4}
\end{equation}
where $W^{m,p}(\Omega,w)$ and $W_0^{m,p}(\Omega,w)$ are weighted Sobolev
spaces associated to a vector of weights
$w=\{w_\alpha\equiv w_\alpha(x)$, $|\alpha|\leq m\}$ on $\Omega$
satisfying some integrability conditions (see sections 2).
 We deal with the case
where $A^{m-1}$ is affine with respect to the top order
derivatives of $u$, i.e, $A^{m-1}u$ is of the form,
\begin{equation}
\begin{aligned}
A^{m-1}u(x)&=\sum_{|\alpha|\leq m-1}(-1)^{|\alpha|}D^\alpha
L_\alpha(x,u,\dots ,\nabla^{m-1}u)\\
&\quad +\sum_{|\alpha|\leq
m-1}\sum_{|\beta|=m}(-1)^{|\alpha|}D^\alpha
C_{\alpha\beta}(x,u,\dots ,\nabla^{m-1}u)D^\beta u
\end{aligned}\label{e1.5}
\end{equation}
where $L_\alpha(x,\eta)$ and $C_{\alpha\beta}(x,\eta)$ are some
real valued functions defined on
$\Omega\times\mathbb{R}^{N_{m-1}}$.
 We will assume the following hypotheses:
 \begin{itemize}
\item[(H1)] For every  $u\in V$ and
any multi-index $|\beta|\leq m-1$, there exists a parameter
$q(\beta)\geq 1$ and a weight function
$\sigma_\beta=\sigma_\beta(x)$ such that,
\begin{gather*}
D^{\beta}u \in L^{q(\beta)}(\Omega,\sigma_\beta),\label{e1.6}
\\
\|D^\beta u(x)\|_{q(\beta),\sigma_\beta}\leq \tilde
c_\beta\|u\|_{m,p,w}\label{e1.7}
\end{gather*}
with some constant $\tilde c_\beta>0$ independent of $u$ and moreover,
the compact imbedding,
\begin{equation}
V\hookrightarrow\hookrightarrow
H^{m-1,q}(\Omega,\sigma)\label{e1.8}
\end{equation}
holds, where $H^{m-1,q}(\Omega,\sigma)=\{u, D^\beta u\in
L^{q(\beta)}(\Omega,\sigma_\beta)\mbox{ for all }|\beta|\leq
m-1\}$.

\item[(H2)] The functions
$\{A_\alpha, |\alpha|=m\}$, $\{L_\alpha, |\alpha|\leq m-1\}$
and $\{C_{\alpha\beta}, |\alpha|\leq m-1 \mbox{ and
}|\beta|=m\}$ are Carath\'eodory functions and there exists
functions $g_\alpha \in L^{p'}(\Omega)$ for all  $|\alpha|=m$,
$\tilde g_\alpha \in L^{q'(\alpha)}(\Omega)$ for all
$|\alpha|\leq m-1$, and $\gamma_{\alpha\beta}\in L^{r_\alpha}(\Omega)$
for all $|\alpha|\leq m-1$ and all $|\beta|=m$ such that
\\
(i) for all  $|\alpha|=m$,
\begin{align*}
&|A_\alpha(x,\eta,\zeta)|\\
&\leq c_\alpha w_\alpha^{1/p}(x)
\Big(g_\alpha(x)+\tilde c_\alpha \sum_{|\beta|=m }
w_\beta^{\frac{1}{p'}}|\zeta_\beta|^{p-1}+\tilde c_\alpha
\sum_{|\beta|\leq m-1}\sigma_\beta^{\frac{1}{p'}}|
\eta_\beta|^{\frac{q(\beta)}{p'}}\Big)
\end{align*}
(ii) for all $|\alpha|\leq m-1$,
\[
|L_\alpha(x,\eta)|
\leq c_\alpha\sigma_\alpha^{\frac{1}{q(\alpha)}}
\Big(\tilde g_\alpha(x)+\tilde c_\alpha \sum_{|\beta|\leq m-1}
\sigma_\beta^{\frac{1}{q'(\alpha)}}|\eta_\beta|^{\frac{q(\beta)}{q'(\alpha)}}
\Big)
\]
(iii) for all $|\alpha|\leq m-1$ and all $|\beta|=m$,
\begin{align*}
&|C_{\alpha\beta}(x,\eta)|\\
&\leq c_{\alpha\beta}\sigma_\alpha^{\frac{1}{q(\alpha)}}
(x)w_\beta^{1/p}(x)\Big(\gamma_{\alpha\beta}(x)+\tilde c_{\alpha\beta}
\sum_{|\lambda|\leq m-1}\sigma_\lambda^{\frac{1}{r_\alpha}}(x)|
\eta_\lambda|^{\frac{q(\lambda)}{r_\alpha}}
\Big)
\end{align*}
for a.e. $x\in \Omega$, some positive constants $c_\alpha$,
$\tilde c_\alpha$ and $\tilde c_{\alpha\beta}$,  every
$(\eta,\zeta)\in\mathbb{R}^{N_{m-1}}\times\mathbb{R}^{N_m}=\mathbb{R}^d$
and some exponent $r_\alpha$ such that
\begin{equation}
\frac{1}{r_\alpha}+\frac{1}{p}+\frac{1}{q(\alpha)}<1\quad
\mbox{for all }|\alpha|\leq m-1. \label{e1.9}
\end{equation}
For the existence of $r_\alpha$ see Remark \ref{rmk2.1} below.
\end{itemize}

 Let us consider the degenerated
boundary value problem (DBVP) associated to the equation,
\begin{equation}
Au=f\in V^*,\label{E}
\end{equation}
where $V^*$ is the dual space of $V$ from \eqref{e1.4}.
 Recently, Drabeck, Kufner and Mustonen
proved in \cite{dr-ku-mu} the existence result for Dirichlet
degenerated problem of second order associated to the operator $A$
of the form,
\begin{equation}
Au(x)=-\sum_{i=1}^N \frac{\partial}{\partial
x_i}a_i(x,u,\nabla u)\label{e1.10}
\end{equation}
where the Carath\'eodory
functions $a_i(x,\eta,\zeta)$ satisfy some simple growth
conditions, that is,
\begin{equation}
|a_i(x,\eta,\zeta)|\leq
c_1w_i^{1/p}(x)\Big(g(x)+\bar
w^{\frac{1}{p'}}(x)|\eta|^{\frac{q}{p'}}+\sum_{j=1}^Nw_j^{\frac{1}{p'}}
|\zeta|^{p-1}\Big)\label{e1.11}
\end{equation}
where the exponent $q$ and the weight function $\bar w(x)$ verify
the so called Hardy-type inequality; i.e,
\begin{equation}
\int_\Omega |u(x)|^q\bar w(x)\,dx\leq c\sum_{i=1}^N \int_\Omega
|D_iu|^pw_i(x)\,dx\label{e1.12}
\end{equation}
and the compact imbedding
\begin{equation}
W_0^{1,p}(\Omega,w)\hookrightarrow\hookrightarrow
L^q(\Omega,\bar w).\label{e1.13}
\end{equation}
The authors have proved that
the mapping $T$ associated to $A$ from \eqref{e1.10} is
pseudo-monotone in $W_0^{1,p}(\Omega,w)$, by assuming only the
so-called weak Leray-Lions condition
\begin{equation}
\sum_{i=1}^N(a_i(x,\eta,
\zeta)-a_i(x,\eta, \bar\zeta))(\zeta_i-\bar\zeta_i)\geq 0.\label{e1.14}
\end{equation}
Our first objective of this paper is to extend
the previous result of \cite{dr-ku-mu} in the general class of
operators $A$ from \eqref{e1.1}, where the lower order part
$A^{m-1}$ is of the form \eqref{e1.5} and where the growth
conditions are of the most general form (H2). More precisely,
we prove the following result.

\begin{theorem} \label{thm1.1}
Assume that (H1), (H2) and that
\begin{equation}
\sum_{|\alpha|=m}(A_\alpha(x,\eta,\zeta)-A_\alpha(x,\eta,\bar\zeta))
(\zeta_\alpha-\bar\zeta_\alpha)\geq 0 \label{w.L-L}
\end{equation}
for a.e. $x\in\Omega$, all $\eta\in \mathbb{R}^{N_{m-1}}$ and all
$(\zeta,\bar\zeta)\in \mathbb{R}^{N_{m}}\times\mathbb{R}^{N_m}$ hold.
Then the mapping $T$ associated to the operator $A$ from \eqref{e1.1}
 and \eqref{e1.5} is pseudo-monotone in $V$.

 If in addition the degeneracy satisfies
\begin{equation}
\sum_{|\alpha|\leq m}A_\alpha(x,\xi)\xi_\alpha\geq
c\sum_{|\alpha|\leq m}w_\alpha(x)|\xi_\alpha|^p, \label{D1}
\end{equation}
for a.e. $x\in \Omega$, some $c>0$ and all $\xi\in
\mathbb{R}^{N_{m-1}}\times\mathbb{R}^{N_m}$, then
the DBVP associated to the equation \eqref{E} has at least one
solution $u\in V$.
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
The statement of Theorem \ref{thm1.1}, is obviously contained in 
 Theorem \ref{thm3.1}  below  (it suffices to take $J=\emptyset$)
where some general situation is considered.
\end{remark}

On the other hand, Drabeck, Kufner and Nikolosi in \cite{dr-ku-ni2}
have studied the existence result for the DBVP from the equation \eqref{E}
with $A$ of the form \eqref{e1.1} and with more general hypotheses
(H1'), (H2'), (H3) (in section 2) and with the so-called Leray-Lions condition
\begin{equation}
\sum_{|\alpha|=m}(A_\alpha(x,\eta, \zeta)-A_\alpha(x,\eta, \bar\zeta))
(\zeta_\alpha-\bar\zeta_\alpha)> 0.\label{L-L}
\end{equation}
The authors have assumed in addition to the previous hypotheses the
 compact imbedding,
\begin{equation}
V\hookrightarrow\hookrightarrow W^{m-1,p}(\Omega,w)\label{e1.15}
\end{equation}
and then, have proved that the mapping $T$ satisfies the condition
$\alpha(V)$ (see definition \ref{def2.2}) and hence used the degree theory
of general mappings of monotone type.
The hypotheses \eqref{e1.15} play an important role in the work
\cite{dr-ku-ni2}, because it is related to some strong converges
 appearing in the $\alpha(V)$ condition.

Our second objective of this paper, is to prove the same result as in
\cite{dr-ku-ni2} without assuming the compact imbedding \eqref{e1.15}.
This is possible by proving the pseudo-monotonicity of the mapping $T$
induced by the operator $A$ from \eqref{e1.1}.
More precisely, we have the following result.

\begin{theorem} \label{thm1.2}
Assume that (H1'), (H2'), (H3) and \eqref{L-L}. Then the
mapping $T$ associated to operator $A$ from \eqref{e1.1} is
pseudo-monotone in $V$. If in  addition the degeneracy \eqref{D1} is
satisfied, then, the DBVP from the equation \eqref{E} has at least
one solution $u\in V$.
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
Theorem \ref{thm1.2} is obviously a consequence of the more general
Theorem \ref{thm3.1} it suffices to take $J^c=\emptyset$).
\end{remark}

Hence, this paper can be seen as an extension of the preceding
papers \cite{dr-ku-mu, dr-ku-ni1,dr-ku-ni2}
(where the second order case without lower order part is
considered in the first paper. The degree theory is used
in the two last papers) and as a continuation of the papers
\cite{ak-az-be1} and  \cite{ak-az-be2} (where the second
order case with lower order part not equal to zero, is studied in
the first paper and where the higher order case with
$A^{m-1}\equiv 0$ or with $A^{m-1}\not\equiv 0$ but under
restrictions $w_\alpha\equiv 1$ for all $|\alpha|\leq m-1$, is
considered in the last paper). Finally, note that our approach
(based on the theory of pseudo-monotone mappings) can be applied
in the case of non reflexive Banach spaces. For example in the
general settings of weighted Orlicz-Sobolev spaces (see
\cite{az} for this direction). This work is divided into five
sections. We start with the introduction of a basic assumptions in
section 2. Next, we give our main general result in section 3,
which is proved in section 4. Finally, we study in section 5, some
particular case (where our basic assumption are satisfied). In our
work, we shall adopt many ideas introduced in \cite{go-mu}, but
the results are generalized and improved.

\section{Preliminaries and basic assumptions}

\subsection{Weighted Sobolev spaces}
Let $\Omega$ be an open subset of $\mathbb{R}^N$ with finite measure.
In the sequel we suppose that the vector of weights, on
$\Omega$, $w=\{w_\alpha (x):\ \ |\alpha|\leq m\}$ satisfies the
integrability conditions:
\begin{gather*}
w_\alpha \in L_{\rm loc}^1(\Omega), \label{e2.1} \\
w_\alpha^{-\frac{1}{p-1}} \in L_{\rm loc}^1(\Omega)\label{e2.2}
\end{gather*}
for any $|\alpha|\leq m$.
We denote by $W^{m,p}(\Omega,w)$ ($1<p<\infty$) the space of all
real-valued functions $u$ such that the derivatives in the sense
of distributions fulfil
$$
D^{\alpha}u \in L^p(\Omega,w_\alpha) \ \
\ \mbox{ for all } |\alpha|\leq m.
$$
The weighted Sobolev space
$W^{m,p}(\Omega,w)$ is normed when equipped by the norm
\begin{equation}
\|u\|_{m,p,w}=\Big(\sum_{|\alpha|\leq m}\int_\Omega |D^\alpha
u|^p w_\alpha\ dx\Big)^{1/p}.\label{e2.3}
\end{equation}
The space $W_0^{m,p}(\Omega,w)$ is  defined as the closure of the set
$C_0^\infty(\Omega)$ with respect to the norm \eqref{e2.3}.
 Note
that the conditions \eqref{e2.1} and \eqref{e2.2} imply that the
spaces $W^{m,p}(\Omega,w)$ and  $W_0^{m,p}(\Omega,w)$ are
reasonably defined and are  reflexive Banach spaces (for more
details see \cite{dr-ku-ni2}). We recall that the dual space
of $W_0^{m,p}(\Omega,w)$ is equivalent to $W^{-m,p'}(\Omega,w^*)$
where $w^*=\{w_\alpha^*=w_\alpha^{1-p'}: |\alpha|\leq m\}$,
with $p'=\frac{p}{p-1}$ is the H\"older's conjugate of $p$.

\subsection{Basic assumptions}
 Let $J$ be a
subset of $\{\alpha\in \mathbb{N}^N,\, |\alpha|=m\}$ and $J^c$ its
complement. We will suppose that the coefficients $A_\alpha$ of
the operator $A$ from \eqref{e1.1} are such that
\begin{equation}
\begin{gathered}
A_\alpha(x,\eta,\zeta)=B_\alpha(x,\eta,\zeta_J)\quad \forall  \alpha\in J,\\
A_\alpha(x,\eta,\zeta)=B_\alpha(x,\eta,\zeta_{J^c})\quad \forall
 \alpha\in J^c, \\
A_\alpha(x,\eta,\zeta)=L_\alpha(x,\eta,\zeta_J)
+{ \sum_{\beta\in J^c}}C_{\alpha\beta}(x,\eta,\zeta_J)\zeta_\beta\quad
\forall  |\alpha|\leq m-1,
\end{gathered}
\label{e2.4}
\end{equation}
for a.e. $x\in \Omega$ and where
$\{B_\alpha, |\alpha|=m\}$, $\{L_\alpha, |\alpha|\leq m-1\}$
and $\{C_{\alpha\beta}, |\alpha|\leq m-1$
and $\beta\in J^c\}$ are some Carath\'eodory  functions and
where $\zeta_I$ denoted $\zeta_I=\{\zeta_\alpha$, $\alpha \in I\}$.
 We denote by $N_I=\mathop{\rm card}\{\alpha\in \mathbb{N}^N,\ \ \ \alpha\in I\}$. Let us
introduce the following modified versions of \eqref{L-L} and
\eqref{w.L-L},
\begin{equation}
\sum_{\alpha\in J}(B_\alpha(x,\eta,
\zeta_J)-B_\alpha(x,\eta,
\bar\zeta_J))(\zeta_\alpha-\bar\zeta_\alpha)>0,\label{L-L-J}
\end{equation}
for a.e $x\in \Omega$, all $\eta\in \mathbb{R}^{N_{m-1}} $ and
all $\zeta_J\neq\bar\zeta_J \in \mathbb{R}^{N_J}$ and
\begin{equation}
\sum_{\alpha\in J^c}(B_\alpha(x,\eta,
\zeta_{J^c})-B_\alpha(x,\eta,
\bar\zeta_{J^c}))(\zeta_\alpha-\bar\zeta_\alpha)\geq
0, \label{w.L-L-Jc}
\end{equation}
for a.e $x\in \Omega$ and all
$(\eta,\zeta_{J^c},\bar\zeta_{J^c})\in
\mathbb{R}^{N_{m-1}}\times\mathbb{R}^{N_{J^c}}\times\mathbb{R}^{N_{J^c}}$.

Let us denote by $m_1=m-\frac{N}{p}$ and suppose that $m_1>0$
i.e, $mp>N$. We denote by $C(\Omega,w_\alpha)$ the weighted
spaces of continuous functions, more precisely
$C(\Omega,w_\alpha)=\{u=u(x) \mbox{ continuous on } \Omega,
\|u\|_{C(\Omega,w_\alpha)}= \sup_{x\in
\Omega}|u(x)w_\alpha(x)|<\infty\}$.
\begin{itemize}
\item[(H1')]  Let $u\in V$.
\begin{enumerate}
\item[(i)] For $|\beta|<m_1$, there is a weight function
$\sigma_\beta=\sigma_\beta(x)$ such that,
$D^{\beta}u \in C(\Omega,\sigma_\beta)$ %\label{e2.5}
and moreover,
\begin{equation}
\sup_{x\in \Omega}|D^\beta u(x)\sigma_\beta(x)|\leq
\tilde c_\beta\|u\|_{m,p,w}\label{e2.6}
\end{equation}
with some constant $\tilde c_\beta>0$ independent of $u$.
When we denote by $k(x,u(x))$ the expression
$ \sum_{|\beta|<m_1}|\sigma_\beta(x)D^\beta u(x)|$,
then, in view of \eqref{e2.6},
\begin{equation}
|k(x,u(x))|\leq c\|u\|_{m,p,w}\quad
\mbox{ for all }u\in V.\label{e2.7}
\end{equation}

\item[(ii)] For $m_1\leq |\beta|\leq m-1$, there is a parameter
$q(\beta)\geq 1$ and a weight function
$\sigma_\beta=\sigma_\beta(x)$ such that
$D^{\beta}u \in L^{q(\beta)}(\Omega,\sigma_\beta)$ %\label{e2.8}
and moreover,
\begin{equation}
\|D^\beta u(x)\|_{q(\beta),\sigma_\beta}\leq \tilde c_\beta\|u\|_{m,p,w}
\label{e2.9}
\end{equation}
for some constant $\tilde c_\beta>0$ independent of $u$.

\item[(iii)] The  imbedding
$V\hookrightarrow\hookrightarrow H^{m-1,q}(\Omega,\sigma)$ %\label{e2.10}
compact,
where $H^{m-1,q}(\Omega,\sigma)=\{u, D^\beta u\in X_\beta, \mbox{ for all
}|\beta|\leq m-1\}$  with $X_\beta=L^{q(\beta)}(\Omega,\sigma_\beta)$
for $m_1\leq |\beta|\leq m-1$ and $X_\beta=C(\Omega,\sigma_\beta)$
for $ |\beta|< m_1$.
\end{enumerate}

\item[(H2')] There exists functions
$g_\alpha \in L^{p'}(\Omega)$ for $|\alpha|=m, \tilde g_\alpha
\in L^{q'(\alpha)}(\Omega)$ for $m_1\leq |\alpha|\leq m-1$,
$\hat g_\alpha \in L^{1}(\Omega)$ for $|\alpha|<m_1$,
$\gamma_{\alpha\beta}\in L^{r_\alpha}(\Omega)$ for all
$|\alpha|\leq m-1$ and $\beta \in J^c$ and some positive constants
$\tilde c_\alpha$ and $\tilde c_{\alpha\beta}$, moreover there
exists a positive continuous, non decreasing function
$G(t)$, $t\geq 0$, such that the following estimates hold:

\noindent(i) For $\alpha\in J$,
\begin{align*}
&|B_\alpha(x,\eta,\zeta_J)|\\
&\leq G(k(x,\kappa))w_\alpha^{1/p} \Big(g_\alpha(x)
  +\tilde c_\alpha \sum_{\beta\in J}w_\beta^{\frac{1}{p'}}
|\zeta_\beta|^{p-1}+\tilde c_\alpha \sum_{m_1\leq|\beta|\leq m-1}
\sigma_\beta^{\frac{1}{p'}}|\eta_\beta|^{\frac{q(\beta)}{p'}}\Big)
\end{align*}

\noindent (ii) for $\alpha\in J^c$,
\begin{align*}
&|B_\alpha(x,\eta,\zeta_{J^c})|\\
&\leq G(k(x,\kappa))w_\alpha^{1/p}\Big(g_\alpha(x)+\tilde c_\alpha
\sum_{\beta\in J^c}w_\beta^{\frac{1}{p'}}|\zeta_\beta|^{p-1}
 +\tilde c_\alpha \sum_{m_1\leq|\beta|\leq m-1}
\sigma_\beta^{\frac{1}{p'}}|\eta_\beta|^{\frac{q(\beta)}{p'}}\Big)
\end{align*}

\noindent (iii) for $m_1\leq|\alpha|\leq m-1$,
\begin{align*}
&|L_\alpha(x,\eta,\zeta_{J})| \\
&\leq G(k(x,\kappa))\sigma_\alpha^{\frac{1}{q(\alpha)}}
\Big(\tilde g_\alpha(x)+\tilde c_\alpha \sum_{\beta\in J}
 w_\beta^{\frac{1}{q'(\alpha)}}|\zeta_\beta|^{\frac{p}{q'(\alpha)}}
 +\tilde c_\alpha \sum_{m_1\leq|\beta|\leq m-1}
\sigma_\beta^{\frac{1}{q'(\alpha)}}|\eta_\beta|^{\frac{q(\beta)}{q'(\alpha)}}
\Big)
\end{align*}

\noindent (iv) for $|\alpha|<m_1$,
\begin{align*}
&|L_\alpha(x,\eta,\zeta_J)|\\
&\leq G(k(x,\kappa))\sigma_\alpha
\Big(\hat g_\alpha(x)+\tilde c_\alpha \sum_{\beta\in J}w_\beta|
\zeta_\beta|^{p}
 +\tilde c_\alpha \sum_{m_1\leq|\beta|\leq m-1}\sigma_\beta|
\eta_\beta|^{q(\beta)}\Big)
\end{align*}

\noindent (v) for $m_1\leq|\alpha|\leq m-1$ and $\beta\in J^c$,
\begin{align*}
&|C_{\alpha\beta}(x,\eta,\zeta_{J})|\\
&\leq G(k(x,\kappa))\sigma_\alpha^{\frac{1}{q(\alpha)}}w_\beta^{1/p}
\Big(\gamma_{\alpha\beta}(x)+\tilde c_{\alpha\beta}
 \sum_{\lambda\in J }w_\lambda^{\frac{1}{r_\alpha}}|\zeta_\lambda|
^{\frac{p}{r_\alpha}}
 +\tilde c_{\alpha\beta} \!\sum_{m_1\leq|\lambda|\leq m-1}\!
\sigma_\lambda^{\frac{1}{r_\alpha}}|\eta_\lambda|^{\frac{q(\lambda)}{r_\alpha}}
\Big)
\end{align*}

\noindent (vi) for $|\alpha|< m_1$ and $\beta\in J^c$,
\begin{align*}
&|C_{\alpha\beta}(x,\eta,\zeta_{J})|\\
&\leq G(k(x,\kappa)) \sigma_\alpha w_\beta^{1/p}
\Big(\gamma_{\alpha\beta}(x)+\tilde c_{\alpha\beta}
\sum_{\lambda\in J }w_\lambda^{\frac{1}{r_\alpha}}
|\zeta_\lambda|^{\frac{p}{r_\alpha}}
 +\tilde c_{\alpha\beta} \sum_{m_1\leq|\lambda|
\leq m-1}\sigma_\lambda^{\frac{1}{r_\alpha}}|\eta_\lambda|
^{\frac{q(\lambda)}{r_\alpha}}
\Big)
\end{align*}
for a.e. $x\in \Omega$, every $\eta\in\mathbb{R}^{N_{m-1}}$ and
every $\zeta_I\in \mathbb{R}^{N_I}$ where $\kappa=\{\eta_\beta,
|\beta|<m_1\}$ and
$$
\frac{1}{r_\alpha}+\frac{1}{p}+\frac{1}{q(\alpha)}<1
$$
for any
$m_1\leq|\alpha|\leq m-1$ and any $\beta\in J^c$ and with
$$
\frac{1}{r_\alpha}+\frac{1}{p}<1
$$
 for any $|\alpha|<m_1$ and
any $\beta\in J^c$. Note that the exponent $q'(\alpha)$ denotes
the H\"older's conjugate of $q(\alpha)$.
\end{itemize}

\begin{remark} \label{rmk2.1} \rm
For all $m_1\leq |\alpha|\leq m-1$, the such $r_\alpha$ satisfying
$\frac{1}{r_\alpha}+\frac{1}{p}+\frac{1}{q(\alpha)}<1$ exists
when $q(\alpha)>p'$. And we can choose $r_\alpha >p'$ when $|\alpha|< m_1$.
\end{remark}

\begin{remark} \label{rmk2.2} \rm
If $m_1 \leq 0$, then the set of multi-indices $\xi_{\beta}$ with
$|\beta|<m_1$ is empty. Then we set $G(t)\equiv 1$ and since the
cases  iv)and vi) in $(H_2')$ are irrelevant, we obtain the
growth condition of type C \cite{dr-ku-ni2}. Further if we do
not differ between $|\alpha|= m$ and $|\alpha|\leq m-1$ i.e, if we
take $\tilde g_\alpha=g_\alpha\in L^{p'}(\Omega)$  we immediately
obtain the growth conditions of type (B) \cite{dr-ku-ni2}.
Finally if we choose $q(\beta) = p$ and
$\sigma_{\beta} = w_{\beta}$, we obtain the growth condition of type A
\cite{dr-ku-ni2}.
\end{remark}

\begin{itemize}
\item[(H3)] Let $G_1$ be a continuous
positive, nonincreasing function on $[0,\infty)$, and let $G_2$ be
a continuous positive, nondecreasing function on $[0,\infty)$, we
will suppose that for every $\xi=(\kappa, \eta, \zeta)\in
\mathbb{R}^d $ and for a.e. $x\in \Omega$ the
ellipticity condition holds
\begin{align*}
&\sum_{|\alpha|=m}A_\alpha(x,\kappa,
\eta, \zeta)\zeta_\alpha\\
&\geq G_1(h(x,\kappa))\sum_{|\beta|=m}w_\beta|\zeta_\beta|^p-G_2
(h(x,\kappa))\sum_{m_1\leq|\beta|\leq m-1}\sigma_\beta|\eta_\beta|^{q(\beta)},
\end{align*}
where
$\kappa=\{\xi_\beta$, $|\beta|<m_1\}\in\mathbb{R}^{d_1}$,
$\eta=\{\xi_\beta$,  $m_1\leq |\beta|\leq m-1\}\in\mathbb{R}^{d_2}$,
$\zeta=\{\xi_\beta$, $|\beta|=m\}\in\mathbb{R}^{N_m}$ and
$d_1+d_2=N_{m-1}$.

\end{itemize}
 Under these assumptions, the differential operator \eqref{e1.1}
generates a mapping $T$ from $V$ to its dual $V^*$ through the
formula
\begin{equation}
\begin{aligned}
\langle Tu,v \rangle
&=\sum_{\alpha \in J}\int_\Omega
B_\alpha(x,\eta(u),\zeta_J(\nabla^mu))D^\alpha v\,dx\\
&\quad +\sum_{\alpha \in J^c}\int_\Omega
B_\alpha(x,\eta(u),\zeta_{J^c}(\nabla^mu))D^\alpha v\,dx\\
&\quad +\sum_{|\alpha|\leq m-1}\int_\Omega
L_\alpha(x,\eta(u),\zeta_J(\nabla^mu))D^\alpha
v\,dx\\
&\quad +\sum_{|\alpha| \leq m-1}\sum_{\beta \in J^c}\int_\Omega
C_{\alpha\beta}(x,\eta(u),\zeta_J(\nabla^mu))D^\beta u D^\alpha
v\,dx,
\end{aligned}\label{e2.11}
\end{equation}
for all $u,v\in V$ and where $\langle .,.\rangle $
denotes the duality pairing between $V^*$ and $V$. The mapping
$T$ is well defined and bounded, this can be easily seen by
H\"older's inequality and the following lemma.

\begin{lemma} \label{lem2.1}
Let $\Omega$ be a subset of $\mathbb{R}^N$ with finite measure and
let $f\in L^p(\Omega,\sigma_1), g\in L^q(\Omega,\sigma_2)$ where
$\sigma_1$ and $\sigma_2$ are weight functions in $\Omega$ and let
$h\in L^r(\Omega,\sigma_1^{-\frac{r}{p}}\sigma_2^{-\frac{r}{q}})$
with
$$
\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\leq 1.
$$
Then $fgh\in L^1(\Omega)$.
\end{lemma}

Indeed. Let $\frac{1}{s}=\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\leq 1$.
By H\"older's inequality we have,
$$
\int_\Omega|fgh|^s\,dx\leq \Big(\int_\Omega f^p\sigma_1\,dx\Big)^{\frac{s}{p}}
\Big(\int_\Omega g^q\sigma_2\,dx\Big)^{\frac{s}{q}}
\Big(\int_\Omega h^r\sigma_1^{-\frac{r}{p}}\sigma_2^{-\frac{r}{q}}\,dx
\Big)^{\frac{s}{r}}<\infty.
$$
Then $fgh\in L^s(\Omega)$, which implies that, $fgh \in L^1(\Omega)$.

Let us recall the following definitions.

\begin{definition} \label{def2.1} \rm
A mapping $T$ from  $X$ to its dual $X^*$, is called
pseudo-monotone, if for every sequence $\{u_n\} \subset X$ with
$u_n \rightharpoonup u$ in $X$ and
$\limsup_{n\to\infty} \langle T u_n,u_n-u\rangle\leq 0$, one has
$$
\liminf_{n\to\infty}\langle Tu_n,u_n-v\rangle \geq
\langle Tu,u-v\rangle \quad \mbox{for all } v\in X.
$$
\end{definition}

\begin{definition} \label{def2.2} \rm
Let $X$ be a reflexive Banach space. The mapping $T$ from $X$ to
$X^*$ is said to satisfy condition $\alpha(X)$ if the assumptions
 $$
u_n \rightharpoonup u \quad \mbox{in } X \quad \mbox{and}\quad
\limsup_{n\to \infty} \langle T\,u_n,u_n-u\rangle \leq 0,
$$
imply
$u_n\to u$ in $X$.
\end{definition}

Obviously, the class $\alpha(X)$ of operators is contained in the class
of pseudo-monotone operators.

\section{Main general result}

The aim of this section, is to prove the following result.

\begin{theorem} \label{thm3.1}
Assume that (H1'), (H2'), (H3), \eqref{L-L-J} and
\eqref{w.L-L-Jc} hold. Then, the  mapping $T$ defined by
\eqref{e2.11} is pseudo-monotone in $V$.
\end{theorem}

\begin{remark} \label{rmk3.1} \rm
\begin{enumerate}
\item 
 When $J=\emptyset$, the previous theorem applies in particular to
operators like \eqref{e1.1} with $A_\alpha$, $|\alpha|\leq m-1$ affine
with respect to $\nabla^mu$. This gives from \eqref{w.L-L} a sufficient
 condition (see Theorem \ref{thm1.1}).
\item When $J=\emptyset$, $m=1$ and $A_0\equiv 0$, we immediately
obtain  \cite[proposition 1]{dr-ku-mu}.
\item When $A_\alpha \equiv 0$ for all $|\alpha|\leq m-1$ and
$J=\emptyset$ (resp. $J^c=\emptyset)$ we obtain Theorem 8.1
(resp. Theorem 8.3) of \cite{az} with some simple the growth conditions.
\end{enumerate}
\end{remark}

\begin{remark} \label{rmk3.2} \rm
Since the hypothesis (H3) concerns only
the terms $L_\alpha$ with $|\alpha|<m_1$ (see Remark \ref{rmk4.1}
below), then the statement of Theorem \ref{thm3.1} remains true without
assuming  (H3), when $m_1\leq 0$.
\end{remark}

\begin{remark} \label{rmk3.3} \rm
If we take $m_1\leq 0$, $q(\beta)=p$ and $\sigma_\beta=w_\beta$,
then $X_\beta=L^p(\Omega,w_\beta)$ for all $|\beta|\leq m-1$,
hence the growth condition (H2') is of the type $A$
(see \cite{dr-ku-ni2}) and the statement of Theorem \ref{thm3.1} remains true
without assuming (H3).
\end{remark}

Applying the previous theorem,
we obtain the following existence results, which generalize the
corresponding (cf. \cite{az,dr-ku-mu}) and extend the
corresponding in \cite{dr-ku-ni1,dr-ku-ni2}.

\begin{corollary} \label{coro3.1}
Assume the hyptheses in Theorem \ref{thm3.1} and  the condtion on the degeneracy
\eqref{D1}. Then the DBVP from the equation \eqref{E} has at least one
solution $u\in V$.
\end{corollary}

\begin{remark} \label{rmk3.4} \rm
If the expression,
$$
\||u|\|_V=\Big(\sum_{|\alpha|=m}\int_\Omega w_\alpha(x)|D^\alpha u|^p \,dx
\Big)^{1/p}
$$
is a norm in $V$ equivalent to the usual norm \eqref{e2.3}
(see section 5 where this fact is verified for
$V=W_0^{m,p}(\Omega,w))$, then we can replace in Corollary \ref{coro3.1},
the degeneracy \eqref{D1} by the weaker condition
\begin{equation}
\sum_{|\alpha|\leq m}A_\alpha(x,\xi)\xi_{\alpha}
\geq c\sum_{|\alpha|= m}w_{\alpha } |\xi_{\alpha }|^{p}.
\label{D2}
\end{equation}
\end{remark}

\section{Proof of Theorem \ref{thm3.1}}

For this goal, we need the following lemmas.

\begin{lemma} \label{lem4.1}
Let $(g_n)_n $ be a sequence of $L^p(\Omega,\tilde\sigma)$ and let
$g\in L^p(\Omega,\tilde\sigma)$ $(1<p<\infty)$, where
$\tilde\sigma$ is a weight function in $\Omega$.
If $g_n \to g$ in measure (in particular a.e in $\Omega $) and
it is bounded in $L^p(\Omega,\tilde\sigma)$, then
$g_n\to g$ in $L^q(\Omega,\tilde\sigma^{\frac{q}{p}})$
 for all $q<p$.
\end{lemma}

\begin{proof}
Let $\varepsilon>0$ and set $A_n=\{x\in \Omega /
|g_n(x)-g(x)|\tilde\sigma^{1/p}(x)\leq
(\frac{\varepsilon}{2\mathop{\rm meas}(\Omega)})^{1/q}\}$. We have
\begin{align*}
\int_\Omega|g_n-g|^q\tilde\sigma^{\frac{q}{p}}\,dx
&=\int_{A_n}|g_n-g|^q \tilde\sigma^{\frac{q}{p}}\,dx
 +\int_{A_n^c}|g_n-g|^q\tilde\sigma^{\frac{q}{p}}\,dx \\
&\leq \frac{\varepsilon}{2}+\int_{A_n^c}|g_n-g|^q
 \tilde\sigma^{\frac{q}{p}}\,dx.
\end{align*}
By H\"older inequality, one can see that
\begin{align*}
\int_{A_n^c}|g_n-g|^q\tilde\sigma^{\frac{q}{p}}\,dx
&\leq \Big(\int_\Omega|g_n-g|^p\tilde\sigma\,dx\Big)^{\frac{q}{p}}
\Big(\mathop{\rm meas}(A_n^c)\Big)^{1-\frac{q}{p}} \\
&\leq M \Big(\mathop{\rm meas}(A_n^c)\Big)^{1-\frac{q}{p}},
\end{align*}
where $M$ is a constant does not depend on $n$.
On the other hand, since $g_n \to g$ in measure,
$\mathop{\rm meas}(A_n^c)\to 0$ as $n\to \infty$. Then there
exists some $n_0\in \mathbb{N}$  such that for all
$n\geq n_0$,
$$
\int_{A_n^c}|g_n-g|^q\tilde\sigma^{\frac{q}{p}}\,dx
\leq \frac{\varepsilon}{2}.
$$
\end{proof}

 The following lemma is a generalization of  \cite[Lemma 3.2]{le-li}
in weighted spaces.

\begin{lemma} \label{lem4.2}
Let $g\in L^q(\Omega,\tilde\sigma)$ and let
$ g_n\in L^q(\Omega,\tilde\sigma)$, with
$\|g_n\|_{q,\tilde\sigma} \leq c$
$(1<q<\infty )$. If $g_n(x)\to g(x)$ a.e. in
$\Omega$, then $g_n\rightharpoonup g$ in
$L^q(\Omega,\tilde\sigma)$, where $\rightharpoonup$ denotes weak
convergence.
\end{lemma}

\begin{proof}
 Since $g_n\tilde\sigma^{\frac{1}{q}}$ is bounded in
$L^q(\Omega)$ and $g_n(x)\tilde\sigma^{\frac{1}{q}}(x)
\to g(x)\tilde\sigma^{\frac{1}{q}}(x)$, a.e.  in
$\Omega$, then by  \cite[lemma 3.2]{le-li},
$$
g_n\tilde\sigma^{\frac{1}{q}} \rightharpoonup
g\tilde\sigma^{\frac{1}{q}}\quad \mbox{ in }L^q(\Omega).
$$
Moreover,
for all $ \varphi \in L^{q'}(\Omega,\tilde\sigma^{1-q'})$, we have
$\varphi \tilde\sigma^{-\frac{1}{q}}\in L^{q'}(\Omega)$. Then
$$
\int_\Omega g_n\varphi\,dx \to \int_\Omega
g\varphi\,dx;\quad \mbox{i.e. } g_n\rightharpoonup g \quad
\mbox{in }L^q(\Omega,\tilde\sigma).
$$
\end{proof}

\subsection*{Proof of Theorem \ref{thm3.1}}
 Let $(u_n)_n$ be a sequence in $V$ such that:
$u_n\rightharpoonup u$  in $V$  %\label{e4.1}
and
\begin{equation} 
\limsup_{n\to \infty}\langle Tu_n,u_n-u\rangle \leq 0,\label{e4.2}
\end{equation}
i.e.,
\begin{align*}
&\limsup_{n\to \infty}\Big\{\int_\Omega \sum_{\alpha\in J }B_\alpha(x,\eta(u_n),
\zeta_J(\nabla^mu_n))(D^\alpha u_n-D^\alpha u)\,dx\\
& +\int_\Omega \sum_{\alpha\in J^c }B_\alpha(x,\eta(u_n),
 \zeta_{J^c}(\nabla^mu_n))(D^\alpha u_n-D^\alpha u)\,dx\\
& +\int_\Omega \sum_{|\alpha|\leq m-1}L_\alpha(x,\eta(u_n),
  \zeta_J(\nabla^mu_n))(D^\alpha u_n-D^\alpha u)\,dx\\
& +\int_\Omega \sum_{|\alpha|\leq m-1}\sum_{\beta\in
J^c}C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n))D^\beta
u_n(D^\alpha u_n-D^\alpha u)\,dx\Big\}\leq 0.
\end{align*}
\textbf{(a)} We shall prove that
\begin{equation}
\langle Tu_n,v\rangle \to \langle Tu,v \rangle \quad{as }n\to\infty\;
\forall v\in V. \label{e4.3}
\end{equation}
By (H1')(iii), the compact imbedding implies that for a subsequence
\begin{equation}
\begin{gathered}
D^\alpha u_n\to D^\alpha u \quad \mbox{in }X_\alpha\\
D^\alpha u_n\to D^\alpha u\quad\mbox{a.e. in }\Omega
\forall  \ |\alpha|\leq m-1.
\end{gathered} \label{e4.4}
\end{equation}

\noindent\textbf{Step (1)} We shall prove that
\begin{equation}
\lim_{n\to \infty}\sum_{|\alpha|\leq m-1}\int_\Omega L_\alpha(x,\eta(u_n),
\zeta_J(\nabla^mu_n))(D^\alpha u_n-D^\alpha u)= 0.\label{e4.5}
\end{equation}
(i) We show that
\begin{equation}
\lim_{n\to \infty}\sum_{m_1\leq|\alpha|\leq m-1}
 \int_\Omega L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))
 (D^\alpha u_n-D^\alpha u)\,dx= 0.\label{e4.6}
\end{equation}
Let $m_1\leq|\alpha|\leq m-1$ be fixed. Thanks to (H2'),
we have
\begin{align*}
&\int_\Omega |L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))(D^\alpha
u_n-D^\alpha u)|\,dx\\
&\leq \int_\Omega G(k(x,u_n(x)))\sigma_\alpha^{\frac{1}{q(\alpha)}}
|(D^\alpha u_n-D^\alpha u)||\tilde g_\alpha|\,dx\\
&\quad +\tilde c_\alpha \sum_{\beta\in J}\int_\Omega G(k(x,u_n(x)))
 \sigma_\alpha^{\frac{1}{q(\alpha)}}|(D^\alpha u_n-D^\alpha u)
 |w_\beta^{\frac{1}{q'(\alpha)}}|D^\beta u_n(x)|^{\frac{p}{q'(\alpha)}}\,dx\\
&\quad +\tilde c_\alpha \sum_{m_1\leq|\beta|\leq
m-1}\int_\Omega
G(k(x,u_n(x)))\sigma_\alpha^{\frac{1}{q(\alpha)}}|(D^\alpha
u_n-D^\alpha u)| \sigma_\beta^{\frac{1}{q'(\alpha)}}|D^\beta
u_n(x)|^{\frac{q(\beta)}{q'(\alpha)}}\,dx.
\end{align*}
By \eqref{e2.7} we have,
$$
G(k(x,u_n(x))\leq G(c\|u_n\|_{m,p,w}).
$$
Applying the H\"older's inequality with exponents $q(\alpha)$ and
$ q'(\alpha)$ we obtain
\begin{align*}
&\int_\Omega |L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))
(D^\alpha u_n-D^\alpha u)|\,dx\\
&\leq G(c\|u_n\|_{m,p,w})\|D^\alpha u_n-D^\alpha u\|_{q(\alpha),\sigma_\alpha}
\Big(\|\tilde g_\alpha\|_{q'(\alpha)}\\
&\quad +\tilde c_\alpha \sum_{\beta\in J}\|D^\beta
u_n\|_{p,w_\beta}^{\frac{p}{q'(\alpha)}}+ \tilde
c_\alpha \sum_{m_1\leq|\beta|\leq m-1}\|D^\beta
u_n\|_{q(\beta),\sigma_\beta}^{\frac{q(\beta)}{q'(\alpha)}}\Big).
\end{align*}
Thanks to \eqref{e2.9}, we have  $\|D^\beta
u_n\|_{q(\beta),\sigma_\beta}\leq \tilde c_\beta\|u_n\|_{m,p,w}$
for all $m_1\leq |\beta|\leq m-1$. Since
$\|D^\beta u_n\|_{p,w_\beta}\leq \|u_n\|_{m,p,w}$ for all $\beta \in J$,
we conclude that
\begin{align*}
&\int_\Omega |L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))(D^\alpha
u_n-D^\alpha u)|\,dx\\
&\leq\|D^\alpha u_n-D^\alpha
u\|_{q(\alpha),\sigma_\alpha}R_\alpha(\|u_n\|_{m,p,w})
\end{align*}
with,
$$
R_\alpha(t)=G(c_1t) \Big(\|\tilde
g_\alpha\|_{q'(\alpha)}+c_2 t^{\frac{p}{q'(\alpha)}}+
c_3 \sum_{m_1\leq|\beta|\leq
m-1}t^{\frac{q(\beta)}{q'(\alpha)}}\Big)
$$
which is a positive continuous function, hence $R_\alpha(\|u_n\|_{m,p,w})$
is bounded.
Moreover, by \eqref{e4.4} we have,
$$
\|D^\alpha u_n-D^\alpha u\|_{q(\alpha),\sigma_\alpha}\to 0\quad
\mbox{as }n\to \infty.
$$
then
$$
\int_\Omega L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))
 (D^\alpha u_n-D^\alpha u)\,dx \to 0,
$$
which yields \eqref{e4.6}.

\noindent(ii) We show that
\begin{equation}
\lim_{n\to \infty}\sum_{|\alpha|<m_1}\int_\Omega L_\alpha(x,\eta(u_n),
\zeta_J(\nabla^mu_n))(D^\alpha u_n-D^\alpha u)\,dx= 0.\label{e4.7}
\end{equation}
Let $|\alpha|<m_1 $ be fixed. Similarly by virtue of (H2'),
\begin{align*}
&\int_\Omega |L_\alpha(x,\eta(u_n),\zeta_J(\nabla^m u_n))
(D^\alpha u_n-D^\alpha u)|\,dx\\
&\leq G(c\|u_n\|_{m,p,w})\sup_{x\in \Omega}
(|(D^\alpha u_n-D^\alpha u)\sigma_\alpha|)
\Big(\|\hat g_\alpha\|_1+\tilde
c_\alpha \sum_{\beta\in J}\|D^\beta u_n\|_{p,w_\beta}^p\\
&\quad + \tilde c_\alpha \sum_{m_1\leq|\beta|\leq m-1}\|D^\beta
u_n\|_{q(\beta),\sigma_\beta}^{q(\beta)}\Big).
\end{align*}
It follows from \eqref{e2.9} and  $\|D^\beta u_n\|_{p,w_\beta}\leq
\|u_n\|_{m,p,w}$ for all $\beta\in J$ that
\begin{align*}
&\int_\Omega |L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))(D^\alpha
u_n-D^\alpha u)|\,dx\\
&\leq\|D^\alpha u_n-D^\alpha
u\|_{C(\Omega,\sigma_\alpha)}\tilde R_\alpha(\|u_n\|_{m,p,w}),
\end{align*}
where
$$
\tilde R_\alpha(t)=G(c_1t) \Big(\|\hat g_\alpha\|_1+c_2
t^p+ c_3 \sum_{m_1\leq|\beta|\leq
m-1}t^{q(\beta)}\Big).
$$
This function is also positive and continuous, hence
$\tilde R_\alpha(\|u_n\|_{m,p,w})$ is bounded.
Since, by \eqref{e4.4} $\|D^\alpha u_n-D^\alpha
u\|_{C(\Omega,\sigma_\alpha)}\to 0$ as
$n\to \infty$, it follows that
$$
\int_\Omega L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))(D^\alpha u_n
-D^\alpha u)\,dx \to 0,
$$
which yields \eqref{e4.7}. Thus, due to \eqref{e4.6} and \eqref{e4.7}
we conclude \eqref{e4.5}.

\noindent\textbf{Step (2)}
We shall prove that
\begin{equation}
\lim_{n\to \infty}\sum_{|\alpha|\leq m-1}\sum_{\beta\in J^c}
\int_{\Omega}C_{\alpha\beta}(x,\eta (u_n),\zeta_J(\nabla^mu_n))
D^{\beta}u_n (D^{\alpha}u_n -D^{\alpha}u)\,dx=0.\label{e4.8}
\end{equation}
(i) Let $m_1 \leq |\alpha|\leq m-1$ and  $\beta\in J^{c}$ be fixed.
And let $s_\alpha$ such that,
$$
\frac{1}{s_\alpha} =
\frac{1}{q(\alpha)} + \frac{1}{p} + \frac{1}{r_\alpha}<1.
$$
By H\"older's inequality, we have
\begin{align*}
&\int_{\Omega}|C_{\alpha\beta}(x,\eta (u_n),\zeta_J(\nabla^mu_n))
 D^{\beta}u_n (D^\alpha u_n -D^\alpha u)|^{s_\alpha}\,dx\\
&\leq \Big(\int_{\Omega}|C_{\alpha\beta}(x,\eta(u_n),
\zeta_J(\nabla^mu_n))|^{r_\alpha}\sigma_\alpha^{-\frac{r_\alpha}
{q(\alpha)}}w_\beta^{-\frac{r_\alpha}{p}}\,dx\Big)^{\frac{s_\alpha}
{r_\alpha}}\\
&\quad\times \Big(\int_{\Omega}|D^{\beta}
u_n|^pw_{\beta}\,dx\Big)^{\frac{s_\alpha}{p}}
\Big(\int_\Omega|D^\alpha u_n -D^\alpha
u|^{q(\alpha)}\sigma_\alpha\,dx\Big)^{\frac{s_\alpha}{q(\alpha)}}.
\end{align*}
By (H2') the sequences
$\{C_{\alpha\beta}(x,\eta (u_n),\zeta_J(\nabla^{m}u_n))$,
$m_1 \leq |\alpha|\leq m-1$, $\beta\in J^{c}\}$
(resp  $\{D^{\beta}u_n ,\ \beta\in J^{c}\})$ remain bounded in
 $L^{r_{\alpha}}(\Omega ,{\sigma}_{\alpha}^{-\frac{r_\alpha}{q(\alpha)}}
 w_\beta^{-\frac{r_\alpha}{p}})$
(resp $L^{p}(\Omega ,w_{\beta})$).
Moreover,
$\|D^{\alpha}u_n -D^{\alpha}u\|_{q(\alpha),
\sigma_\alpha}^{s_\alpha}\to 0$ as $n\to\infty$.
Then
$$
\lim_{n\to \infty}\int_{\Omega}|C_{\alpha\beta}(x,\eta
(u_n),\zeta_J(\nabla^mu_n))D^{\beta}u_n (D^{\alpha}u_n
-D^{\alpha}u)|^{s_\alpha}\,dx =0.
$$
Consequently,
$$
\lim_{n\to \infty}\int_{\Omega}|C_{\alpha\beta}(x,\eta
(u_n),\zeta_J(\nabla^{m}u_n))D^{\beta}u_n (D^{\alpha}u_n
-D^{\alpha}u)|\,dx =0,
$$
i.e,
\begin{equation}
\lim_{n\to
\infty}\sum_{m_1\leq|\alpha|\leq m-1}\sum_{\beta\in
J^c}\int_{\Omega}C_{\alpha\beta}(x,\eta
(u_n),\zeta_J(\nabla^mu_n))D^{\beta}u_n (D^{\alpha}u_n
-D^{\alpha}u)=0.\label{e4.9}
\end{equation}

\noindent(ii) Let  $|\alpha|<m_1$ and
$\beta\in J^c$ be fixed. By (H2') the sequences
$$
\{{\sigma}_{\alpha}^{-1}C_{\alpha\beta}(x,\eta
(u_n),\zeta_J(\nabla^mu_n))D^{\beta}u_n,  |\alpha|<m_1 ,
\beta\in J^{c}\}
$$ 
remain bounded in $L^{s_\alpha} (\Omega)$ with
$\frac{1}{s_\alpha} =\frac{1}{p} + \frac{1}{r_\alpha}<1$. Indeed,
 \begin{align*}
 &\int_\Omega|\sigma_\alpha^{-1}C_{\alpha\beta}(x,\eta (u_n),
\zeta_J(\nabla^mu_n))D^\beta u_n |^{s_\alpha}\,dx\\
&\leq \Big(\int_\Omega|C_{\alpha\beta}(x,\eta (u_n),\zeta_J(\nabla^mu_n))
|^{r_\alpha}\sigma_\alpha^{-r_\alpha}
w_\beta^{-\frac{r_\alpha}{p}}\,dx\Big)^{\frac{s_\alpha}{r_\alpha}}
\Big(\int_\Omega|D^\beta u_n|^pw_\beta\,dx\Big)^{\frac{s_\alpha}{p}}.
\end{align*}
The right hand side is bounded because
$\{C_{\alpha\beta}(x,\eta (u_n),\zeta_J(\nabla^mu_n))\}$ is bounded
in $L^{r}(\Omega ,{\sigma}_{\alpha}^{-r_\alpha}w_{\beta}^{-\frac{r_\alpha}{p}})$
 and $\{D^{\beta}u_n\}$ is bounded in $L^p (\Omega,w_{\beta})$.\\
Thanks to $s_\alpha\geq 1$, the sequences
$\{{\sigma}_{\alpha}^{-1}C_{\alpha\beta}(x,\eta
(u_n),\zeta_J(\nabla^mu_n))D^{\beta}u_n,  |\alpha| <m_1 ,
\beta\in J^{c}\}$ remain bounded in $L^1 (\Omega)$. Since
\begin{align*}
&\int_{\Omega}|C_{\alpha\beta}(x,\eta (u_n),
 \zeta_J(\nabla^mu_n))D^{\beta}u_n (D^{\alpha}u_n -D^{\alpha}u)|\,dx\\
&\leq \sup_{x\in \Omega }(|(D^{\alpha}u_n -D^{\alpha}u){\sigma}_{\alpha}|)
\int_{\Omega}|C_{\alpha\beta}(x,\eta (u_n),\zeta_J(\nabla^mu_n))
D^{\beta}u_{n}{\sigma}_{\alpha}^{-1}|\,dx
\end{align*}
it follows that
$$
\lim_{n\to \infty}\int_{\Omega}|C_{\alpha\beta}(x,\eta
(u_n),\zeta_J(\nabla^mu_n))D^{\beta}u_n (D^{\alpha}u_n
-D^{\alpha}u)|\,dx = 0
$$
(because ${ \sup_{x\in \Omega }(|(D^{\alpha}u_n
-D^{\alpha}u){\sigma}_{\alpha}|)\to 0 )}$.
 Which gives
\begin{equation}
\lim_{n\to \infty}\sum_{|\alpha|< m_1}
\sum_{\beta\in J^c}\int_{\Omega}C_{\alpha\beta}(x,\eta (u_n),
\zeta_J(\nabla^mu_n))D^{\beta}u_n (D^{\alpha}u_n -D^{\alpha}u)\,dx=0.
\label{e4.10}
\end{equation}
Combining \eqref{e4.9} and \eqref{e4.10} we obtain \eqref{e4.8}.

\noindent\textbf{Step (3)}
We shall prove that
\begin{equation}
\begin{aligned}
&\lim_{n\to \infty} \sum_{\alpha\in J}\int_\Omega B_\alpha(x,\eta(u_n),
\zeta_J(\nabla^mu_n))\\
&-B_\alpha(x,\eta(u_n), \zeta_J(\nabla^mu)))(D^\alpha u_n-D^\alpha
u)\,dx=0
\end{aligned} \label{e4.11}
\end{equation}
 and that
\begin{equation}
\begin{aligned}
&\lim_{n\to \infty}\int_\Omega \sum_{\alpha\in J^c}(B_\alpha(x,\eta(u_n),
\zeta_{J^c}(\nabla^mu_n))\\
&-B_\alpha(x,\eta(u_n),\zeta_{J^c}(\nabla^mu)))(D^\alpha u_n-D^\alpha u)\,dx=0.
\end{aligned}\label{e4.12}
\end{equation}
Combining \eqref{e4.2}, \eqref{e4.5} and \eqref{e4.8} one obtain
\begin{equation}
\begin{aligned}
&\limsup_{n\to \infty}\sum_{\alpha\in J}\int_\Omega B_\alpha(x,\eta(u_n),
 \zeta_J(\nabla^mu_n))(D^\alpha u_n-D^\alpha u)\,dx\\
&+\sum_{\alpha\in J^c}\int_\Omega B_\alpha(x,\eta(u_n),
 \zeta_{J^c}(\nabla^mu_n))(D^\alpha u_n-D^\alpha u)\,dx\leq 0.
\end{aligned} \label{e4.13}
\end{equation}
Thanks to \eqref{e4.4} and (H2') one deduce that
\begin{gather*}
B_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu))\to B_\alpha(x,\eta(u),
\zeta_J(\nabla^mu))\quad \mbox{in }L^{p'}(\Omega,w_\alpha^*),\; \alpha\in J\\
B_\alpha(x,\eta(u_n),\zeta_{J^c}(\nabla^mu))\to
B_\alpha(x,\eta(u),\zeta_{J^c}(\nabla^mu))\quad
\mbox{in }L^{p'}(\Omega,w_\alpha^*),\; \alpha\in J^c.
\end{gather*}
Since $D^\alpha u_n\rightharpoonup D^\alpha u$ in
$L^p(\Omega,w_\alpha) $ for all $|\alpha|=m$, one can write
\begin{equation}
\begin{gathered}
 \lim_{n\to \infty}\int_\Omega \sum_{\alpha\in J}B_\alpha(x,\eta(u_n),
\zeta_J(\nabla^mu))(D^\alpha u_n-D^\alpha u)\,dx=0\,, \\
 \lim_{n\to \infty}\int_\Omega \sum_{\alpha\in
J^c}B_\alpha(x,\eta(u_n),\zeta_{J^c}(\nabla^mu))(D^\alpha
u_n-D^\alpha u)\,dx=0\,.
\end{gathered}\label{e4.14}
\end{equation}
Combining \eqref{e4.13}, \eqref{e4.14}, $\eqref{L-L-J}$ and
\eqref{w.L-L-Jc} we conclude the assertions \eqref{e4.11} and \eqref{e4.12}.

\noindent\textbf{Step (4)}
To prove the relation \eqref{e4.3}, it
suffices to show the following assertions:
\textbf{(i)} For every $v\in V$,
\begin{equation}
\begin{aligned}
&\lim_{n\to \infty}\int_\Omega \sum_{\alpha\in
J}B_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))D^\alpha v \,dx\\
&=\int_\Omega \sum_{\alpha\in
J}B_\alpha(x,\eta(u),\zeta_J(\nabla^mu))D^\alpha v \,dx.
\end{aligned} \label{e4.15}
\end{equation}
\textbf{(ii)} For every $v\in V$,
\begin{equation}
\begin{aligned}
&\lim_{n\to \infty}\int_\Omega \sum_{m_1\leq |\alpha|\leq
m-1}L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))D^\alpha v \,dx\\
&=\int_\Omega \sum_{m_1\leq |\alpha|\leq m-1}
L_\alpha(x,\eta(u),\zeta_J(\nabla^mu))D^\alpha v \,dx.
\end{aligned} \label{e4.16}
\end{equation}

\textbf{(iii)} For every $v\in V$,
\begin{equation}
\begin{aligned}
&\lim_{n\to \infty}\int_\Omega \sum_{|\alpha|<
m_1}L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))D^\alpha v\,dx\\
&=\int_\Omega \sum_{|\alpha|<
m_1}L_\alpha(x,\eta(u),\zeta_J(\nabla^mu))D^\alpha v \,dx.
\end{aligned}\label{e4.17}
\end{equation}

\textbf{(iv)} For every $v\in V$,
\begin{equation}
\begin{aligned}
&\lim_{n\to\infty} \int_\Omega
\sum_{|\alpha|\leq m-1}\sum_{\beta\in J^c}
C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n))D^\beta u_nD^\alpha v\\
&= \int_\Omega \sum_{|\alpha|\leq m-1}\sum_{\beta\in J^c}
C_{\alpha\beta}(x,\eta(u),\zeta_J(\nabla^mu))D^\beta uD^\alpha v.
\end{aligned} \label{e4.18}
\end{equation}

\textbf{(v)} For every $v\in V$,
\begin{equation}
\begin{aligned}
&\lim_{n\to\infty}\int_\Omega \sum_{\alpha\in
J^c}(B_\alpha(x,\eta(u_n),\zeta_{J^c}(\nabla^mu_n))D^\alpha v\,dx\\
&= \int_\Omega\sum_{\alpha\in
J^c}(B_\alpha(x,\eta(u),\zeta_{J^c}(\nabla^mu))D^\alpha v\,dx.
\end{aligned} \label{e4.19}
\end{equation}

\begin{proof}[Proof of assertions (i)and (ii)]
Invoking  Landes \cite[lemma 6]{la}, we obtain
from \eqref{e4.11} and the strict monotonicity \eqref{L-L-J} that
\begin{equation}
D^\alpha u_n\to D^\alpha u\quad\text{a.e  in $\Omega$ for each }
\alpha\in J,\label{e4.20}
\end{equation}
which gives
\begin{gather*}
B_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))\to
B_\alpha(x,\eta(u), \zeta_J(\nabla^mu))\quad\text{a.e. in }\Omega\;
\forall\alpha\in J\,,
\\
L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))\to
L_\alpha(x,\eta(u),\zeta_J(\nabla^mu))\\
\quad \text{ a.e in }\Omega\; \forall  m_1\leq|\alpha|\leq m-1.
\end{gather*}
>From the growth condition (H2'), the sequence
$\{B_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n)),\alpha\in J\}$
(resp $\{L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))\ \
m_1\leq |\alpha|\leq m-1\}$) are bounded in
$L^{p'}(\Omega,w_\alpha^*)$ (resp.
$L^{q'(\alpha)}(\Omega,\sigma_\alpha^*))$, hence by Lemma \ref{lem4.2} we
have
$$
B_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))\rightharpoonup B_\alpha(x,\eta(u),
\zeta_J(\nabla^mu))
$$
in $L^{p'}(\Omega,w_\alpha^*)$ for all $\alpha\in J$
and
$$
L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))\rightharpoonup L_\alpha(x,\eta(u),
\zeta_J(\nabla^mu))
$$
in $L^{q'(\alpha)}(\Omega,\sigma_\alpha^*)$
for all $m_1\leq |\alpha|\leq m-1$,
which implies (i) and (ii).
\end{proof}

\begin{proof}[Proof of assertion (iii)]
In virtue of the growth condition (H2') we have for all $v\in V$ and
all $|\alpha|<m_1$
\begin{align*}
|L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))D^\alpha v| 
&\leq |D^\alpha v|G(k(x,\eta(u_n)))\sigma_\alpha
\Big(\hat g_\alpha(x)
 +\tilde c_\alpha\sum_{\beta\in J}w_\beta |D^\beta u_n|^p\\
&\quad + \tilde c_\alpha\sum_{m_1\leq|\beta|\leq m-1}\sigma_\beta |D^\beta
u_n|^{q(\beta)}\Big).
\end{align*}
Since $G(k(x,\eta (u_n)))\leq c_1$ and
$ \sup_{x\in \Omega}(|D^{\alpha}v\sigma_{\alpha}|)\leq c_2$
for all $|\alpha|<m_1$, where $c_i (i = 1,2)$ are some positive constants,
it follows that
\begin{align*}
&|L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))D^\alpha v|\\
& \leq c\Big(\hat g_\alpha(x)+c_\alpha\sum_{\beta\in J}w_\beta |D^\beta
u_n|^p+ c_\alpha\sum_{m_1\leq|\beta|\leq m-1}\sigma_\beta |D^\beta
u_n|^{q(\beta)}\Big)=g_n.
\end{align*}
 It follows from \eqref{e4.4} and \eqref{e4.20}
that
$$
L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))\to
L_\alpha(x,\eta(u),\zeta_J(\nabla^mu))\quad \mbox{a.e. in }\Omega\;
\forall  |\alpha|< m_1
$$
and
$$
g_n\to g=c \Big(\hat g_\alpha(x)+c_\alpha\sum_{\beta\in J} w_\beta |D^\beta
u|^p+ c_\alpha\sum_{m_1\leq|\beta|\leq m-1}\sigma_\beta |D^\beta
u|^{q(\beta)}\Big)a.e \mbox{a.e. in }\Omega.
$$

\begin{lemma} \label{lem4.3}
$D^\beta u_n\to D^\beta u $ as $n\to\infty$ in
$L^p(\Omega, w_\beta)$ for all $\beta\in J$.
\end{lemma}

By \eqref{e4.4} and Lemma \ref{lem4.3} we obtain
$$
\int_\Omega g_n\,dx\to \int_\Omega g\,dx.
$$
By the generalized Lebesgue theorem we have,
$$
\int_\Omega L_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu_n))D^\alpha v\,dx
\to \int_\Omega
L_\alpha(x,\eta(u),\zeta_J(\nabla^mu))D^\alpha v\,dx
$$
for all $|\alpha|<m_1$ which implies \eqref{e4.17}.
\end{proof}

\begin{proof}[Proof of assertion (iv)]
 By \eqref{e4.4} and \eqref{e4.20} we have for each $|\alpha|\leq m-1$
and each $\beta\in J^c$,
$$
C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n))\to
C_{\alpha\beta}(x,\eta(u),\zeta_J(\nabla^mu))\quad \mbox{a.e. in
}\Omega.
$$
So, from (H2') the sequences
$\{C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n)), m_1\leq
|\alpha|\leq m-1 \mbox{ and } \beta\in J^c\}$
(resp.
$\{C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n)), |\alpha|<
m_1 \mbox{ and  }\beta\in J^c\})$ remain bounded in
$L^{r_\alpha}(\Omega , {\sigma}_\alpha^{-\frac{{r_\alpha}}{q(\alpha)}}
 w_\beta^{-\frac{{r_\alpha}}{p}})$  (resp.  $L^{{r_\alpha}}(\Omega ,
{\sigma}_\alpha^{-r_\alpha} w_\beta^{-\frac{{r_\alpha}}{p}})$).
Then  Lemma \ref{lem4.1} yields
$$
C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n))\to
C_{\alpha\beta}(x,\eta(u),\zeta_J(\nabla^mu))
$$
 in $L^{q}(\Omega , {\sigma}_\alpha^{-\frac{q}{q(\alpha)}}
w_\beta^{-\frac{q}{p}})$ for all $q< r_\alpha$, all
$m_1\leq |\alpha|\leq m-1$ and all $\beta\in J^c$.
Lemma \ref{lem4.1} also yields
$$
C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n))\to C_{\alpha\beta}
(x,\eta(u),\zeta_J(\nabla^mu))
$$
in  $L^q(\Omega , {\sigma}_\alpha^{-q} w_\beta^{-\frac{q}{p}})$
 for all $q<r_\alpha$, all $|\alpha|< m_1$  and all
$\beta\in J^c$.

Let $s_\alpha$ such that $\frac{1}{s_\alpha}=\frac{1}{p}+\frac{1}{q(\alpha)}$.
Remark that $r_\alpha>s_\alpha'=\frac{s_\alpha}{s_\alpha-1}$ for
$m_1\leq |\alpha|\leq m-1$ and since $p'<r_\alpha$ for $|\alpha|< m_1$ one
has
$$
C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n)) \to
C_{\alpha\beta}(x,\eta(u),\zeta_J(\nabla^mu))
$$
 in $L^{s_\alpha'}(\Omega , {\sigma}_\alpha^{-\frac{s_\alpha'}{q(\alpha)}}
w_\beta^{-\frac{s_\alpha'}{p}})$
for all $m_1\leq |\alpha|\leq m-1$. Also one has
\begin{equation}
C_{\alpha\beta}(x,\eta(u_n),\zeta_J(\nabla^mu_n))\sigma_\alpha^{-1}
\to
C_{\alpha\beta}(x,\eta(u),\zeta_J(\nabla^mu))\sigma_\alpha^{-1}
\label{e4.21}
\end{equation}
in $L^{p'}(\Omega , w_\beta^{-\frac{p'}{p}})$
for all $|\alpha|< m_1$.

\begin{lemma} \label{lem4.4}
For all $v\in V$, one has
\begin{enumerate}
\item $D^\beta u_nD^\alpha v \rightharpoonup D^\beta u D^\alpha v$
 in $L^{s_\alpha}(\Omega ,\sigma_\alpha^{\frac{s_\alpha}{q(\alpha)}}
w_\beta^{\frac{s_\alpha}{p}})$ for each $ m_1\leq |\alpha|\leq m-1$
and each $\beta\in J^c$.

\item $D^\beta u_nD^\alpha v\sigma_\alpha\rightharpoonup D^\beta u
D^\alpha v\sigma_\alpha$  in
$ L^p(\Omega , w_\beta)
\mbox{ for each }\ |\alpha|< m_1$ and each $\beta\in J^c$.
\end{enumerate}
\end{lemma}

In view of \eqref{e4.21} and Lemma \ref{lem4.4} we conclude \eqref{e4.18}.
\end{proof}

\begin{proof}[Proof of assertion (v)]
First we show that
\begin{equation}
\int_\Omega \sum_{\alpha\in J^c}(B_\alpha(x,\eta(u),
\zeta_{J^c}(v))-h_\alpha)(v_\alpha-D^\alpha u)\,dx\geq 0 \label{e4.22}
\end{equation}
for all $v=(v_\alpha)\in \prod_{|\alpha|=m}L^p(\Omega,w_\alpha)$,
where $h_\alpha$ stands for the weak limit of
$\{B_\alpha(x,\eta(u_n),\zeta_{J^c}(\nabla^mu_n)),\alpha\in
J^c\}$ in $L^{p'}(\Omega,w_\alpha^*)$. Indeed by \eqref{e4.12} we have,
$$
\limsup_{n\to \infty}\int_\Omega \sum_{\alpha\in J^c}B_\alpha(x,\eta(u_n),
\zeta_{J^c}(\nabla^mu_n))(D^\alpha u_n-D^\alpha u)\,dx\leq 0,
$$
implies 
\begin{equation}
\limsup_{n\to \infty}\int_\Omega \sum_{\alpha\in J^c}B_\alpha(x,\eta(u_n),
\zeta_{J^c}(\nabla^mu_n))D^\alpha u_n\,dx
\leq \int_\Omega \sum_{\alpha\in J^c}h_\alpha D^\alpha u\,dx
\label{e4.23}
\end{equation}
and from weak Leray-Lions condition \eqref{w.L-L-Jc},  for
any $v=(v_\alpha)\in { \prod_{|\alpha|=m}L^p(\Omega,w_\alpha)}$,
we obtain
\begin{align*}
&\int_\Omega\sum_{\alpha\in J^c}B_\alpha(x,\eta(u_n),
\zeta_{J^c}(\nabla^mu_n))D^\alpha u_n \,dx\\
&\geq \int_\Omega\sum_{\alpha\in J^c}B_\alpha(x,\eta(u_n),
\zeta_{J^c}(\nabla^mu_n))v_\alpha \,dx\\
&\quad +\int_\Omega\sum_{\alpha\in
J^c}B_\alpha(x,\eta(u_n),\zeta_{J^c}(v))(D^\alpha
u_n-v_\alpha)\,dx.
\end{align*}
Letting $n\to\infty$ we conclude by \eqref{e4.23} that
 $$
\int_\Omega\sum_{\alpha\in J^c}h_\alpha D^\alpha u
\,dx\geq\int_\Omega\sum_{\alpha\in J^c}h_\alpha v_\alpha\,dx
+\int_\Omega\sum_{\alpha\in
J^c}B_\alpha(x,\eta(u),\zeta_{J^c}(v))(D^\alpha u-v_\alpha)\,dx
$$
and hence \eqref{e4.22} follows. Choosing $v=D^\alpha u+t\hat w$ with
$t>0$, $\hat w=(\hat w_\alpha)\in {\prod_{|\alpha|=m}L^p(\Omega,w_\alpha)}$
and letting $t\to 0$ we obtain
$h_\alpha =B_\alpha(x,\eta(u),\zeta_{J^c}(\nabla^mu))$
a.e. in $\Omega$ which implies \eqref{e4.19}.
\end{proof}

\textbf{(b)} We shall prove that
\begin{equation}
\liminf_{n \to\infty}\langle Tu_n,u_n \rangle \geq
\langle Tu,u\rangle .\label{e4.24}
\end{equation}
In view of monotonicity condition
$\eqref{L-L-J}$ and \eqref{w.L-L-Jc} we have
\begin{align*}
&\int_\Omega \sum_{\alpha\in J}B_\alpha(x,\eta(u_n),
 \zeta_J(\nabla^mu_n))D^\alpha u_n
 +\int_\Omega \sum_{\alpha\in J^c}B_\alpha(x,\eta(u_n),
 \zeta_{J^c}(\nabla^mu_n))D^\alpha u_n \\
&\geq\int_\Omega \sum_{\alpha\in J}B_\alpha(x,\eta(u_n),
 \zeta_J(\nabla^mu_n))D^\alpha u \\
&\quad +\int_\Omega \sum_{\alpha\in J}B_\alpha(x,\eta(u_n),\zeta_J(\nabla^mu))
 (D^\alpha u_n-D^\alpha u)\\
&\quad +\int_\Omega \sum_{\alpha\in
 J^c}B_\alpha(x,\eta(u_n),\zeta_{J^c}(\nabla^mu_n))D^\alpha u \\
&\quad +\int_\Omega \sum_{\alpha\in J^c}B_\alpha(x,\eta(u_n),
 \zeta_{J^c}(\nabla^mu))(D^\alpha u_n-D^\alpha u).
\end{align*}
Letting $n\to \infty $ and using \eqref{e4.5} and \eqref{e4.8}we
obtain \eqref{e4.24}.

\begin{proof}[Proof of Lemma \ref{lem4.3}]
Let $E$ be a measurable subset of $\Omega$, in view of steps 1, 2, 3 and 4
 in \cite[Lemma 2.7]{dr-ku-ni2}, we obtain
\begin{equation}
\lim_{\mathop{\rm meas}E\to 0}\int_E \sum_{\beta\in J}w_\beta
|D^\beta u_n(x)|^p\,dx=0\label{e4.25}
\end{equation}
uniformly with respect to $n\in \mathbb{N}$, i.e the sequence
$(w_\beta |D^\beta u_n|^p)$ is equi-integrable.
 And due to \eqref{e4.20} we have
$$
w_\beta
|D^\beta u_n|^p\to w_\beta |D^\beta u|^p\quad\text{a.e. }
\forall \beta\in J.
$$
 Since $\mathop{\rm meas}(\Omega)<\infty$,  by
Vitali's theorem we obtain
$D^\beta u_n\to D^\beta u $ in $L^p(\Omega,w_\beta)$ for all $\beta\in J$.
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem4.4}]
(1) Let $m_1\leq |\alpha|\leq m-1$ and $\beta \in J^c$ fixed.
Let $\varphi\in L^{s_\alpha'}(\Omega,
\sigma_\alpha^{-\frac{s_\alpha'}{q(\alpha)}}w_\beta^{-\frac{s_\alpha'}{p}})$.\\
Since, $\frac{1}{p'}=\frac{1}{s_\alpha'}+\frac{1}{q(\alpha)}$,
 by H\"older's inequality we obtain
$$
\int_\Omega|D^\alpha v\varphi|^{p'}w_\beta^{1-p'}\leq
\Big(\int_\Omega|D^\alpha
v|^{q(\alpha)}\sigma_\alpha\Big)^{\frac{p'}{q(\alpha)}}
\Big(\int_\Omega|\varphi|^{s_\alpha'}w_\beta^{\frac{s_\alpha'(1-p')}{p'}}
\sigma_\alpha^{\frac{-s_\alpha'}{q(\alpha)}}
\Big)^{\frac{p'}{s_\alpha'}}<\infty
$$
(because $\frac{s_\alpha'(1-p')}{p'}=\frac{-s_\alpha'}{p})$.
 Then $D^\alpha v\varphi\in L^{p'}(\Omega,w_\beta^{1-p'})$ and since
$D^\beta u_n \rightharpoonup D^\beta u$ in $L^p(\Omega,w_\alpha)$,
we have
$$
\int_\Omega D^\beta u_nD^\alpha v \varphi\to
\int_\Omega D^\beta u_nD^\alpha v \varphi\quad \mbox{for all }
\varphi\in
L^{s_\alpha'}(\Omega,\sigma_\alpha^{-\frac{s_\alpha'}{q(\alpha)}}
w_\alpha^{-\frac{s_\alpha'}{p}});
$$
i.e.,
$$
D^\beta u_nD^\alpha v \rightharpoonup D^\beta uD^\alpha v
\in L^{s_\alpha}(\Omega,\sigma_\alpha^{\frac{s_\alpha}{q(\alpha)}}
w_\beta^{\frac{s_\alpha}{p}})
$$
for all $m_1\leq |\alpha|\leq m-1$  and all
$\beta\in J^c$.

(2) Let $|\alpha|<m_1$ and $\beta\in J^c$ and let $\varphi\in
L^{p'}(\Omega, w_\beta^*)$. Thanks to $ D^\alpha v\in C(\Omega,
\sigma_\alpha)\ \ \forall v\in V$, we have
$D^\alpha v\sigma_\alpha\varphi\in L^{p'}(\Omega, w_\beta^*)$. Since
$D^\beta u_n \rightharpoonup D^\beta u$ in $L^p(\Omega,w_\alpha)$,
we have
$$
\int_\Omega D^\beta u_nD^\alpha
v\sigma_\alpha\varphi\,dx\to \int_\Omega D^\beta u
D^\alpha v\sigma_\alpha\varphi\,dx\quad \mbox{ for all }\varphi\in
L^{p'}(\Omega,w_\beta^*).
$$
\end{proof}

\begin{remark} \label{rmk4.1} \rm
Note that, the ellepticity condition (H3) is only used to prove
\eqref{e4.25} (see step 3  in \cite[lemma 2.7]{dr-ku-ni2}, which concerns
only the equality \eqref{e4.17} corresponding to a terms $L_\alpha$ with
$|\alpha|<m_1$).
\end{remark}

\section{Specific case}

Let $\Omega$ be a bounded open subset of $\mathbb{R}^N$ satisfying the
cone condition.
In the sequel we assume in addition that the collection of weight
functions $w=\{w_\alpha(x), |\alpha|\leq m\}$ satisfies
$w_\alpha(x)=1$ for all $|\alpha|\leq m-1$, and the integrability
condition: There exists
$\nu\in ]\frac{N}{P},\infty[\cap [\frac{1}{P-1},\infty[$
such that
\begin{equation}
w_\alpha^{-\nu}\in L^1(\Omega) \quad \forall |\alpha|=m.\label{e5.1}
\end{equation}
Note that \eqref{e5.1} is stronger than \eqref{e2.2}.
Assumptions \eqref{e2.1} and \eqref{e5.1} imply
$$
\||u|\|_V=\Big(\sum_{|\alpha|=m}\int_\Omega |D^\alpha u|^pw_\alpha(x)
\,dx\Big)^{1/p}
$$
is a norm defined on $V=W_0^{m,p}(\Omega,w)$ and it's equivalent
to \eqref{e2.3}.
Let
\begin{equation}
m_1=\frac{mp\nu-N(\nu+1)}{p\nu}=m-\frac{N}{p_1}\quad\text{with}\quad
p_1=\frac{p\nu}{\nu+1}.\label{e5.2}
\end{equation}

\begin{remark}[\cite{dr-ku-ni1}] \label{rmk5.1}\rm
Under the above assumption  the following continuous imbeddings hold:
(i) For $k<m_1$,
$$
W^{m,p}(\Omega,w)\hookrightarrow C^k(\bar\Omega).
$$
(ii) For $k=m_1$, with arbitrary $r, 1<r<\infty$,
 $$
W^{m,p}(\Omega,w)\hookrightarrow W^{k,r}(\Omega)\,.
$$
(iii) For $k>m_1$,
 $$
W^{m,p}(\Omega,w)\hookrightarrow W^{k,r_k}(\Omega)
$$
where $r_k$ satisfies  $1<r_k\leq q_k=\frac{p\nu N}{N(\nu+1)-p\nu(m-k)}$.

Moreover the imbedding (i) and (ii) are compact and (iii) is compact
if $r_k<q_k$.
\end{remark}

Now, we define
$$
H^{m-1,q}(\Omega,\sigma)={ \prod_{|\beta|\leq m-1}X_\beta},
$$
where $X_\beta=L^{q(\beta)}(\Omega,\sigma_\beta),q(\beta)>1$
 for $m_1\leq | \beta|\leq m-1$
and $X_\beta=C^{|\beta|}(\Omega,\sigma_\beta)$ for $|\beta|<m_1$.

Also we define the assumption
\begin{itemize}
\item[(H4)] Let
$$
1<q(\beta)<q_{|\beta|} =\frac{p\nu N}{N(\nu+1)-p\nu(m-|\beta|)}
$$
for $m_1<|\beta|\leq m-1$ and $q(\beta)$ arbitrary if $|\beta|=m_1$
and $\sigma_\beta\equiv 1$ for all $\beta\leq m-1$.
\end{itemize}

\begin{remark} \label{rmk5.2} \rm
If (H4) is satisfied, then by Remark \ref{rmk5.1},
$$
W^{m,p}(\Omega,w)\hookrightarrow \hookrightarrow
H^{m-1,q}(\Omega)
$$
which implies immediately that (H2')(iii)
with $\sigma\equiv 1$.
\end{remark}

\begin{theorem} \label{thm5.1}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^d$. And
assume that \eqref{e2.1}, \eqref{e5.1}, (H1'), (H2')(,i,ii), (H3),
(H4), \eqref{L-L-J} and \eqref{w.L-L-Jc} are satisfied.
Then the operator $T$ defined in \eqref{e2.11} is pseudo-monotone
in $V=W_0^{m,p}(\Omega,w)$.

If in addition the degeneracy \eqref{D2} is satisfied, then the
degenerate boundary-value problem from  \eqref{E} has at least one
 solution $u\in V$.
\end{theorem}

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\end{document}