Two positive solutions for p-Laplacian equations with convex and concave non-linearities and a non-smooth potential

Abstract

We consider a non-linear elliptic problem driven by the p-Laplacian and a non-smooth potential (hemivariational inequality). We assume that the non-linearity involves the combined effects of both concave (i.e. p-sublinear) and convex (i.e. p-superlinear) terms. We show the existence of two positive solutions, using non-smooth critical point theory and the Ekeland variational principle.

Keywords:

• p-Laplacian
• generalized subdifferential
• concave and convex non-linear terms
• non-smooth critical point theory
• Ekeland variational principle
• positive solutions

AMS Subject Classifications::

• 35J65
• 35J70

Acknowledgements

The authors wish to thank the referee for his/her corrections and remarks.