NONTRIVIAL SOLUTIONS FOR SEMILINEAR EQUATIONS WITH A DISCONTINUOUS NONLINEAR TERM

We study the existence of at least one nontrivial solution for semi linear equations of the form Lu is an element of Nu in the case where N interacts with a finite number of eigenvalues of finite multiplicity of L. It is assumed that L is a self-adjoint closed densely defined linear operator in Hilbert space L-2 having infinite dimensional kernel and N is a bounded set-valued operator generated by certain nonlinearity g. The method of approach is to use a topological degree theory for a class of operators associated with the given problem. As an application, we consider the periodic Dirichlet problem for semilinear wave equations with a discontinuous nonlinear term.