Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 12, pp. 1-19.

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

Authors: Bernard K. Bonzi (Univ. de Ouagadougou, Burkina Faso)
         Ismael Nyanquini (Univ. Polytech. de Bobo Dioulasso, Burkina Faso)
	 Stanislas Ouaro  (Univ. de Ouagadougou, Burkina Faso)

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.