Index theory and multiple solutions for asymptotically linear wave equation

Abstract

In this paper, we consider the existence and multiplicity of periodic solutions for the asymptotically linear wave equations. We will establish an index theory for the linear wave equation and define a relative Morse index to measure the difference between the linearizations of the nonlinearity at the origin and at the infinity. Taking advantage of a generalized linking theorem for the strong indefinite functionals, we obtain existence and multiplicity of periodic solutions and the nonlinearity is not assumed to be $C^1$.

Keywords:

• Asymptotically linear wave equation
• essential spectrum
• relative morse index

2020 Mathematics Subject Classifications:

• 35L05
• 58E05
• 19K56

Disclosure statement

No potential conflict of interest was reported by the author(s).