Optimal Hardy–Sobolev inequalities on compact Riemannian manifolds

Best constant and Mountain-Pass solutions for a supercritical Hardy-Sobolev problem in the presence of symmetries

“…We denote ū = u • Π, where Π is the projection of x on the (n − k)-Riemannian manifold N := B r0 (Gx 0 )/G ′ . By (12), there exists…”

“…We use the following version of the Mountain-Pass Lemma by Ambrosetti-Rabinowitz seen in Jaber [12]: Concerning terminology, we say that a sequence (u N ) N ∈N ∈ E is a Palais-Smale sequence (PS) for J ∈ C 1 (E) at the level β ∈ R if J(u N ) → β and J ′ (u N ) → 0 strongly in the dual E ′ Theorem 13 (Ambrosetti-Rabinowitz [1]). Let J ∈ C 1 (E, R), where (E, | • ∥ E ) is a Banach space.…”

“…The existence of u is a consequence of the boundedness of (u N ) and the compactness of the subcritical embeddings. Passing to the limit in J ′ q (u N ) = o(1) yields J ′ q (u) = 0: we refer to Jaber [12] and Mesmar [14] for details.…”

“…Note that the case s = 0 was handled by Faget [7] and the case k = 0 was handled by Jaber [12]. From this inclusion, we define the first best constant of the Hardy-Sobolev inequalities in the presence of isometries.…”

Asymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy–Sobolev critical exponent

“…In the interior singularity case, the remainder term of the Hardy-Sobolev inequality is studied by [18]. The optimal Hardy-Sobolev inequality on compact Riemannian manifold is also studied due to [15].…”