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 ${ℰ}_{\lambda }(u)=L(u)‐(J_1\circ T)(u)‐\lambda (J_2\circ S)(u)$ , where $L:X\to \mathbb{R}$ is a sequentially weakly lower semicontinuous C ¹ functional, $J_1:Y\to \mathbb{R},J_2:Z\to \mathbb{R}$ (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 ${ℰ}_{{\lambda }^{*}}$ 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.