Existence and multiplicity of solutions for a quasilinear equation involving the p(x)-Laplace operator

Abstract

The purpose of this paper is to study the following nonlinear problem involving the p(x)-Laplace operator:

where is a bounded domain with boundary, is a positive parameter, p(x),  and f(x, u) are assumed to satisfy the assumptions (H1)–(H4) in the introduction. We employ variational techniques in order to show the existence of a number such that problem admits two solutions for one solution for and no solutions for

Keywords:

• p(x)-Laplace operator
• quasilinear equation
• variational methods
• generalized Lebesgue Sobolev spaces

AMS Subject Classifications:

• 35J20
• 35J60
• 35J70
• 47J10
• 46E35

Acknowledgements

I want to thank the anonymous referees for carefully reading this paper and their useful comments.

Notes

No potential conflict of interest was reported by the authors.