On the ground state to Hamiltonian elliptic system with Choquard's nonlinear term (Q2246567)

Summary: In the present paper, we consider the following Hamiltonian elliptic system with Choquard's nonlinear term \[ \begin{cases} -\varDelta u + V(x) u = \int_\varOmega(G(v(y))/|x - y|^\beta) dy g(v) \text{ in } \varOmega,\\ -\varDelta v + V(x) v = \int_\varOmega(F(u(y))/|x - y|^\alpha) dy f(u) \text{ in } \varOmega,\\ u = 0 , v = 0 \text{ on } \partial \varOmega, \end{cases} \] where \(\varOmega\subset \mathbb{R}^N\) is a bounded domain with a smooth boundary, \(0<\alpha<N\), \(0<\beta<N\), and \(F\) is the primitive of \(f\), similarly for \(G\). By establishing a strongly indefinite variational setting, we prove that the above problem has a ground state solution.