Critical Points for Non Differentiable Functionals

In this paper we deal with the existence and multiplicity of critical points for non differentiable integral functionals defined in the Sobolev space W1,p(Ω) (p > 1) by: 0 where Ω is a bounded open set of RN, with N ≥ 3 and p ≤ N. Under natural assump- tions F turns out to be not Frech ́et differentiable on W1,p(Ω), thus classical critical point theory cannot be applied. The existence of a critical point of F has been proved in [1] by means of a suitable extension of the Ambrosetti-Rabinowitz minimax result. Here we get existence and multiplicity of critical points of F applying a generalization of a symmetric version of the Mountain-Pass theorem proved in [10]. We will follow the same procedure of [7] where the quasilinear case has been treated.