On the Fu\v{c}ik spectrum of the wave operator and an asymptotically linear problem

TL;DR

This paper investigates the Fučík spectrum of the wave operator and an asymptotically linear nonlinear wave equation, establishing existence of spectral curves and solutions using a dual variational approach.

Contribution

It provides new existence results for Fučík spectrum curves and solutions of nonlinear wave equations, extending known results to a broader class of operators and conditions.

Findings

Existence of two families of Fučík spectrum curves intersecting at eigenvalues.

Application of dual variational methods to the wave operator and other operators.

Existence of solutions for the nonlinear wave equation with sublinear forcing term.

Abstract

We study generalized solutions of the nonlinear wave equation $$u_{tt}-u_{ss}=au^+-bu^-+p(s,t,u),$$ 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 $p\equiv 0$ 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 Fu\v{c}ik 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 Fu\v{c}ik curves passing through $(\lambda,\lambda)\neq(0,0)$ and $p$ is a continuous function, sublinear at infinity.