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 Eλ(u) = L(u)− (J1 ◦T )(u)−λ(J2 ◦ S)(u), where L : X → R is a sequentially weakly lower semicontinuous C functional, J1 : Y → R, J2 : Z → 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 Eλ∗ 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. 2010 Mathematics Subject Classification: 58K05, 47J30, 58E05, 34A60, 47J22.