Multi-bubble blow-up analysis for an almost critical problem

Abstract

Consider a smooth, bounded domain $\Omega \subset R^n$ with $n\ge 4$ and a smooth positive function V. We analyze the asymptotic behavior of a sequence of positive solutions $u_{\epsilon }$ to the equation $-\Delta u+V\left(x\right)u=u^{\frac{n+2}{n-2}-\epsilon }$ in Ω with zero Dirichlet boundary conditions, as $\epsilon \to 0$. 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.