Opuscula Math. 41, no. 2 (2021), 187-204
https://doi.org/10.7494/OpMath.2021.41.2.187

Opuscula Mathematica

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

Idowu Esther Ijaodoro

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)
• https://orcid.org/0000-0003-1206-8312
• Université de Thies, UFR des Sciences et Techniques, Département de Mathématiques, Thies, Senegal
• Communicated by Patrizia Pucci.
• Revised: 2021-02-15.
• Accepted: 2021-02-17.
• Published online: 2021-03-17.

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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