The Choquard-type Brézis–Nirenberg problem in 4D

Abstract

In this paper, we focus on the following Choquard-type Brézis–Nirenberg problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\displaystyle \Bigg (\int \limits _{\Omega }\frac{u^{4-\frac{\alpha }{2}}(y)}{|x-y|^\alpha }dy\Bigg )u^{3-\frac{\alpha }{2}}+\varepsilon u, \ \ & \hbox {in}\ \Omega ,\\ u>0,\ \ & \hbox {in}\ \Omega ,\\ u=0, \ \ & \hbox {on}\ \partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^4\), \(\alpha \in (0,4)\), \(4-\frac{\alpha }{2}\) is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, and \(\varepsilon >0\) is a small parameter. By applying the reduction argument, we prove the existence of solutions, which blow up and concentrate around the critical points of the Robin function as \(\varepsilon \rightarrow 0\).