@article{Yang2024, 
author = {Jie Yang and Lintao Liu and Haibo Chen},
title = {Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {4},
pages = {8353-8370},
keywords = {fractional Laplace operator, Kirchhoff type problems, sign-changing, Nehari manifold, variational methods},
url = {https://www.sciopen.com/article/10.3934/math.2024406},
doi = {10.3934/math.2024406},
abstract = {This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator    (  −  Δ      )          s               {                                        (                          1              +              α                              ∫                                                                            R                                                              3                                                                                                  |                            (              −              Δ                              )                                                      s                    2                                                              u                                                |                                                  2                                            d              x                        )                    (          −          Δ                      )                          s                                u          +          μ          K          (          x          )          u          =          g          (          x          )                      |                    u                                    |                                      p              −              2                                u          ,                                      i            n                                                        R                                      3                                ,                                      u          ∈                      H                          s                                (                                    R                                      3                                )          ,                                  where    α  ,    μ  >  0,    s  ∈  [      3    4    ,  1  ),    2  <  p  <  4. By filtration of the Nehari manifold and variational techniques, we obtained the existence of one and two positive solutions under some conditions imposed on    K and    g.}
}