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.

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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

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