Uniqueness, symmetry and convergence of positive ground state solutions of the Choquard type equation on a ball

Abstract

This paper is concerned with the qualitative properties of the positive ground state solutions to the nonlocal Choquard type equation on a ball BR. First, we prove the radial symmetry of all the positive ground state solutions by using Talenti's inequality. Next we develop the Newton's Theorem and then resort to the contraction mapping principle to establish the uniqueness of the positive ground state solution. Finally, by constructing cut-off functions and applying energy comparison method, we show the convergence of the unique positive ground state solutions as the radius R→∞. Our results extend and complement the existing ones in the literature.