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

We prove the existence of infinitely many classical periodic solutions for a class of degenerate semilinear wave equations: $$u_{tt}-u_{xx}+\mathrm{\mid }u{\mathrm{\mid }}^{s-1}u=f(x,t),$$ 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.