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.
Categories: math.AP