Ground state solutions and multiple solutions for a p(x)-Laplacian equation in with periodic data

Abstract

In this work, we consider the following p(x)-Laplacian equation in

where f, V and p(x) are periodic in . Under some appropriate assumptions, we prove the existence of the ground state solutions via the generalized Nehari method due to Szulkin and Weth. Moreover, if f is odd in u, infinitely many pairs of geometrically distinct solutions are given. To the best of our knowledge, our results are new even in the constant exponent case.

Keywords:

• p(x)-Laplacian equation
• variable exponent Sobolev spaces
• ground state solutions
• multiplicity of solutions
• Nehari manifold

AMS Subject Classifications:

• 35A15
• 35J60

Acknowledgements

The authors would like to express their gratefulness to the referees for useful comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is Supported by NSFC [grant number 11301181]; China Postdoctoral Science Foundation, the second author is supported by Supported by NSFC [grant number 11301182]; Science and Technology Commission of Shanghai Municipality [grant number 13ZR1453400]; The third author is supported by Research Foundation of Heilongjiang Educational Committee [grant number 12541667].