@article{He2023, 
author = {Shuwen He and Xiaobo Wen},
title = {Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {3},
pages = {6432-6446},
keywords = {variational methods, steep potential well, Kirchhoff-type problem},
url = {https://www.sciopen.com/article/10.3934/math.2023325},
doi = {10.3934/math.2023325},
abstract = {In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well                                 {                                                    −                                  (                                a                +                b                                  ∫                                                                                    R                                            3                                                                                        |                                ∇                u                                                      |                                    2                                d                x                                  )                                Δ                u                +                λ                V                (                x                )                u                =                f                (                x                ,                u                )                +                g                (                x                )                                  |                                u                                                      |                                                        q                    −                    2                                                  u                                                in                                                                      R                                    3                                ,                                                                                                          u                ∈                                  H                  1                                (                                                      R                                    3                                )                ,                                                                  where    a and    b are positive constants,    λ  >  0 is a parameter,    1  <  q  <  2, the potential    V  ∈  C  (            R        3    ,      R    ) and        V          −      1        (  0  ) has a nonempty interior. The functions    f and    g are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as    λ  →  ∞ is also explored.}
}