Existence of ground state solutions of the coupled nonlinear Hartree type system with attractive couplings

Abstract

In this paper we prove the existence and nonexistence of positive ground state solutions of coupled nonlinear Hartree type system attractive couplings. On the one hand, we prove the existence of positive ground state solution by using the Morse theory. On the other hand, we find the exact conditions to guarantee the existence of positive ground state solution with two coupling system. Moreover, we show that these conditions are better than the previous results (Wang and Shi in Calc Var Partial Differ Equ 56:168, 2017).