Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent

Abstract

In this paper, we study that the nearly critical nonlocal problem \begin{equation*} \left\lbrace \begin{aligned} &-Δu=(|x|^{-{(n-2)}}\ast u^{p-ε})u^{p-1-ε} \quad \mbox{in}\quad Ω, &u>0\quad \mbox{in}\quad\hspace{1mm} Ω, &u=0\quad \mbox{on}\hspace{2.5mm}\partialΩ, \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 $\epsilon \to 0$. 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 $\Omega$.

This paper has been accepted for publication in Ann. Sc. Norm. Super. Pisa Cl. Sci