Resonant ( p, 2)-equations with concave terms

Abstract

We consider a nonlinear, nonhomogeneous parametric elliptic Dirichlet equation driven by the sum of a p-Laplacian and a Laplacian (so-called (p,2)-equation) and with a nonlinearity involving a concave term which enters with a negative sign. By applying variational methods along with truncation and comparison techniques as well as Morse theory (critical groups), we show that the problem under consideration has at least five nontrivial solutions (four of them have constant sign) for all sufficiently small values of the parameter.

Keywords:

• concave term
• resonance
• nonlinear regularity
• nonlinear maximum principle
• critical groups

AMS Subject Classifications:

• 35J20
• 35J60
• 35J92
• 58E05

Acknowledgement

The authors wish to thank the anonymous referee for his/her corrections and remarks that improved the paper considerably.