Choquard equations with critical nonlinearities

TL;DR

This paper investigates critical Choquard equations with nonlinearities at the Hardy-Littlewood-Sobolev critical exponents, establishing existence of groundstate solutions using approximation and Pohozaev methods, and analyzing solution regularity.

Contribution

It introduces a novel approach combining subcritical approximation and Pohozaev constraints to prove groundstate solutions for critical Choquard equations.

Findings

Existence of positive, radially nonincreasing groundstate solutions.

Development of regularity results for solutions.

Derivation of Pohožaev identities for general Choquard equations.

Abstract

In this paper, we study the Brezis-Nirenberg type problem for Choquard equations in $\mathbb{R}^N$ \begin{equation*} -\Delta u+u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u+\lambda|u|^{q-2}u \quad \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq 3,\ \alpha\in(0,N)$, $\lambda>0$, $q\in (2,\frac{2N}{N-2}]$, $p=\frac{N+\alpha}{N}$ or $\frac{N+\alpha}{N-2}$ are the critical exponents in the sense of Hardy-Littlewood-Sobolev inequality and $I_\alpha$ is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Poho\v{z}aev constraint method, we obtain a positive and radially nonincreasing groundstate solution in $H^1(\mathbb{R}^N)$ for the problem. To the end, the regularity and the Poho\v{z}aev identity of solutions to a general Choquard equation are obtained.