@article{Zhang2025, 
author = {Shiyong Zhang and Qiongfen Zhang},
title = {Normalized solution for a kind of coupled Kirchhoff systems},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {2},
pages = {600-612},
keywords = {normalized solutions, nonlinearity, coupled Kirchhoff equation, Pohožaev manifold},
url = {https://www.sciopen.com/article/10.3934/era.2025028},
doi = {10.3934/era.2025028},
abstract = {In this paper, we investigate the existence of a normalized solution for the following Kirchhoff system in the entire space              R        N   (   N  ≥  3):                                 {                                                    −                                  (                                      1                    +                                          ∫                                                                                                    R                                                    N                                                                                                            |                                        ∇                    u                                                                  |                                            2                                        d                    x                                    )                                Δ                u                =                                  λ                  1                                u                +                                  μ                  1                                                  |                                u                                                      |                                                        p                    −                    2                                                  u                +                β                                  r                  1                                                  |                                u                                                      |                                                                              r                      1                                        −                    2                                                  u                                  |                                v                                                      |                                                                              r                      2                                                                      ,                                                                    −                                  (                                      1                    +                                          ∫                                                                                                    R                                                    N                                                                                                            |                                        ∇                    v                                                                  |                                            2                                        d                    x                                    )                                Δ                v                =                                  λ                  2                                v                +                                  μ                  2                                                  |                                v                                                      |                                                        q                    −                    2                                                  v                +                β                                  r                  2                                                  |                                u                                                      |                                                                              r                      1                                                                                        |                                v                                                      |                                                                              r                      2                                        −                    2                                                  v                ,                                                                                                                                                              (                              P                          )            under the constraints        ∫                            R                N                  |    u            |        2    d  x  =      m    1      and        ∫                            R                N                  |    v            |        2    d  x  =      m    2  , where        m    1    ,      m    2    >  0 are prescribed. The parameters        μ    1    ,      μ    2    ,  β  >  0,    2  ≤  p  ,  q  <  2  +      8    N  ,        r    1    ,      r    2    >  1  , and satisfy              r      1        +            r      2        =      2    ∗    =            2      N              N      −      2      . The frequencies        λ    1    ,      λ    2   appear as Lagrange multipliers. With the help of the Pohožaev manifold and the minimization of the energy functional over a combination of the mass constraints and the closed balls, we obtain a positive ground state solution to (P). We mainly extend the results of Yang (Normalized ground state solutions for Kirchhoff-type systems) concerning the above problem from a single critical to a coupled critical nonlinearity.}
}