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

Abstract

We study generalized solutions of the nonlinear wave equation u t t − u s s = 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 ≡ 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 ) . 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 ) is not ‘between’ the Fučik curves passing through ( λ , λ ) ≠ ( 0 , 0 ) and p is a continuous function, sublinear at infinity.