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Set-Valued and Variational Analysis

, Volume 20, Issue 3, pp 417–443 | Cite as

Multiple Solutions for Nonlinear Coercive Problems with a Nonhomogeneous Differential Operator and a Nonsmooth Potential

  • Leszek GasińskiEmail author
  • Nikolaos S. Papageorgiou
Open Access
Article

Abstract

We consider a nonlinear elliptic problem driven by a nonlinear nonhomogeneous differential operator and a nonsmooth potential. We prove two multiplicity theorems for problems with coercive energy functional. In both theorems we produce three nontrivial smooth solutions. In the second multiplicity theorem, we provide precise sign information for all three solutions (the first positive, the second negative and the third nodal). Out approach is variational, based on the nonsmooth critical point theory. We also prove an auxiliary result relating smooth and Sobolev local minimizer for a large class of locally Lipschitz functionals.

Keywords

Locally Lipschitz function Generalized subdifferential Palais-Smale condition Mountain pass theorem Second deformation theorem Nodal solutions 

Mathematics Subject Classifications (2010)

35J20 35J70 

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© The Author(s) 2011

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer Science, Institute of Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Department of MathematicsNational Technical UniversityAthensGreece

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