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Influence of an lp –perturbation on Hardy-Sobolev inequality with singularity a curve

Ijaodoro Idowu Esther, Thiam El Hadji Abdoulaye

Opuscula Mathematica

2021

Vol. 41, no. 2

187--204

We consider a bounded domain Ω of [formula],[formula] h and b continuous functions on Ω. Let Γ be a closed curve contained in Ω. We study existence of positive solutions [formula] to the perturbed Hardy-Sobolev equation: [formula] where [formula] is the critical Hardy-Sobolev exponent [formula] and [formula] is the distance function to Γ. We show that the existence of minimizers does not depend on the local geometry of Γ nor on the potential h. For N = 3, the existence of ground-state solution may depends on the trace of the regular part of the Green function of —Δ + h and or on b. This is due to the perturbative term of order 1 + δ.