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 ${\mathcal{E}_\lambda(u)=L(u)-(J_1\circ T)(u)-\lambda (J_2\circ S)(u)}$ , 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 λ > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each λ > 0 and there exists λ* > 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.