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

TL;DR

This paper investigates the symmetry, uniqueness, and convergence of positive ground state solutions to a nonlocal Choquard equation on a ball, providing new insights into their qualitative properties and behavior as the domain expands.

Contribution

It establishes the radial symmetry, uniqueness, and convergence of solutions, extending and improving previous results in the study of Choquard equations.

Findings

Positive ground state solutions are radially symmetric.

Solutions are unique under the given conditions.

Solutions converge as the domain radius tends to infinity.

Abstract

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