Opuscula Math. 41, no. 2 (2021), 187-204

Opuscula Mathematica

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

Idowu Esther Ijaodoro
El Hadji Abdoulaye Thiam

Abstract. We consider a bounded domain \(\Omega\) of \(\mathbb{R}^N\), \(N \geq 3\), \(h\) and \(b\) continuous functions on \(\Omega\). Let \(\Gamma\) be a closed curve contained in \(\Omega\). We study existence of positive solutions \(u \in H^1_0(\Omega)\) to the perturbed Hardy-Sobolev equation: \[-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,\] where \(2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}\) is the critical Hardy-Sobolev exponent, \(\sigma\in [0,2)\), \(0\lt\delta\lt\frac{4}{N-2}\) and \(\rho_{\Gamma}\) is the distance function to \(\Gamma\). We show that the existence of minimizers does not depend on the local geometry of \(\Gamma\) 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 \(-\Delta+h\) and or on \(b\). This is due to the perturbative term of order \(1+\delta\).

Keywords: Hardy-Sobolev inequality, positive minimizers, parametrized curve, mass, Green function.

Mathematics Subject Classification: 35J91, 35J20, 35J75.

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  • El Hadji Abdoulaye Thiam (corresponding author)
  • ORCID iD https://orcid.org/0000-0003-1206-8312
  • Université de Thies, UFR des Sciences et Techniques, Département de Mathématiques, Thies, Senegal
  • Communicated by Patrizia Pucci.
  • Received: 2020-07-15.
  • Revised: 2021-02-15.
  • Accepted: 2021-02-17.
  • Published online: 2021-03-17.
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Cite this article as:
Idowu Esther Ijaodoro, El Hadji Abdoulaye Thiam, Influence of an Lp-perturbation on Hardy-Sobolev inequality with singularity a curve, Opuscula Math. 41, no. 2 (2021), 187-204, https://doi.org/10.7494/OpMath.2021.41.2.187

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