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Multiplicity Results for Nonlinear Neumann Problems

Published online by Cambridge University Press:  20 November 2018

Michael Filippakis ,
Leszek Gasiński  and
Nikolaos S. Papageorgiou
Michael Filippakis
Affiliation:
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece
Leszek Gasiński
Affiliation:
Jagiellonian University, Institute of Computer Science, ul. Nawojki 11, 30072 Cracow, Poland email: gasinski@softlab.ii.uj.edu.pl
Nikolaos S. Papageorgiou
Affiliation:
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece email:npapg@math.ntua.gr
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Abstract

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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 five multiplicity results, four 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

35J2035J6035J85Nonsmooth critical point theorylocally Lipschitz functionClarke subdifferentialNeumann problemstrong resonancesecond deformation theoremnonsmooth symmetric mountain pass theoremp-Laplacian.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 1 , 01 February 2006 , pp. 64 - 92
Copyright
Copyright © Canadian Mathematical Society 2006

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