The existence of least energy and high energy solutions to the Kirchhoff type problem in high dimensions

Abstract In this paper, we study the following Kirchhoff type problem with critical growth: (0.1) − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = β f ( x ) | u | r − 2 u − g ( x ) | u | q − 2 u + u 2 ⁎ − 1 in R N , where N ≥ 4 , a, b > 0 are constants, β > 0 is a parameter and 2 r q 2 ⁎ . By using variational methods, we obtain the existence of least energy solutions and high energy solutions for certain parameter ranges. Also, we obtain the existence of sign-changing solutions.