Hardy–Sobolev equations on compact Riemannian manifolds

Abstract

Let (M,g) be a compact Riemannian Manifold of dimension n≥3, x0∈M, and s∈(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∈C0(M) to the critical equation Δgu+a(x)u=u2⋆(s)−1dg(x,x0)sin M where Δg≔−divg(∇) is the Laplace–Beltrami operator, and dg is the Riemannian distance on (M,g). Via a minimization method in the spirit of Aubin, we prove existence in dimension n≥4 when the potential a is sufficiently below the scalar curvature at x0. In dimension n=3, we use a global argument and we prove existence when the mass of the linear operator Δg+a is positive at x0. As a byproduct of our analysis, we compute the best first constant for the related Riemannian Hardy–Sobolev inequality.