Title: Hardy-Sobolev equations with asymptotically vanishing singularity: Blow-up analysis for the minimal energy,
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 Schrödinger 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 $0 \in \partial \Omega$.