Existence of solutions for a nonlinear wave equation

Abstract

We prove the existence of a strong solution of a periodic-Dirichlet problem
for the semilinear wave equation with irrational period and with
nonlinearity satisfying some general growth conditions locally around $0$.
We construct a new variational method, called a dual method, and using
relations between critical points and critical values of the primal action
and the dual action functionals we prove that the solution exists. The dual
functional which we define is different from the ones known so far in that
it depends on two dual variables.