Multiple positive solutions for a class of quasilinear problems with distinct potentials

Abstract

In this paper, we concern with the following quasilinear problemwhere is the -Laplacian of for , is a small parameter, is a continuous superlinear and subcritical nonlinearity. Suppose that has at least one minimum and has at least one maximum. We first prove that there are two families of positive solutions for small, which concentrate, respectively, on the set of minimal points of and the set of maximal points of . In addition, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

Keywords:

• variational method
• quasilinear equations
• positive ground state solutions

Acknowledgements

The authors would like to thank the referees for giving valuable comments and suggestions.

Notes

This work was supported by Natural Science Foundation of China [grant number 11201186]; Natural Science Foundation of Jiangsu Province [grant number BK2012282]; Jiangsu University foundation [grant number 11JDG117]; China Postdoctoral Science Foundation funded project [grant number 2012M511199], [grant number 2013T60499]; NSFC [grant number 11071038], [grant number 11171135].