@article{Zhang2023, 
author = {Jinguo Zhang and Shuhai Zhu},
title = {On criticality coupled sub-Laplacian systems with Hardy type potentials on Stratified Lie groups},
year = {2023},
journal = {Communications in Analysis and Mechanics},
volume = {15},
number = {2},
pages = {70-90},
keywords = {Sub-Laplacian system, critical exponents, Hardy-type potential, Carnot groups},
url = {https://www.sciopen.com/article/10.3934/cam.2023005},
doi = {10.3934/cam.2023005},
abstract = {In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group         {                                              −                      Δ                                          G                                              u          =                                                    ψ                                  α                                                            |                            u                                                |                                                                      2                    ∗                                    (                  α                  )                  −                  2                                            u                                      d              (              z                              )                                  α                                                              +                                    p                              1                                                                    2                ∗                            (              γ              )                                                                          ψ                                  γ                                                            |                            u                                                |                                                                      p                                          1                                                        −                  2                                            u                              |                            v                                                |                                                                      p                                          2                                                                                                          d              (              z              ,                              z                                  0                                                            )                                  γ                                                              +          λ          h          (          z          )                                                    ψ                                  σ                                                            |                            u                                                |                                                  q                  −                  2                                            u                                      d              (              z                              )                                  σ                                                                                                  in                              Ω          ,                                                        −                      Δ                                          G                                              v          =                                                    ψ                                  β                                                            |                            v                                                |                                                                      2                    ∗                                    (                  β                  )                  −                  2                                            v                                      d              (              z                              )                                  β                                                              +                                    p                              2                                                                    2                ∗                            (              γ              )                                                                          ψ                                  γ                                                            |                            u                                                |                                                                      p                                          1                                                                                                  |                            v                                                |                                                                      p                                          2                                                        −                  2                                            v                                      d              (              z              ,                              z                                  0                                                            )                                  γ                                                              +          λ          h          (          z          )                                                    ψ                                  σ                                                            |                            v                                                |                                                  q                  −                  2                                            v                                      d              (              z                              )                                  σ                                                                                                  in                              Ω          ,                                                        u          =          v          =          0                                              on                              ∂          Ω          ,                        where    −      Δ                  G             is a sub-Laplacian on Carnot group        G  ,    α  ,  β  ,  γ  ,  σ  ∈  [  0  ,  2  ),    d is the        Δ                  G            -natural gauge,    ψ  =      |        ∇                  G              d      |   and        ∇                  G             is the horizontal gradient associated to        Δ                  G            . The positive parameters    λ,    q satisfy    0  <  λ  <  ∞,    1  <  q  <  2, and        p          1      ,        p          2        >  1 with        p          1        +      p          2        =      2    ∗    (  γ  ), here        2    ∗    (  α  )  :=            2      (      Q      −      α      )              Q      −      2      ,        2    ∗    (  β  )  :=            2      (      Q      −      β      )              Q      −      2       and        2    ∗    (  γ  )  =            2      (      Q      −      γ      )              Q      −      2       are the critical Hardy-Sobolev exponents,    Q is the homogeneous dimension of the space        G  . By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.}
}