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 L-2(Omega)-functions, Omega subset of R-N an arbitrary domain, and g : Omega x R --> R is a "jumping nonlinearity" in the sense that the limits lim(s-->-infinity) g(x,s)/s = a, lim(s-->infinity)g(x,s)/s = b exist and "jump" over the principal eigenvalue of the s S 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 R-N. (C) 2002 Elsevier Science (USA). All rights reserved.