Doubly critical problems involving Sub-Laplace operator on Carnot group
This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group $ \mathbb{G} $: \begin{document}$ -\Delta_{\mathbb{G}}u-\mu \frac{\psi^{2}(\xi) u }{\text{d}(\xi)^2} = \vert u\vert^{p-2}u +\psi^{\alpha}(\xi)\frac{\vert u\vert^{2^*(\alpha)-2}u}{\...
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AIMS Press
2024-08-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024229 |
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| author | Shuhai Zhu |
| author_facet | Shuhai Zhu |
| author_sort | Shuhai Zhu |
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| description | This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group $ \mathbb{G} $: \begin{document}$ -\Delta_{\mathbb{G}}u-\mu \frac{\psi^{2}(\xi) u }{\text{d}(\xi)^2} = \vert u\vert^{p-2}u +\psi^{\alpha}(\xi)\frac{\vert u\vert^{2^*(\alpha)-2}u}{\text{d}(\xi)^{\alpha}}, \quad u\in S^{1, 2}(\mathbb{G}). $\end{document} Here, $ p\in (1, 2^*] $, $ \alpha\in (0, 2) $, $ \mu\in [0, \mu_{\mathbb{G}}) $, $ 2^* = \frac{2Q}{Q-2} $, and $ 2^*(\alpha) = \frac{2(Q-\alpha)}{Q-2} $. By means of variational techniques, we extended the arguments developed in [1]. In addition, we also established the existence result for the subelliptic system which involved sub-Laplacian and critical homogeneous terms. |
| format | Article |
| id | doaj-art-2d0f6041909845588b459a85a41c5220 |
| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-2d0f6041909845588b459a85a41c52202025-01-23T07:51:27ZengAIMS PressElectronic Research Archive2688-15942024-08-013284969499010.3934/era.2024229Doubly critical problems involving Sub-Laplace operator on Carnot groupShuhai Zhu0College of Basic Science, Ningbo University of Finance and Economics, Ningbo 315175, ChinaThis paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group $ \mathbb{G} $: \begin{document}$ -\Delta_{\mathbb{G}}u-\mu \frac{\psi^{2}(\xi) u }{\text{d}(\xi)^2} = \vert u\vert^{p-2}u +\psi^{\alpha}(\xi)\frac{\vert u\vert^{2^*(\alpha)-2}u}{\text{d}(\xi)^{\alpha}}, \quad u\in S^{1, 2}(\mathbb{G}). $\end{document} Here, $ p\in (1, 2^*] $, $ \alpha\in (0, 2) $, $ \mu\in [0, \mu_{\mathbb{G}}) $, $ 2^* = \frac{2Q}{Q-2} $, and $ 2^*(\alpha) = \frac{2(Q-\alpha)}{Q-2} $. By means of variational techniques, we extended the arguments developed in [1]. In addition, we also established the existence result for the subelliptic system which involved sub-Laplacian and critical homogeneous terms.https://www.aimspress.com/article/doi/10.3934/era.2024229doubly critical problemcarnot grouphardy potentialpohozaev identityvariational method |
| spellingShingle | Shuhai Zhu Doubly critical problems involving Sub-Laplace operator on Carnot group Electronic Research Archive doubly critical problem carnot group hardy potential pohozaev identity variational method |
| title | Doubly critical problems involving Sub-Laplace operator on Carnot group |
| title_full | Doubly critical problems involving Sub-Laplace operator on Carnot group |
| title_fullStr | Doubly critical problems involving Sub-Laplace operator on Carnot group |
| title_full_unstemmed | Doubly critical problems involving Sub-Laplace operator on Carnot group |
| title_short | Doubly critical problems involving Sub-Laplace operator on Carnot group |
| title_sort | doubly critical problems involving sub laplace operator on carnot group |
| topic | doubly critical problem carnot group hardy potential pohozaev identity variational method |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024229 |
| work_keys_str_mv | AT shuhaizhu doublycriticalproblemsinvolvingsublaplaceoperatoroncarnotgroup |