Doubly critical problems involving Sub-Laplace operator on Carnot group

Shuhai Zhu; Shuhai Zhu

doi:10.3934/era.2024229

Electronic Research Archive

2024, Volume 32, Issue 8: 4969-4990. doi: 10.3934/era.2024229

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Doubly critical problems involving Sub-Laplace operator on Carnot group

Shuhai Zhu ^,

College of Basic Science, Ningbo University of Finance and Economics, Ningbo 315175, China

Received: 01 April 2024 Revised: 18 July 2024 Accepted: 31 July 2024 Published: 16 August 2024

This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group $ \mathbb{G} $:
$ -\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}). $
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.
- doubly critical problem,
- carnot group,
- hardy potential,
- pohozaev identity,
- variational method
Citation: Shuhai Zhu. Doubly critical problems involving Sub-Laplace operator on Carnot group[J]. Electronic Research Archive, 2024, 32(8): 4969-4990. doi: 10.3934/era.2024229

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Abstract

This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group $ \mathbb{G} $:

$ -\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}). $

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.

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