The Brezis–Nirenberg type double critical problem for the Choquard equation

Abstract

In this paper, we study the following Choquard equation

$$\begin{aligned} -\Delta u=\alpha |u|^{2^*-2}u+\beta \left( I_\mu *|u|^{2_\mu ^*}\right) |u|^{2_\mu ^* -2}u +\lambda u,\quad in\,\,\Omega , \end{aligned}$$

where $\Omega$ is a bounded domain of ${\mathbb {R}}^N$ with Lipschitz boundary, $N\ge 3,$ $\alpha ,\beta ,\lambda$ are real parameters satisfying suitable conditions, $2^* =\frac{2N}{N-2}$ is the critical exponent for the embedding of $H_0^1 (\Omega )$ to $L^p (\Omega ),$ $2_\mu ^* =\frac{2N-\mu }{N-2}$ is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Using variational methods, we show the existence of nontrivial solutions for the Choquard equation with double critical exponents.