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

TL;DR

This paper investigates the asymptotic behavior of least energy solutions to a nearly critical nonlocal nonlinear Hartree equation, revealing their blow-up characteristics and the precise location of blow-up points as the parameter approaches zero.

Contribution

It provides a detailed analysis of the blow-up behavior and the exact blow-up rates for solutions near the critical exponent, including the characterization of blow-up points via Robin's function.

Findings

Solutions blow up at exactly one point as epsilon approaches zero.

The blow-up point is characterized as a global maximum of Robin's function.

Exact rates and shape of blow-up are determined.

Abstract

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 $\Omega$.