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

TL;DR

This paper investigates the existence and blow-up behavior of solutions to a nonlinear Choquard-type Brezis-Nirenberg problem in three dimensions, identifying conditions under which solutions concentrate at critical points as parameters vary.

Contribution

It introduces a reduction method to find solutions that blow up at critical points, extending the understanding of nonlinear Choquard equations with boundary conditions.

Findings

Existence of solutions near a critical parameter value

Solutions blow up and concentrate at critical points of the Robin function

Extension to zero Neumann boundary conditions

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.