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Advances in Pure and Applied Mathematics

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Variational analysis for an indefinite quasilinear problem with variable exponent

Mabrouk Bouslimi
  • Institut Préparatoire aux Études d'Ingénieurs de Tunis, Université de Tunis, Rue Jawaher Lel Nehru, 1008 Montfleury, Tunis, Tunisia
  • :
/ Khaled Kefi
  • Institut Supérieur du Transport et de la Logistique de Sousse, Université de Sousse, 12 Rue Abdallah Ibn Zoubeir, 4029-Sousse, Tunisia
  • :
/ Felician-Dumitru Preda
  • Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie 13, 050711 Bucharest, Romania
  • :
Published Online: 2012-01-19 | DOI: https://doi.org/10.1515/apam.2011.011

Abstract.

We study the nonlinear boundary value problem - div ((|u(x)|p1(x)-2+|u(x)|p2(x)-2u(x) =V1(x)|u|q(x)-2u-V2(x)|u|(x)-2u in , u=0 on , where is a bounded domain in N with smooth boundary, , are positive real numbers, p1, p2, q, are continuous functions on , V1 and V2 are weight functions in generalized Lebesgue spaces Ls1(x)() and Ls2(x)(), respectively, such that V1>0 in an open set 0 with |0|>0 and V20 on . We prove, under appropriate conditions that for any >0 there exists * sufficiently small such that for any (0,*) the above nonhomogeneous quasilinear problem has a nontrivial positive weak solution. The proof relies on some variational arguments based on Ekeland's variational principle.

Keywords.: p(x)$p(x)$-Laplace operator; generalized Sobolev spaces; mountain pass theorem; Ekeland's variational principle; weak solution


Received: 2011-04-20

Accepted: 2011-06-09

Published Online: 2012-01-19

Published in Print: 2012-01-01


Citation Information: Advances in Pure and Applied Mathematics. Volume 3, Issue 1, Pages 67–83, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152, DOI: https://doi.org/10.1515/apam.2011.011, January 2012

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