Infinitely many solutions for elliptic system with Hamiltonian type
Jia Zhang, Weimin Zhang
TL;DR
The paper tackles the existence of infinitely many solutions for a strongly indefinite elliptic Hamiltonian system in a bounded domain by converting the problem to a dual variational form using the Legendre–Fenchel transform. It develops a two-variable space decomposition and applies Fountain and dual Fountain theorems, augmented by a $\mathbb{Z}_2$-cohomological index to manage intersection properties, under subcritical growth assumptions on $H$. The main results establish two multiplicity regimes: in the superlinear case, there are infinitely many solutions with energies $\to+\infty$, and in the sublinear/coercive regime, there are infinitely many solutions with negative energies $\to0^-$. As a byproduct, the Lane–Emden system under subcritical growth also admits infinitely many solutions, broadening multiplicity results for Hamiltonian systems with strong indefiniteness and linking structures.
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: \[ \begin{cases} \begin{aligned} -Δu&=H_v(u, v) \,\quad&&\text{in}~Ω,\\ -Δv&=H_u(u, v) \,\quad&&\text{in}~Ω,\\ u,\,v&=0~~&&\text{on} ~ \partialΩ,\\ \end{aligned} \end{cases} \] where $N\ge 1$, $Ω\subset \mathbb{R}^N$ 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.