On the existence of multiple solutions for semilinear equations with monotone nonlinearities crossing a finite number of eigenvalues

Abstract

LET 0 be a bounded domain in R” (n 2 1) with smooth boundary. In this paper we study in L,(Q) nonlinear equations of the form Lu = N(u), (1.1) where L is a densely defined closed self-adjoint linear operator with closed range and N is a Nemytskii operator generated by a Caratheodory function (w, s) + g(w, s) from Q x R to lR with g( *, 0) = 0. We are interested in the existence of multiple solutions of (1.1) in the case where N interacts with a finite number of eigenvalues of finite multiplicity of L. It is essential in our study that the kernel of L may be infinite dimensional and that L has a pure point spectrum which may be unbounded both from above and below. Hence the results obtained can be applied to the problem of finding periodic solutions for semilinear wave equations of the form