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

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 $0 \in \partial \Omega$.