A multiplicity result for the jumping nonlinearity problem

We consider a class of nonlinear problems of the form Au+g(x,u)=f, where A is an unbounded self-adjoint operator on a Hilbert space H of L2(Ω)-functions, an arbitrary domain, and is a “jumping nonlinearity” in the sense that the limits , exist and “jump” over the principal eigenvalue of the operator −A. Under rather general conditions on the operator L and for suitable a<b, we prove some multiplicity results. Applications are given to the wave equation, and elliptic equations in the whole space .