In this paper, we study that the nearly critical nonlocal problem \begin{ equation*} \left\lbrace \begin{ aligned} &-\Delta u=(|x|^{ -{ (n-2)}}\ast u^{ p-\epsilon})u^{ p-1-\epsilon} \quad \mbox{ in}\quad \Omega, &u>0\quad \mbox{ in}\quad\hspace{ 1mm} \Omega, &u=0\quad \mbox{ on}\hspace{ 2.5mm}\partial\Omega, \end{ aligned} \right. \end{ equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{ R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $\epsilon>0$ is a small parameter and $p=\frac{ n+2}{ n-2}$ is energy-critical exponent. We study the asymptotic behavior of least energy solutions as $\epsilon\rightarrow0$. These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied. Finally, in order to further locate the blowing-up point $x_0$, we prove that $x_0$ is a global maximum point of the Robin's function of $\Omega$.
Categories: math.AP