Ground state solutions for Hamiltonian elliptic system with inverse square potential (Q524623)

The paper under review deals with ground state solutions for Hamiltonian elliptic systems with gradient term and inverse square potential. Precisely, the authors study the Hamiltonian elliptic system \[ \begin{cases} -\Delta u+ \vec{b}(x)\cdot\nabla u+ V(x)u-\frac{\mu}{|x|^2}v=H_v(x,u,v),\\ -\Delta v- \vec{b}(x)\cdot\nabla v+ V(x)u-\frac{\mu}{|x|^2}u=H_u(x,u,v) \end{cases} \] for all \(x\in \mathbb{R}^N,\) \(N\geq3.\) Here \(\mu\in\mathbb{R}\) is a parameter, and the functions \(\vec{b}\in C^1(\mathbb{R}^N;\mathbb{R}^N,\) \(V\in C^1(\mathbb{R}^N;\mathbb{R})\) and \(H\in C^1(\mathbb{R}^N\times\mathbb{R}^2;\mathbb{R})\) are \(1\)-periodic in \(x.\) Imposing suitable conditions on the data and employing variational methods, the authors prove that the system admits a ground state solution for small enough \(\mu\geq0.\) A comparison between the energy of ground state solutions is also provided in the cases \(\mu>0\) and \(\mu=0.\) Moreover, the convergence of the ground state solutions is studied when \(\mu\to 0^+.\)