Blow-up of solutions of critical elliptic equations in three dimensions

TL;DR

This paper analyzes the blow-up behavior of positive solutions to critical elliptic equations in three dimensions, determining the blow-up rate and concentration points, and confirming a longstanding conjecture.

Contribution

It provides the exact blow-up rate and location for solutions of critical elliptic equations, confirming a conjecture by Brézis and Peletier and extending results to perturbed equations.

Findings

Derived the exact blow-up rate of solutions.

Identified the concentration point location.

Confirmed a conjecture of Brézis and Peletier (1989).

Abstract

We describe the asymptotic behavior of positive solutions $u_\epsilon$ of the equation $-\Delta u + au = 3\,u^{5-\epsilon}$ in $\Omega\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_\epsilon$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Br\'ezis and Peletier (1989). Similar results are also obtained for solutions of the equation $-\Delta u + (a+\epsilon V) u = 3\,u^5$ in $\Omega$.