On number of solutions for one class of elliptic equations with a spectral parameter and discontinuous nonlinearity

We consider the question of existence of Dirichlet’s problem solution for the Laplace equation with a spectral parameter and discontinuous on a phase variable nonlinearity. Using the variational method, we prove a theorem about a number of solutions. We result an example of discontinuous nonlinearity that satisfies to conditions of the theorem for which there is unique semiregular solution of this boundary problem.