, Volume 20, Issue 3, pp 417-443,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 18 Oct 2011

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

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.

This research has been partially supported by the Ministry of Science and Higher Education of Poland under Grants no. N201 542438 and N201 604640.