\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
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|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|