A Doubly Critical Elliptic Problem with Submanifold Singularities

Abstract

Let N ≥ 4 and Ω be a bounded domain in ℝᴺ containing a smooth closed submanifold Σ of dimension k with 2 ≤ k ≤ N-2. We study the existence of positive solutions to a doubly critical Hardy–Sobolev elliptic equation whose singular weights are powers of the distance to Σ, with two distinct critical exponents 2s₁^* and 2s₂^* corresponding to the parameters s₁ and s₂ with 0 ≤ s₂ < s₁ < 2. Using the mountain pass lemma and asymptotic expansions of the energy functional in Fermi coordinates, we prove that a positive solution exists whenever a geometric quantity involving the mean curvature H, the scalar curvature Rg of Σ, and the potential h satisfies a negativity condition at some point of Σ.