The minimizing problem involving $p$-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent
In this paper, we study the minimizing problem $$ S_{p,1,\alpha,\mu}:= \inf_{u\in W^{1,p}(\mathbb{R}^{N})\setminus\{0\}} \frac{ \int_{\mathbb{R}^{N}}|\nabla u|^{p}\mathrm{d}x - \mu \int_{\mathbb{R}^{N}} \frac{|u|^{p}}{|x|^{p}} \mathrm{d}x} {\left( \int_{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}} \frac{|u...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2018-08-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=6753 |
| _version_ | 1797830543355674624 |
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| author | Yu Su Haibo Chen |
| author_facet | Yu Su Haibo Chen |
| author_sort | Yu Su |
| collection | DOAJ |
| description | In this paper, we study the minimizing problem
$$
S_{p,1,\alpha,\mu}:=
\inf_{u\in W^{1,p}(\mathbb{R}^{N})\setminus\{0\}}
\frac{
\int_{\mathbb{R}^{N}}|\nabla u|^{p}\mathrm{d}x
-
\mu
\int_{\mathbb{R}^{N}}
\frac{|u|^{p}}{|x|^{p}}
\mathrm{d}x}
{\left(
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{p^{*}_{\alpha}}|u(y)|^{p^{*}_{\alpha}}}{|x-y|^{\alpha}}
\mathrm{d}x
\mathrm{d}y
\right)^{\frac{p}{2\cdot p^{*}_{\alpha}}}},
$$
where $N\geqslant3$, $p\in(1,N)$, $\mu\in\big[0,\big(\frac{N-p}{p}\big)^{p} \big)$, $\alpha\in(0,N)$ and $p^{*}_{\alpha}=\frac{p}{2}\big(\frac{2N-\alpha}{N-p}\big)$ is the Hardy–Littlewood–Sobolev upper critical exponent. Firstly, by using refinement of Hardy–Littlewood–Sobolev inequality, we prove that $S_{p,1,\alpha,\mu}$ is achieved in $\mathbb{R}^{N}$ by a radially symmetric, nonincreasing and nonnegative function. Secondly, we give a estimation of extremal function. |
| first_indexed | 2024-04-09T13:37:46Z |
| format | Article |
| id | doaj.art-e7f3f6464e6f4e349cb8809e781728bf |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:46Z |
| publishDate | 2018-08-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-e7f3f6464e6f4e349cb8809e781728bf2023-05-09T07:53:08ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752018-08-0120187411610.14232/ejqtde.2018.1.746753The minimizing problem involving $p$-Laplacian and Hardy–Littlewood–Sobolev upper critical exponentYu Su0Haibo Chen1Central South University, Changsha, Hunan, ChinaCentral South University, Changsha, Hunan, ChinaIn this paper, we study the minimizing problem $$ S_{p,1,\alpha,\mu}:= \inf_{u\in W^{1,p}(\mathbb{R}^{N})\setminus\{0\}} \frac{ \int_{\mathbb{R}^{N}}|\nabla u|^{p}\mathrm{d}x - \mu \int_{\mathbb{R}^{N}} \frac{|u|^{p}}{|x|^{p}} \mathrm{d}x} {\left( \int_{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}} \frac{|u(x)|^{p^{*}_{\alpha}}|u(y)|^{p^{*}_{\alpha}}}{|x-y|^{\alpha}} \mathrm{d}x \mathrm{d}y \right)^{\frac{p}{2\cdot p^{*}_{\alpha}}}}, $$ where $N\geqslant3$, $p\in(1,N)$, $\mu\in\big[0,\big(\frac{N-p}{p}\big)^{p} \big)$, $\alpha\in(0,N)$ and $p^{*}_{\alpha}=\frac{p}{2}\big(\frac{2N-\alpha}{N-p}\big)$ is the Hardy–Littlewood–Sobolev upper critical exponent. Firstly, by using refinement of Hardy–Littlewood–Sobolev inequality, we prove that $S_{p,1,\alpha,\mu}$ is achieved in $\mathbb{R}^{N}$ by a radially symmetric, nonincreasing and nonnegative function. Secondly, we give a estimation of extremal function.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=6753refinement of hardy–littlewood–sobolev inequalityhardy–littlewood–sobolev upper critical exponent$p$-laplacianminimizing |
| spellingShingle | Yu Su Haibo Chen The minimizing problem involving $p$-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent Electronic Journal of Qualitative Theory of Differential Equations refinement of hardy–littlewood–sobolev inequality hardy–littlewood–sobolev upper critical exponent $p$-laplacian minimizing |
| title | The minimizing problem involving $p$-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent |
| title_full | The minimizing problem involving $p$-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent |
| title_fullStr | The minimizing problem involving $p$-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent |
| title_full_unstemmed | The minimizing problem involving $p$-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent |
| title_short | The minimizing problem involving $p$-Laplacian and Hardy–Littlewood–Sobolev upper critical exponent |
| title_sort | minimizing problem involving p laplacian and hardy littlewood sobolev upper critical exponent |
| topic | refinement of hardy–littlewood–sobolev inequality hardy–littlewood–sobolev upper critical exponent $p$-laplacian minimizing |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=6753 |
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