Abstract. In this paper we prove some abstract minimax principles for nonsmooth locally Lipschitz energy functionals and then we use those abstract results to study semilinear and quasilinear hemivariational inequalities at resonance. We permit the possibility of strong resonance at $\pm\infty $ and using a variational approach, based on the nonsmooth critical point theory of Chang, we prove the existence of nontrivial solutions and multiple solutions for semilinear and quasilinear hemivariational inequalities at resonance.

AMS Subject Classification (1991): 35J20, 35J85, 35R70

Keyword(s): hemivariational inequalities, strong resonance, locally Lipschitz functional, subdifferential, nonsmooth Cerami condition, critical point, minimax principle, nonsmooth Saddle Point Theorem, Ekeland variational principle, Rayleigh quotient, principal eigenvalue, p-Laplacian

Received March 3, 2000, and in revised form April 18, 2002. (Registered under 2865/2009.)