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

Abstract

Consider a smooth, bounded domain $\tilde{A}˜\subset {\mathbb{R}}^n$ with $n\ge 4$ and a smooth positive function $V$. We analyze the asymptotic behavior of a sequence of positive solutions $u_\e$ to the equation $-Δu +V(x)u =u^{\frac{n+2}{n-2}-\e}$ in $\tilde{A}˜$ 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.

29 pages