Multiplicity Result for Nonlinear Neumann Problems |
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 C^{1}-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.