Variational methods for breather solutions of nonlinear wave equations

We construct infinitely many real-valued, time-periodic breather solutions of the nonlinear wave equation $$\partial^2_t U-\Delta U=Q(x)|U|^{p-2}U\quad\text{ on }\mathbb{T}\times\mathbb{R}^N$$ with suitable $N\ge2, p > 2$ and localized nonnegative $Q$. These solutions are obtained from critical points of a dual functional and they are weakly localized in space. Our abstract framework allows to find similar existence results for the Klein-Gordon equation or
biharmonic wave equations.