Hardy-Sobolev inequality with singularity a curve

2017/02/07 by Fall, Mouhamed Moustapha, Thiam, El hadji Abdoulaye · 2 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1702.02202

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∈ H¹₀(Ω) to the equation -Δu+h u=ρ^-σ_Γu^{2^*_σ-1} \textrm in Ω where 2^*_σ:=(2(N-σ))/(N-2), 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 -Δ+h is positive at a point of the curve.

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• A Nonlinear elliptic PDE with curve singularity on the boundary
• On solutions to Hardy-Sobolev equations on Riemannian manifolds

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