@article{Shao2024, 
author = {Liuyang Shao and Haibo Chen and Yicheng Pang and Yingmin Wang},
title = {Multiplicity of nontrivial solutions for a class of fractional Kirchhoff equations},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {2},
pages = {4135-4160},
keywords = {fractional Kirchhoff, varied method, multiplicity, nontrivial solution},
url = {https://www.sciopen.com/article/10.3934/math.2024203},
doi = {10.3934/math.2024203},
abstract = {In this article, we study a class of fractional Kirchhoff with a superlinear nonlinearity:         {                            M          (                      ∫                                                            R                                                  N                                                                          |                    (          −          △                      )                                          α                2                                              u                                    |                                      2                                d          x          )          (          −          △                      )                          α                                u          +          λ          V          (          x          )          u          =          f          (          x          ,          u          )                              in                                                        R                                      N                                ,                                      u          ∈                      H                          α                                (                                    R                                      N                                )          ,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  N          ≥          1          ,                                                                                          (          1.1          )                        where    λ  >  0 is a parameter,    a and    b are positive numbers satisfying    M  (  t  )  =  a  m  (  t  )  +  b,    m  :            R              +        →            R              +       is continuous.    V  :            R              N        ×      R    →      R   is continuous.    f satisfies        lim                  |            t              |            →      ∞        f  (  x  ,  t  )      /        |    t            |              k      −      1        =  Q  (  x  ) uniformly in    x  ∈            R              N       for each    2  <  k  <      2          α              ∗        ,  (      2          α              ∗        =            2      N              N      −      2      α        ). We investigated the effects of functions    m and    Q on the solution. By applying the variational method, we obtain the existence of multiple solutions. Furthermore, it is worth mentioning that the ground state solution has also been obtained.}
}