Normalized ground states solutions for nonautonomous Choquard equations

TL;DR

This paper investigates the existence and stability of normalized ground state solutions for a class of nonautonomous Choquard equations, identifying critical exponents and extending previous results in the field.

Contribution

It establishes the existence of ground state solutions for certain parameter ranges and analyzes their stability, extending prior work to nonautonomous cases with variable coefficients.

Findings

Existence of ground state solutions for p in (2_{*,μ}, 𝑝̄)

Orbital stability of these ground states

Existence of non-minimizer solutions for p in (𝑝̄, 2^*_{μ})

Abstract

In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation: $$-\Delta u-\lambda u=\left(\frac{1}{|x|^{\mu}}\ast A|u|^{p}\right)A|u|^{p-2}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c,\quad u\in H^1(\mathbb{R}^N,\mathbb{R}),$$ where $c>0$, $0<\mu<N$, $\lambda\in\mathbb{R}$, $A\in C^1(\mathbb{R}^N,\mathbb{R})$. For $p\in(2_{*,\mu}, \bar{p})$, we prove that the Choquard equation possesses ground state normalized solutions, and the set of ground states is orbitally stable. For $p\in (\bar{p},2^*_\mu)$, we find a normalized solution, which is not a global minimizer. $2^*_\mu$ and $2_{*,\mu}$ are the upper and lower critical exponents due to the Hardy-Littlewood-Sobolev inequality, respectively. $\bar{p}$ is $L^2-$critical exponent. Our results generalize and extend some related results.