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

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