Set-Valued and Variational Analysis

, Volume 20, Issue 3, pp 417–443

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

Open Access

DOI: 10.1007/s11228-011-0198-4

Cite this article as:
Gasiński, L. & Papageorgiou, N.S. Set-Valued Var. Anal (2012) 20: 417. doi:10.1007/s11228-011-0198-4


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.


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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Copyright information

© The Author(s) 2011

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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