Electron. J. Diff. Equ., Vol. 2013 (2013), No. 220, pp. 1-13.

Anisotropic problems with variable exponents and constant Dirichlet conditions

Maria-Magdalena Boureanu, Cristian Udrea, Diana-Nicoleta Udrea

Abstract:
We study a general class of anisotropic problems involving $\vec p(\cdot)$-Laplace type operators. We search for weak solutions that are constant on the boundary by introducing a new subspace of the anisotropic Sobolev space with variable exponent and by proving that it is a reflexive Banach space. Our argumentation for the existence of weak solutions is mainly based on a variant of the mountain pass theorem of Ambrosetti and Rabinowitz.

Submitted April 8, 2013. Published October 4, 2013.
Math Subject Classifications: 35J25, 46E35, 35D30, 35J20.
Key Words: Anisotropic variable exponent Sobolev spaces; Dirichlet problem; existence of weak solutions; mountain pass theorem.

A corrigendum was posted on December 23, 2013. It adds a condition to the original problem, and adds six references. See the last page of this article.

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Maria-Magdalena Boureanu
Department of Applied Mathematics
University of Craiova
200585 Craiova, Romania
email: mmboureanu@yahoo.com
Cristian Udrea
Department of Applied Mathematics
University of Craiova
200585 Craiova, Romania
email: udrea.cristian2013@yahoo.com
Diana-Nicoleta Udrea
Department of Mathematics
University of Craiova
200585 Craiova, Romania
email: diannannicole@yahoo.com

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