Solutions of minimal period of a wave equation via a generalization of a Hofer's theorem (Q910584)

Consider the following semilinear wave equation \[ (1)\quad u_{tt}- u_{xx}+g(u,t,x)=0,\quad t\in {\mathbb{R}},\quad x\in [0,\pi] \] under boundary and periodicity conditions \[ (2)\quad u(t,0)=u(t,\pi)=0;\quad u(t,x)=u(t+T,x),\quad t\in {\mathbb{R}},\quad x\in [0,\pi], \] where T is a rational multiple of \(\pi\). In the present paper the author studies the existence of solutions of (1)-(2) with minimal period. The main result is established via a generalization of the Hofer's theorem and an application of Ambrosetti-Rabinowitz mountain pass theorem, which gives either a solution of (1)-(2) with minimal period T or a solution which is an accumulation point of periodic solutions.