Nontrivial solutions for a multivalued problem with strong resonance

Abstract

The Mountain-Pass Theorem of Ambrosetti and Rabinowitz (see [1]) and the Saddle Point Theorem of Rabinowitz (see [21]) are very important tools in the critical point theory of C1-functional. That is why it is natural to ask us what happens if the functional fails to be differentiable. The first who considered such a case were Aubin and Clarke (see [6]) and Chang (see [12]),who gave suitable variants of the Mountain-Pass Theorem for locally Lipschitz functionals which are denned on reflexive Banach spaces. For this aim they replaced the usual gradient with a generalized one, which was firstly defined by Clarke (see [13], [14]).As observed by Brezis (see [12, p. 114]), these abstract critical point theorems remain valid in non-reflexive Banach spaces.