Electron. J. Diff. Equ., Vol. 2012 (2012), No. 12, pp. 1-19.

Existence and uniqueness of weak and entropy solutions for homogeneous Neumann boundary-value problems involving variable exponents

Bernard K. Bonzi, Ismael Nyanquini, Stanislas Ouaro

Abstract:
In this article we study the nonlinear homogeneous Neumann boundary-value problem
$$\displaylines{ b(u)-\hbox{div} a(x,\nabla u)=f\quad \hbox{in } \Omega\cr a(x,\nabla u).\eta=0 \quad\hbox{on }\partial \Omega, }$$
where $\Omega$ is a smooth bounded open domain in $\mathbb{R}^{N}$, $N \geq 3$ and $\eta$ the outer unit normal vector on $\partial\Omega$. We prove the existence and uniqueness of a weak solution for $f \in L^{\infty}(\Omega)$ and the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

Submitted March 13, 2011. Published January 17, 2012.
Math Subject Classifications: 35J20, 35J25, 35D30, 35B38, 35J60.
Key Words: Elliptic equation; weak solution; entropy solution; Leray-Lions operator; variable exponent.

Show me the PDF file (325 KB), TEX file, and other files for this article.

Bernard K. Bonzi
Laboratoire d'Analyse Mathématique des Equations (LAME)
UFR. Sciences Exactes et Appliquées, Université de Ouagadougou
03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso
email: bonzib@univ-ouaga.bf
Ismael Nyanquini
Laboratoire d'Analyse Mathématique des Equations (LAME)
Institut des Sciences Exactes et Appliquées
Université Polytechnique de Bobo Dioulasso
01 BP 1091 Bobo-Dioulasso 01, Bobo Dioulasso, Burkina Faso
email: nyanquis@yahoo.fr
Stanislas Ouaro
Laboratoire d'Analyse Mathématique des Equations (LAME)
UFR. Sciences Exactes et Appliquées, Université de Ouagadougou
03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso
email: souaro@univ-ouaga.bf, ouaro@yahoo.fr

Return to the EJDE web page