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

Abstract

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 $${\mathcal{E}_\lambda(u)=L(u)-(J_1\circ T)(u)-\lambda (J_2\circ S)(u)}$$ , where $${L:X \rightarrow \mathbb R}$$ is a sequentially weakly lower semicontinuous C 1 functional, $${J_1:Y\rightarrow\mathbb R, J_2:Z\rightarrow \mathbb R}$$ (Y, Z Banach spaces) are two locally Lipschitz functionals, T : X ? Y, S : X ? Z are linear and compact operators and ? > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each ? > 0 and there exists ?* > 0 such that the functional $${\mathcal{E}_{\lambda^\ast}}$$ 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.