Existence of periodic solution for a class of beam equation via variational methods
The paper is concerned with a class of beam equations, or generalized membrane equations, of the form \begin{align*} &u_{tt}+\Delta_x^2 u = f(x,t,u),\quad &&\text{in }Q=\Omega\times\mathbb{R},\\ &u(x,t) = D_{ii}u(x,t) = 0 &&\text{on }\partial\Omega,\\ &u(x,t) = u(x,t+2\pi) &&\text{in }Q, \end{align*} where \(\Omega=(0,2\pi)^N\) and \(f\in C^1(Q\times\mathbb{R}^2;\mathbb{R})\) is \(2\pi\)-periodic in \(t\), superlinear in \(u\). Under various growth conditions on the nonlinearity, the authors obtain the existence of a ground state, and more generally, the existence of a nontrivial solution.
scientific article; zbMATH DE number 7493669
scientific article