### Existence of solutions for a nonlinear wave equation

DOI: http://dx.doi.org/10.12775/TMNA.2006.034

#### 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.

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.

#### Keywords

Duality; variational method; existence of solutions; semilinear wave equation

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