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

For a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ and smooth functions $a$ and $V$, we consider the asymptotic behavior of a sequence of positive solutions $u_{\epsilon }$ to $-\mathrm{\Delta }{u}_{\epsilon }+(a+\epsilon V)u_{\epsilon }=u_{\epsilon }^5$ on $\mathrm{\Omega }$ 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 $N=3$.