A multiplicity results for a singular equation involving the p(x)-Laplace operator

Abstract

The purpose of this paper is to study the singular problem involving the p(x)-Laplace operator:(Section.Display)

where be a bounded domain with boundary, is a positive parameter and p(x),  and f(x, u) are assumed to satisfy the assumptions (H0)–(H4) in the Introduction. We employ variational techniques in order to show the existence of a number such that problem has two solutions for one solution for and no solutions for To obtain multiple (at least two distinct, positive) solutions of problem , we need to prove two new results: a regularity result for solutions to problem in with some and a strong comparison principle.

Keywords:

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

AMS Subject Classifications:

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

Acknowledgements

The authors would like to thank the anonymous referees for their carefully reading this paper and their useful comments.

Notes

No potential conflict of interest was reported by the authors.