Compactness for the Hardy-Sobolev equation on manifolds

Let $(M, g)$ be a closed Riemannian manifold of dimension $n \geq 3$, and let $h \in C^1(M)$ be such that the operator $\Delta_g + h$ is coercive. Fix $x_0 \in M$ and $s \in (0, 2)$. We obtain uniform bounds on the solutions of the critical \emph{Hardy-Sobolev equation}: \begin{equation}\label{HS0} \tag{{\color{MainRed}HS}} \left\{\begin{array}{ll} \Delta_{g}u + hu = \frac{u^{\crits-1}}{d_{g}(\xo,x)^{s}} & \hbox{ in }M\setminus\{\xo\}, \\ \qquad u > 0 &\hbox{ in }M\setminus\{\xo\}, \end{array}\right. \end{equation} where $\Delta_{g}:=-\diver_{g}(\nabla)$ and $\crits:=2(n-s)/(n-2)$. More precisely, we assume $h(x_0)<\frac{(n-2)(6-s)}{12(2n-2-s)}\mathrm{Scal}_g(x_0),$ when $n \geq 4$, and $h\le\frac{1}{8}\sg$, $h(\xo)<\frac{1}{8}\sg(\xo)$ when $n = 3$. Here, $\mathrm{Scal}_g$ denotes the scalar curvature of $(M, g)$. These conditions were introduced in \cite{HCA4}, and shown to be optimal in \cite{CAR} for a single bubble configuration when $n\ge7$ . \noindent We do not assume any bounds on the energy or the Sobolev norm of the solutions.