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Variational analysis for an indefinite quasilinear problem with variable exponent

  • Mabrouk Bouslimi EMAIL logo , Khaled Kefi and Felician-Dumitru Preda
Published/Copyright: January 19, 2012
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Advances in Pure and Applied Mathematics
From the journal Volume 3 Issue 1

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.: -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-January

© 2012 by Walter de Gruyter Berlin Boston

Articles in the same Issue

  1. Prelims
  2. Warfield invariants of normed unit groups in abelian group rings
  3. Opdam's hypergeometric functions: product formula and convolution structure in dimension 1
  4. On generalized Laplace equation and nonlinear operators
  5. Steady state solutions to the conserved Kuramoto–Sivashinsky equation
  6. Variational analysis for an indefinite quasilinear problem with variable exponent
  7. A probabilistic counterpart of the Askey scheme for continuous polynomials
  8. A characterisation of the Weyl transform
https://doi.org/10.1515/apam.2011.011
Keywords for this article
-Laplace operator; generalized Sobolev spaces; mountain pass theorem; Ekeland's variational principle; weak solution

Articles in the same Issue

  1. Prelims
  2. Warfield invariants of normed unit groups in abelian group rings
  3. Opdam's hypergeometric functions: product formula and convolution structure in dimension 1
  4. On generalized Laplace equation and nonlinear operators
  5. Steady state solutions to the conserved Kuramoto–Sivashinsky equation
  6. Variational analysis for an indefinite quasilinear problem with variable exponent
  7. A probabilistic counterpart of the Askey scheme for continuous polynomials
  8. A characterisation of the Weyl transform
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