Classification and qualitative analysis of positive solutions of the nonlinear Hartree type system

In this paper, we focus on the qualitative analysis of positive solutions for a specific class of static coupled nonlinear Hartree type systems. In the initial step, we convert these equations into integral systems that incorporate the Riesz potentials. Following that, we acquire integrability outcomes and decay characteristics for the integrable solutions of these integral systems, employing the regularity lifting lemma as our second approach. Lastly, as a third step, we establish Liouville-type results and conduct a comprehensive classification of positive solutions for the integral systems in RN using the moving plane method. To the best of our knowledge, this kind of Liouville type result is sharp and new even for Choquard equation. As an application, we deduce the sharp and new Liouville type results for the elliptic system with quadratic nonlinearity.