Asymptotic behavior of least energy solutions to the nonlinear Hartree equation near critical exponent
Silvia Cingolani, Minbo Yang, Shunneng Zhao
TL;DR
This work analyzes least-energy positive solutions to a nonlocal Hartree equation with energy-critical exponent on a smooth bounded domain in dimensions 3–5, as the perturbation ε→0. The authors develop a comprehensive blow-up framework built on rescaled bubbles, the Kelvin transform, and sharp Pohozaev-type identities, showing the solution concentrates at a single interior point x0. They prove x0 is a critical point (indeed a global maximum) of the Robin function φ(x)=H(x,x), and provide detailed asymptotics: the solution near blow-up resembles a bubble W[0,1], with the gradient concentrating as a multiple of the Green function G(·,x0), and global energy expansions tied to φ(x0). The results extend the classical Brezis–Nirenberg picture to a nonlocal Choquard-type problem, and include an extension to a μ-parameterized nonlocal interaction with precise energy corrections and blow-up profiles, yielding sharp upper and lower bounds for the least-energy level S_H^ε. The analysis highlights the Robin function as the key selector of blow-up location and delivers precise quantitative blow-up rates and asymptotic profiles, with implications for nonlocal elliptic equations and related critical phenomena.
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 $Ω$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $ε>0$ 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 $ε\rightarrow0$. 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 $Ω$.