Infinitely many solutions for elliptic system with Hamiltonian type

TL;DR

This paper proves the existence of infinitely many solutions for a Hamiltonian-type elliptic system using variational methods, with solutions' energies diverging or approaching zero depending on the nonlinearity's nature.

Contribution

It introduces a novel application of the Fountain theorem and dual Fountain theorem to establish multiple solutions for Hamiltonian elliptic systems with subcritical growth.

Findings

Infinite solutions when H is superlinear with diverging energies.

Infinite solutions when H is sublinear with energies approaching zero.

Extension to Lane-Emden systems under subcritical growth.

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.