Abstract

We study the boundary value problem −div(a(x,∇u))=f(x,u)-div(a(x,\nabla u))=f(x,u) in Ω\Omega, u=0u=0 on ∂Ω\partial \Omega, where Ω\Omega is a smooth bounded domain in RN\mathbb{R}^{N} and div(a(x,∇u))div(a(x,\nabla u)) is a p(x)p(x)-Laplace type operator. We obtain the existence and uniqueness of an entropy solution for L1L^{1}-data ff independent of uu, the existence of weak energy solution for general data ff dependent of uu where the variable exponent p(.)p(.) is not necessarily continuous.

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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

souaro@univ-ouaga.bf

Sado Traoré

Laboratoire d'Analyse Mathématique des Equations (LAME), Institut des Sciences Exactes et Appliquées, Université de Bobo Dioulasso, 01 BP 1091 Bobo-Dioulasso 01, Burkina Faso

sado@univ-ouaga.bf

Keywords

  • Generalized Lebesgue-Sobolev spaces
  • weak energy solution
  • entropy solution
  • p(x)-Laplace operator
  • electrorheological fluids

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Suggested citation

S. Ouaro, S. Traoré. “Weak and Entropy Solutions to Nonlinear Elliptic Problems with Variable Exponent.” Journal of Convex Analysis 16 (2009), No. 2, 523–541. https://doi.org/10.68381/jca16029

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.