Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition

Abstract

We discuss the problem – div(a(x, ∇u)) = m(x)|u| ^r(x)−2 u + n(x)|u| ^s(x)−2 u in Ω, where Ω is a bounded domain with smooth boundary in ℝ ^N (N ≥ 2), and div(a(x, ∇u)) is a p(x)-Laplace type operator with 1 < r(x) < p(x) < s(x). We show the existence of infinitely many nontrivial weak solutions in . Our approach relies on the theory of the variable exponent Lebesgue and Sobolev spaces combined with adequate variational methods and a variation of the Mountain Pass lemma and critical point theory.

Keywords:

• p(x)-Laplace operator
• nonuniform elliptic equations
• critical point
• multiple solutions
• Ekeland's variational principle
• Mountain Pass theorem

AMS Subject Classification:

• 35J60
• 35B30
• 35B40

View correction statement:

Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition

Acknowledgements

The authors thank the referees for their valuable suggestions and helpful corrections, which have improved the presentation of this article. This research project was supported by DUBAP -10-FF-15, Dicle University, Turkey.