Hardy-Sobolev inequality with singularity a curve

Abstract

Abstract. We consider a bounded domain Ω of R N , N ≥ 3, and h a continuous function on Ω. Let Γ be a closed curve contained in Ω. We study existence of positive solutions u ∈ H 1 0 (Ω) to the equation, σ ∈ (0, 2), and ρ Γ is the distance function to Γ. For N ≥ 4, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case N = 3, we obtain existence of ground-state solution provided the trace of the regular part of the Green of −∆ + h is positive at a point of the curve.

Mass effect on an elliptic PDE involving two Hardy-Sobolev critical exponents

“…where A ∈ R 2 is the vector curvature of Γ and |g| stands for the determinant of g, see [1] for more details related to this parametrization.…”

“…We refer also to [4] for existence of mountain pass solution to a Hardy-Sobolev equation with an additional perturbation term. For the Hardy-Sobolev equations on domains with singularity a curve, we refer to the papers of the author and Fall [1] and the author and Ijaodoro [2]. We also suggest to the interested readers the nice work of Schoen-Yau [5] and [6] for more details related to the positive mass theorem.…”