Multiplicity and regularity of periodic solutions for a class of degenerate semilinear wave equations

TL;DR

This paper establishes the existence of infinitely many classical periodic solutions for a class of degenerate semilinear wave equations with a focus on the case s=3, using new a priori estimates for minimax values.

Contribution

It introduces a novel approach with upper a priori estimates for minimax values of a perturbed indefinite functional to prove multiple solutions.

Findings

Existence of infinitely many classical solutions for the wave equation.

New upper a priori estimates for minimax values.

Application to the case s=3 posed by Brézis.

Abstract

We prove the existence of infinitely many classical periodic solutions for a class of degenerate semilinear wave equations: \[ u_{tt}-u_{xx}+|u|^{s-1}u=f(x,t), \] for all $s>1$. In particular we prove the existence of infinitely many classical solutions for the case $s=3$ posed by Br\'ezis in \cite{BrezisBAMS}. The proof relies on a new upper a priori estimates for minimax values of, a pertubed from symmetry, strongly indefinite functional,depending on a small parameter.