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 (Ω)-functions, Ω R N an arbitrary domain, and g:ΩxR->R is a ''jumping nonlinearity'' in the sense that the limits lim s - > - ~ g(x,s)s=a, lim s - > ~ g(x,s)s=b 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 R N .