Existence of Three Nontrivial Solutions of an Elliptic Boundary-Value Problem with Discontinuous Nonlinearity in the Case of Strong Resonance

We consider a strongly resonant homogeneous Dirichlet problem for elliptic-type equations with discontinuous nonlinearity in the phase variable. Using the variational method, we prove an existence theorem for at least three nontrivial solutions of the problem under consideration; at least two of these are semiregular. The resulting theorem is applied to the eigenvalue problem for elliptic-type equations with discontinuous nonlinearity with positive spectral parameter. An example of a discontinuous nonlinearity satisfying all the assumptions of the theorem is given.