Compactness for the Hardy-Sobolev equation on manifolds

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\ge 3$, and let $h\in C^1(M)$ be such that the operator ${\mathrm{\Delta }}_g+h$ 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} Δ_{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 ${\mathrm{\Delta }}_g:=-{\text{diver}}_g(\mathrm{\nabla })$ and $\text{crits}:=2(n-s)\mathrm{/}(n-2)$. More precisely, we assume $h(x_0)<\frac{(n-2)(6-s)}{12(2n-2-s)}{\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}}_g(x_0),$ when $n\ge 4$, and $h\le \frac{1}{8}\text{sg}$, $h(\text{xo})<\frac{1}{8}\text{sg}(\text{xo})$ when $n=3$. Here, ${\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}}_g$ 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 $n\ge 7$ . \noindent We do not assume any bounds on the energy or the Sobolev norm of the solutions.