Journal of Convex Analysis 16 (2009), No. 2, 523--541
Copyright Heldermann Verlag 2009
Weak and Entropy Solutions to Nonlinear Elliptic Problems with Variable Exponent
Stanislas Ouaro
Laboratoire d'Analyse Mathématique des Equations, Institut des Sciences Exactes et Appliquées, Université de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso
souaro@univ-ouaga.bf
Sado Traore
Laboratoire d'Analyse Mathématique des Equations, Institut des Sciences Exactes et Appliquées, Université de Bobo Dioulasso, 01 BP 1091, Bobo-Dioulasso 01, Burkina Faso
sado@univ-ouaga.bf
[Abstract-pdf]
We study the boundary value problem $-div(a(x,\nabla u))=f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ and $div(a(x,\nabla u))$ is a $p(x)$-Laplace type operator. We obtain the existence and uniqueness of an entropy solution for $L^{1}$-data $f$ independent of $u$, the existence of weak energy solution for general data $f$ dependent of $u$ where the variable exponent $p(.)$ is not necessarily continuous.
Keywords: Generalized Lebesgue-Sobolev spaces, weak energy solution, entropy solution, p(x)-Laplace operator, electrorheological fluids.
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