Multi-Bubble Blow-up Analysis for an Almost Critical Problem

TL;DR

This paper investigates the detailed blow-up behavior of solutions to a nearly critical elliptic PDE in bounded domains, identifying precise blow-up rates and concentration points as the perturbation parameter approaches zero.

Contribution

It provides a comprehensive analysis of multiple blow-up points and their locations for solutions to an almost critical elliptic problem, extending previous single blow-up results.

Findings

Determined the exact blow-up rate of solutions as epsilon approaches zero.

Characterized the interior concentration points for multiple blow-up solutions.

Developed a delicate analytical method based on the gradient of the Euler-Lagrange functional.

Abstract

Consider a smooth, bounded domain $\O\subset \mathbb{R}^n$ with $n\geq 4$ and a smooth positive function $V$. We analyze the asymptotic behavior of a sequence of positive solutions $u_\e$ to the equation $-\Delta u +V(x)u =u^{\frac{n+2}{n-2}-\e}$ in $\O$ with zero Dirichlet boundary conditions, as $\e\to 0$. We determine the precise blow-up rate and characterize the locations of interior concentration points in the general case of multiple blow-up, providing an exhaustive description of interior blow-up phenomena of this equation. Our result is established through a delicate analysis of the gradient of the corresponding Euler-Lagrange functional.