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

Bonzi, Bernard K. ; Nyanquini, Ismael ; Ouaro, Stanislas

Electronic Journal of Differential Equations, Tome 2012 (2012).

Résumé

Summary: 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 $\mathrm{\Omega }$ is a smooth bounded open domain in ${\mathbb{R}}^N,N\ge 3$ and $\eta$ the outer unit normal vector on $\partial \mathrm{\Omega }$. We prove the existence and uniqueness of a weak solution for $f\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ 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.

Détail

Classification : 35J20, 35J25, 35D30, 35B38, 35J60
Mots-clés : elliptic equation, weak solution, entropy solution, Leray-lions operator, variable exponent