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

Abstract

We describe the asymptotic behavior of positive solutions $u_{\epsilon }$ of the equation $-\Delta u+au=3u^{5-\epsilon }$ 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 Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation $-\Delta u+(a+\epsilon V)u=3u^5$ in $\Omega$.

accepted for publication in Analysis & PDE

Cited by in corpus (2)

• Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension
• Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions