Ground state solutions to generalized nonlinear wave equations with infinite-dimensional kernel

TL;DR

This paper establishes the existence of time-periodic ground state solutions for a class of generalized nonlinear wave equations on Riemannian manifolds, addressing the challenges posed by infinite-dimensional kernels and degeneracies.

Contribution

It introduces a novel variational method with a new saddle point reduction technique to handle doubly degenerate nonlinear wave equations with infinite-dimensional kernels.

Findings

Existence of ground state solutions in doubly degenerate settings

Characterization of ground state energy via a minimax principle

Application of a new variational approach to complex nonlinear wave equations

Abstract

The present paper is devoted to existence results for time-periodic solutions of generalized nonlinear wave equations in a closed Riemannian manifold M. Our main focus lies on the doubly degenerate setting where the associated generalized wave operator has an infinite dimensional kernel and the nonlinearity may vanish on open subsets of M. To deal with this setting, we apply a direct variational approach based on a new variant of the nonlinear saddle point reduction to the associated Nehari-Pankov set. This allows us to find ground state solutions and to characterize the associated ground state energy by a fairly simple minimax principle.