Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions

In this paper we deal with a class of non-differentiable functionals defined on a real reflexive Banach space X and depending on a real parameter of the form , where is a sequentially weakly lower semicontinuous C (1) functional, (Y, Z Banach spaces) are two locally Lipschitz functionals, T : X -> Y, S : X -> Z are linear and compact operators and lambda > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each lambda > 0 and there exists lambda* > 0 such that the functional possesses at least four nonsmooth critical points. As an application, we study a nonhomogeneous differential inclusion involving the p(x)-Laplace operator whose weak solutions are exactly the nonsmooth critical points of some "energy functional" which satisfies the conditions required in our main result.