On the Fučik spectrum of the wave operator and an asymptotically linear problem

We study generalized solutions of the nonlinear wave equationutt−uss=au+−bu−+p(s,t,u),utt−uss=au+−bu−+p(s,t,u), with periodic conditions in t and homogeneous Dirichlet conditions in s  , under the assumption that the ratio of the period to the length of the interval is two. When p≡0p≡0 and λ   is a nonzero eigenvalue of the wave operator, we give a proof of the existence of two families of curves (which may coincide) in the Fučik spectrum intersecting at (λ,λ)(λ,λ). This result is known for some classes of self-adjoint operators (which does not cover the situation we consider here), but in a smaller region than ours. Our approach is based on a dual variational formulation and is also applicable to other operators, such as the Laplacian. In addition, we prove an existence result for the non-homogeneous situation, when the pair (a,b)(a,b) is not ‘between’ the Fučik curves passing through (λ,λ)≠(0,0)(λ,λ)≠(0,0) and p is a continuous function, sublinear at infinity.