Hardy–Sobolev equations on compact Riemannian manifolds

Abstract

Let (M, g) be a compact Riemannian Manifold of dimension n ≥ 3, x 0 ∈ M , and s ∈ (0, 2). We let 2 ⋆ (s) := 2(n−s) n−2

Optimal Hardy–Sobolev inequalities on compact Riemannian manifolds

“…Note that if ∂M = φ then we get the same result as Theorem 4 of Jaber [20] providing that x 0 ∈ M \ ∂M and for all α > 0, u α|∂M ≡ 0.…”

“…The existence of such a sequence of u α follows from the hypothesis of Section 2 and the existence Theorem 4 of Jaber [20]. It follows from the regularity and the maximum principle of [20] that, for any α > 0, u α ∈ C 0,β (M ) ∩ C 2,γ loc (M \ {x 0 }), β ∈ (0, min(1, 2 − s)), γ ∈ (0, 1), and u α > 0.…”

“…When s ∈ (0, 2), the author proved in [20] that A 0 (M, g, x 0 , s) = K(n, s). In particular, for any > 0, there exists B > 0 such that we have:…”

Hardy-Sobolev inequality with singularity a curve

“…The Hardy-Sobolev inequality with interior singularity on Riemannian manifolds have been studied by Jaber [18] and Thiam [24]. Here also the impact of the scalar curvature at the point singularity plays an important role for the existence of minimizers in higher dimensions N ≥ 4.…”

“…Here also the impact of the scalar curvature at the point singularity plays an important role for the existence of minimizers in higher dimensions N ≥ 4. The paper [18] contains also existence result under positive mass condition for N = 3.…”