Infinitely many solutions for elliptic system with Hamiltonian type

Abstract

In this paper, we use Legendre-Fenchel transform and a space decomposition to carry out Fountain theorem and dual Fountain theorem for the following elliptic system of Hamiltonian type: \[ \begin{cases} \begin{aligned} -\Delta u&=H_v(u, v) \,\quad&&\text{in}~\Omega,\\ -\Delta v&=H_u(u, v) \,\quad&&\text{in}~\Omega,\\ u,\,v&=0~~&&\text{on} ~ \partial\Omega,\\ \end{aligned} \end{cases} \] where $N\ge 1$, $\Omega \subset \mathbb{R}^N$ is a bounded domain and $H\in C^1( \mathbb{R}^2)$ is strictly convex, even and subcritical. We mainly present two results: (i) When $H$ is superlinear, the system has infinitely many solutions, whose energies tend to infinity. (ii) When $H$ is sublinear, the system has infinitely many solutions, whose energies are negative and tend to 0. As a byproduct, the Lane-Emden system under subcritical growth has infinitely many solutions.