Eigenvalue Problems for Hemivariational Inequalities

Abstract

We consider a semilinear eigenvalue problem with a nonsmooth potential (hemivariational inequality). Using a nonsmooth analog of the local Ambrosetti–Rabinowitz condition (AR-condition), we show that the problem has a nontrivial smooth solution. In the scalar case, we show that we can relax the local AR-condition. Finally, for the resonant λ = λ1 problem, using the nonsmooth version of the local linking theorem, we show that the problem has at least two nontrivial solutions. Our approach is variational, using minimax methods from the nonsmooth critical point theory.