Preprint
(2003)

Author(s):  
       Michael Filippakis
       National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

       L. Gasiński, gasinski(at)softlab.ii.uj.edu.pl, http://www.ii.uj.edu.pl/~gasinski/
       Jagiellonian University, Institute of Computer Science, Nawojki 11, 30-072 Cracow, Poland

       Nikolaos S. Papageorgiou
       National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

Pages: 26

Abstract: In this paper we study nonlinear elliptic problems of Neumann type driven by the p-Laplacian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove three multiplicity results, two for problems with nonsmooth potential and one for problems with a C¹-potential. In the last part for nonsmooth problems in which the potential eventually exhibits a strict super-p-growth, under a symmetry condition we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative p-Laplacian with Neumann boundary condition.

Keywords: nonsmooth critical point theory, locally Lipschitz function, Clarke subdifferential, Neumann problem, strong resonance, second deformation theorem, nonsmooth symmetric mountain pass theorem, p-Laplacian.

Published: Canadian Journal of Mathematics, 58:1 (2006) 64-92.