# Hardy-Sobolev equations with asymptotically vanishing singularity: Blow-up analysis for the minimal energy

## TL;DR

This paper analyzes the asymptotic behavior of solutions to Hardy-Sobolev equations with a vanishing singularity, characterizing blow-up phenomena and their localization near boundary points.

## Contribution

It provides a detailed blow-up analysis for solutions with asymptotically vanishing singularity, including pointwise control and precise blow-up rates, extending understanding of critical growth problems.

## Key findings

- Bounded solutions converge to minimizers of the stationary Schrödinger equation.
- Blow-up solutions are localized at at most one point, either interior or boundary.
- Precise blow-up rates are derived using Pohozaev identity.

## Abstract

We study the asymptotic behavior of a sequence of positive solutions $(u_{\epsilon})_{\epsilon >0}$ as $\epsilon \to 0$ to the family of equations \begin{equation*} \left\{\begin{array}{ll} \Delta u_{\epsilon}+a(x)u_{\epsilon}= \frac{u_{\epsilon}^{2^*(s_{\epsilon})-1}}{|x|^{s_{\epsilon}}}& \hbox{ in }\Omega\\ u_{\epsilon}=0 & \hbox{ on }\partial\Omega. \end{array}\right. \end{equation*} where $(s_{\epsilon})_{\epsilon >0}$ is a sequence of positive real numbers such that $\lim \limits_{\epsilon \rightarrow 0} s_{\epsilon}=0$, $2^{*}(s_{\epsilon}):= \frac{2(n-s_{\epsilon})}{n-2}$ and $\Omega \subset \mathbb{R}^{n}$ is a bounded smooth domain such that $0 \in \partial \Omega$. When the sequence $(u_{\epsilon})_{\epsilon >0}$ is uniformly bounded in $L^{\infty}$, then upto a subsequence it converges strongly to a minimizing solution of the stationary Schr\"{o}dinger equation with critical growth. In case the sequence blows up, we obtain strong pointwise control on the blow up sequence, and then using the Pohozaev identity localize the point of singularity, which in this case can at most be one, and derive precise blow up rates. In particular when $n=3$ or $a\equiv 0$ then blow up can occur only at an interior point of $\Omega$ or the point $0 \in \partial \Omega$.