Diff. Geom, Math. Phys., PDE Seminar: Hassan Jaber

Let (M,g) be a compact Riemannian Manifold without boundry of dimension n \geq 3, x_0 \in M, and s \in (0,2). We let \crit: = \frac{2(n-s)}{n-2} be the critical Hardy-Sobolev exponent. I investigate the influence of geometry on the existence of positive distributional solutions u \in C^0(M) for the critical equation

 

\Delta_g u+a(x) u=\frac{u^{\crit-1}}{d_g(x,x_0)^s}  \;\; \hbox{ in} \ M.

 

Via a minimization method, I prove existence in dimension n\geq 4 when the potential a is sufficiently below the scalar curvature at x_0. In dimension n=3, using a global argument, i prove existence when the mass of the linear operator \Delta_g + a is positive at x_0. On the other hand, by using a Blow-up around x_0, i prove that the sharp constant of the related Hardy-Sobolev inequalities on (M,g), which is equal to the one of the Euclidean Hardy-Sobolev inequalities, is achieved for all compact Riemannian Manifold of dimension n \geq 3 with or without boundary.