Blowing-up solutions for the Choquard type Brezis-Nirenberg problem in dimension three

Abstract

In this paper, we are interested in the existence of solutions for the following Choquard type Brezis-Nirenberg problem \begin{align*} \left\{ \begin{array}{ll} -\Delta u=\displaystyle\Big(\int\limits_{\Omega}\frac{u^{6-\alpha}(y)}{|x-y|^\alpha}dy\Big)u^{5-\alpha}+\lambda u, \ \ &\mbox{in}\ \Omega, u=0, \ \ &\mbox{on}\ \partial \Omega, \end{array} \right. \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$, $\alpha\in (0,3)$, $6-\alpha$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, and $\lambda$ is a real positive parameter. By applying the reduction argument, we find and characterize a positive value $\lambda_0$ such that if $\lambda-\lambda_0>0$ is small enough, then the above problem admits a solution, which blows up and concentrates at the critical point of the Robin function as $\lambda\rightarrow \lambda_0$. Moreover, we consider the above problem under zero Neumann boundary condition.