Electron. J. Diff. Equ., Vol. 2013 (2013), No. 68, pp. 1-27.

Existence and regularity of entropy solutions for strongly nonlinear p(x)-elliptic equations

Elhoussine Azroul, Hassane Hjiaj, Abdelfattah Touzani

Abstract:
This article is devoted to study the existence of solutions for the strongly nonlinear p(x)-elliptic problem
$$\displaylines{ - \operatorname{div} a(x,u,\nabla u) + g(x,u,\nabla u) = f- \operatorname{div} \phi(u) \quad \hbox{in } \Omega, \cr u = 0 \quad \hbox{on } \partial\Omega, }$$
with $ f\in L^1(\Omega) $ and $ \phi \in C^{0}(\mathbb{R}^{N})$, also we will give some regularity results for these solutions.

Submitted May 22, 2012. Published March 8, 2013.
Math Subject Classifications: 35J20, 35J25, 35J60.
Key Words: Sobolev spaces with variable exponents; entropy solutions; strongly nonlinear elliptic equations; boundary value problems.

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Elhoussine Azroul
University of Fez, Faculty of Sciences Dhar El Mahraz
Laboratory LAMA, Department of Mathematics
B.P. 1796 Atlas Fez, Morocco
email: azroul_elhoussine@yahoo.fr
Hassane Hjiaj
University of Fez, Faculty of Sciences Dhar El Mahraz
Laboratory LAMA, Department of Mathematics
B.P. 1796 Atlas Fez, Morocco
email: hjiajhassane@yahoo.fr
Abdelfattah Touzani
University of Fez, Faculty of Sciences Dhar El Mahraz
Laboratory LAMA, Department of Mathematics
B.P. 1796 Atlas Fez, Morocco
email: atouzani07@gmail.com

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