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Research Article | Open Access

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

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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 $G$ :

$- Δ_{G} u - μ \frac{ψ^{2} (ξ) u}{d (ξ)^{2}} = | u |^{p - 2} u + ψ^{α} (ξ) \frac{| u |^{2^{*} (α) - 2} u}{d (ξ)^{α}}, u \in S^{1, 2} (G) .$

Here, $p \in (1, 2^{*}]$ , $α \in (0, 2)$ , $μ \in [0, μ_{G})$ , $2^{*} = \frac{2 Q}{Q - 2}$ , and $2^{*} (α) = \frac{2 (Q - α)}{Q - 2}$ . 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.

Keywords

doubly critical problem carnot group hardy potential pohozaev identity variational method

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Electronic Research Archive

Volume 32 Issue 8,
August 2024

Pages 4969-4990

DOI: 10.3934/era.2024229

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Zhu S. Doubly critical problems involving Sub-Laplace operator on Carnot group. Electronic Research Archive, 2024, 32(8): 4969-4990. https://doi.org/10.3934/era.2024229

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Received: 01 April 2024

Revised: 18 July 2024

Accepted: 31 July 2024

Published: 16 August 2024

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)