Choquard equations with critical nonlinearities

Abstract

In this paper, we study the Brezis-Nirenberg type problem for Choquard equations in ${\mathbb{R}}^N$ \begin{equation*} -Δu+u=(I_α\ast|u|^{p})|u|^{p-2}u+λ|u|^{q-2}u \quad \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\ge 3,\alpha \in (0,N)$, $\lambda >0$, $q\in (2,\frac{2N}{N-2}]$, $p=\frac{N+\alpha }{N}$ 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 Pohožaev 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 Pohožaev identity of solutions to a general Choquard equation are obtained.