@article{Hao2023, 
author = {Zhiwei Hao and Huiqin Zheng},
title = {Existence and multiplicity of solutions for fractional    p  (  x  )-Kirchhoff-type problems},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {6},
pages = {3309-3321},
keywords = {the symmetryic mountain pass theorem, Kirchhoff-type problem, fractional p(x)-Laplacian, fractional Sobolev space with variable exponents},
url = {https://www.sciopen.com/article/10.3934/era.2023167},
doi = {10.3934/era.2023167},
abstract = {In this paper, we deal with the existence and multiplicity of solutions for fractional    p  (  x  )-Kirchhoff-type problems as follows:         {                            M                      (                                ∫            Q                                1                          p              (              x              ,              y              )                                                                          |                            v              (              x              )              −              v              (              y              )                                                |                                                  p                  (                  x                  ,                  y                  )                                                                                    |                            x              −              y                                                |                                                  d                  +                  s                  p                  (                  x                  ,                  y                  )                                                              d          x          d          y                      )                    (          −                      Δ                          p              (              x              )                                            )            s                    v          (          x          )                                                                                                                      =          λ                      |                    v          (          x          )                                    |                                      r              (              x              )              −              2                                v          (          x          )          ,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              in                              Ω          ,                                      v          =          0          ,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                in                                              R                        d                    ∖          Ω          ,                        where    (  −      △          p      (      x      )            )    s   is the fractional    p  (  x  )-Laplacian. Different from the previous ones which have recently appeared, we weaken the condition of    M and obtain the existence and multiplicity of solutions via the symmetric mountain pass theorem and the theory of the fractional Sobolev space with variable exponents.}
}