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

Abstract

We study generalized solutions of the nonlinear wave equation $$u_{tt}-u_{ss}=a{u}^{+}-b{u}^{-}+p(s,t,u),$$ 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 $p\equiv 0$ and $\lambda$ 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 $(\lambda ,\lambda )$. 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 nonhomogeneous situation, when the pair $(a,b)$ is not `between' the Fučik curves passing through $(\lambda ,\lambda )\ne (0,0)$ and $p$ is a continuous function, sublinear at infinity.