Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions

Abstract

For a smooth bounded domain $\Omega \subset {\mathbb{R}}^3$ and smooth functions $a$ and $V$, we consider the asymptotic behavior of a sequence of positive solutions $u_{\epsilon }$ to $-\Delta u_{\epsilon }+(a+\epsilon V)u_{\epsilon }=u_{\epsilon }^5$ on $\Omega$ with zero Dirichlet boundary conditions, which blow up as $\epsilon \to 0$. We derive the sharp blow-up rate and characterize the location of concentration points in the general case of multiple blow-up, thereby obtaining a complete picture of blow-up phenomena in the framework of the Brezis-Peletier conjecture in dimension $N=3$.

arXiv commentv4: We correct an inaccuracy related to the proof of Prop. 3.4: To guarantee that the error function $Q_{i,\epsilon }$ is in $L^{\infty }$, one needs to subtract all partial second derivatives and not only a radial term; see definition (3.1). The conclusions of the paper are unchanged

References in corpus (2)

• Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem
• Blow-up of solutions of critical elliptic equations in three dimensions