Multispike solutions for the Brezis-Nirenberg problem in dimension three

TL;DR

This paper constructs solutions with multiple concentration points for a nonlinear PDE in three dimensions, revealing complex bubbling phenomena as a parameter approaches a critical value.

Contribution

It introduces new multispike solutions for the Brezis-Nirenberg problem in three dimensions, characterized by their bubbling behavior at multiple points.

Findings

Existence of solutions with multiple bubbling points.

Characterization of the bubbling points via Green function.

Behavior of solutions as the parameter approaches a critical value.

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 $-\Delta - \lambda$.