ACTA issues

Nontrivial solutions for semilinear hemivariational inequalities resonant at higher eigenvalues

Michael Filippakis, Leszek GasiƄski, Nikolaos S. Papageorgiou

Acta Sci. Math. (Szeged) 71:3-4(2005), 581-602

Abstract. We consider semilinear elliptic equations with nonsmooth potential and resonant at high parts of the spectrum of $\left(-\Delta,H^1_0(Z)\right )$. Asymptotically at infinity we permit double resonance of ${\partial j(z,\zeta )\over\zeta }$ between two consecutive eigenvalues. The resonance is complete in the higher part of the spectrum, incomplete in the lower part. We also permit resonance asymptotically at zero. Using a variational approach based on nonsmooth critical point theory, we prove the existence of a nontrivial solution.

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

Keyword(s): Eigenvalues, resonance, unique continuation property, orthogonality, locally Lipschitz function, Clarke subdifferential, nonsmooth critical point theory, nonsmooth linking theorem

Received November 24, 2003. (Registered under 5888/2009.)