Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues

Abstract

We consider a semilinear elliptic equation with a nonsmooth, locally \hbox{Lipschitz} potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and an extension of the Castro--Lazer--Thews reduction method to a nonsmooth setting, which we develop here using tools from nonsmooth analysis.