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 and a generalized version of the Ekeland variational principle. At the end of the paper we also show that the nonsmooth (PS)-condition implies the coercivity of the functionaj 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." J. Math. Soc. Japan 53 (1) 17 - 34, January, 2001. https://doi.org/10.2969/jmsj/05310017