Uniqueness of positive radial solutions of Choquard type equations

TL;DR

This paper proves the existence and uniqueness of positive radial solutions for a class of Choquard equations with non-variational structure, extending previous results from the case p=2 to p in [1,2], using shooting methods.

Contribution

It introduces a novel approach using shooting methods and differential inequalities to establish uniqueness for Choquard equations with p in [1,2], where variational methods are not applicable.

Findings

Proved uniqueness of positive radial solutions for p in [1,2].

Extended existing results from p=2 to a broader range of p.

Developed new differential inequalities for analysis.

Abstract

In this paper, we consider the following Choquard type equation \begin{equation} \left\{\begin{aligned} &-\Delta u+\lambda u=\gamma(\Phi_N(|x|)\ast|u|^p)u \ \ \mbox{in $\mathbb{R}^N$}, \\ &\lim\limits_{|x|\to\infty}u(x)=0,\\ \end{aligned}\right. \end{equation} where $N\geq2,\lambda>0,\gamma>0, p\in[1,2]$ and $\Phi_N(|x|)$ denotes the fundamental solution of the Laplacian $-\Delta$ on $\mathbb{R}^N$. This equation does not have a variational frame when $p\neq 2.$ Instead of variational methods, we prove the existence and uniqueness of positive radial solutions of the above equation via the shooting method by establishing some new differential inequalities. The proofs are based on an analysis of the corresponding system of second-order differential equations, and our results extend the existing ones in the literature from $p=2$ to $p\in[1,2]$.