Multispike solutions for the Brezis-Nirenberg problem in dimension three

Abstract

We consider the problem $\Delta u+\lambda u+u^5=0$, $u>0$, in a smooth bounded domain $\Omega$ in ${\mathbb{R}}^3$, under zero Dirichlet boundary conditions. We obtain solutions to this problem exhibiting multiple bubbling behavior at $k$ different points of the domain as $\lambda$ tends to a special positive value ${\lambda }_0$, which we characterize in terms of the Green function of $-\Delta -\lambda$.

38 pages