Abstract and Applied Analysis
Volume 5 (2000), Issue 2, Pages 119-135
Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems
National Technical University, Department of Mathematics, Zografou Campus, Athens 157 80, Greece
Received 22 August 1999
Copyright © 2000 Nikolaos C. Kourogenis and Nikolaos S. Papageorgiou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais-Smale (PS)-condition implies the coercivity of the
functional, extending this way a well-known result of the “smooth” case.