Hardy-Sobolev Equations on Compact Riemannian Manifolds

Abstract

Let (M,g) be a compact Riemannien Manifold of dimension n > 2, x_0 in M a fix and singular point and s in (0,2). We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. we investigate the existence of positive distributional solutions u in C^0(M) to the critical equation Δ_g u + a(x) u = u^{2*(s)-1}/ d_g(x,x_0)^s in M where Δ_g := - div_g(\nabla) is the Laplace-Beltrami operator, and d_g is the Riemannian distance on (M,g). Via a minimization method in the spirit of Aubin, we prove existence in dimension n > 3 when the potential a is sufficiently below the scalar curvature at x_0. In dimension n = 3, we use a global argument and we prove existence when the mass of the linear operator Δ_g + a is positive at x_0. As a byproduct of our analysis, we compute the best first constant for the related Riemannian Hardy-Sobolev inequality.