Multiple solutions for strongly resonant nonlinear elliptic problems with discontinuities

We examine a nonlinear strongly resonant elliptic problem driven by the p-Laplacian and with a discontinuous nonlinearity. We assume that the discontinuity points are countable and at them the nonlinearity has an upward jump discontinuity. We show that the problem has at least two nontrivial solutions without using a multivalued interpretation of the problem as it is often the case in the literature. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions.