Periodic solutions for a class of first order super-quadratic Hamiltonian system
The authors consider the unbounded Hamiltonian system \[ \left\{ \begin{aligned} \partial_t u-\Delta_x u+V(x) u= H_v(t,x,u,v)\\ -\partial_t v-\Delta_xv+V(x)v=H_n(t,x,u,v) \end{aligned} \right. \] for \((t,x)\in\mathbb{R}\times\Omega\), \(\Omega\subset\mathbb{R}^N\) a bounded smooth domain. Here \(V\in C^0(\overline{\Omega})\) and \(0\) is not a Dirichlet eigenvalue of \(-\Delta_x+V(x)\). The nonlinearity \(H\in C^1(\mathbb{R}\times\overline{\Omega} \times\mathbb{R}^2)\) is \(T\)-periodic in \(t\) and superquadratic in \(z=(u,v)\) for \(z\) near \(0\) and near \(\infty\). The main theorem states the existence of a solution \(z(t,x)\not\equiv 0\) which is \(T\)-periodic in \(t\) and satisfies Dirichlet boundary conditions in the \(x\)-variable. The proof is based on variational methods for strongly indefinite functionals due to \textit{S. Luan} and \textit{A. Mao} [Nonlinear Anal., Theory Methods Appl. 61, No. 8 (A), 1413--1426 (2005; Zbl 1071.34039)] and \textit{Y. Ding} and the reviewer [Math. Z. 240, No. 2, 289--310 (2002; Zbl 1008.37040)].
scientific article; zbMATH DE number 5141603