Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve
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: \[-\Delt...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2021-03-01
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| Series: | Opuscula Mathematica |
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| Online Access: | https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4109.pdf |
| _version_ | 1818876344758435840 |
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| author | Idowu Esther Ijaodoro El Hadji Abdoulaye Thiam |
| author_facet | Idowu Esther Ijaodoro El Hadji Abdoulaye Thiam |
| author_sort | Idowu Esther Ijaodoro |
| collection | DOAJ |
| description | 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 \(1+\delta\). |
| first_indexed | 2024-12-19T13:40:54Z |
| format | Article |
| id | doaj.art-f8ef745882864fd9ad27b5466308ad8d |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-19T13:40:54Z |
| publishDate | 2021-03-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-f8ef745882864fd9ad27b5466308ad8d2022-12-21T20:19:01ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742021-03-01412187204https://doi.org/10.7494/OpMath.2021.41.2.1874109Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curveIdowu Esther Ijaodoro0https://orcid.org/0000-0002-9550-9090El Hadji Abdoulaye Thiam1https://orcid.org/0000-0003-1206-8312African Institute for Mathematical Sciences in Senegal, KM 2, Route de Joal, B.P. 14 18, Mbour, SenegalUniversité de Thies, UFR des Sciences et Techniques, Département de Mathématiques, Thies, SenegalWe 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 \(1+\delta\).https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4109.pdfhardy-sobolev inequalitypositive minimizersparametrized curvemassgreen function |
| spellingShingle | Idowu Esther Ijaodoro El Hadji Abdoulaye Thiam Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve Opuscula Mathematica hardy-sobolev inequality positive minimizers parametrized curve mass green function |
| title | Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve |
| title_full | Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve |
| title_fullStr | Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve |
| title_full_unstemmed | Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve |
| title_short | Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve |
| title_sort | influence of an l p perturbation on hardy sobolev inequality with singularity a curve |
| topic | hardy-sobolev inequality positive minimizers parametrized curve mass green function |
| url | https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4109.pdf |
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