Hardy–Sobolev inequality with higher dimensional singularity

Abstract

Abstract For N ≥ 4 {N\geq 4} , we let Ω be a smooth bounded domain of ℝ N {\mathbb{R}^{N}} , Γ a smooth closed submanifold of Ω of dimension k, with 1 ≤ k ≤ N - 2 {1\leq k\leq N-2} , and h a continuous function defined on Ω. We denote by ρ Γ ⁢ ( ⋅ ) := dist ⁡ ( ⋅ , Γ ) {\rho_{\Gamma}(\,\cdot\,):=\operatorname{dist}(\,\cdot\,,\Gamma)} the distance function to Γ. For σ ∈ ( 0 , 2 ) {\sigma\in(0,2)} , we study the existence of positive solutions u ∈ H 0 1 ⁢ ( Ω ) {u\in H^{1}_{0}(\Omega)} to the nonlinear equation - Δ ⁢ u + h ⁢ u = ρ Γ - σ ⁢ u 2 * ⁢ ( σ ) - 1 in ⁢ Ω , -\Delta u+hu=\rho_{\Gamma}^{-\sigma}u^{2^{*}(\sigma)-1}\quad\text{in }\Omega, where 2 * ⁢ ( σ ) := 2 ⁢ ( N - σ ) N - 2 {2^{*}(\sigma):=\frac{2(N-\sigma)}{N-2}} is the critical Hardy–Sobolev exponent. In particular, we prove the existence of solution under the influence of the local geometry of Γ and the potential h.