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.
Nikolaos C. Kourogenis. Nikolaos S. Papageorgiou. "Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems." Abstr. Appl. Anal. 5 (2) 119 - 135, 2000. https://doi.org/10.1155/S1085337500000269