Existence and Asymptotic Behavior of Localized Nodal Solutions for a Class of Kirchhoff-Type Equations

In this paper, we study the existence and asymptotic behavior of localized nodal solutions for the following Kirchhoff-type equation - ε 2 a + ε b ∫ R 3 | ∇ u | 2 d x Δ u + V ( x ) u = | u | p - 2 u , x ∈ R 3 , where a > 0 , b > 0 and 4 < p < 6 . Under only a local condition that V has a local trapping potential well, when ε > 0 is sufficiently small, we construct the existence of a sequence of localized nodal solutions concentrating around the local minimum points of the potential function V by using variational method and penalization approach. Moreover, we regard b as a parameter and study the asymptotic behavior of the nodal solutions as b ↘ 0 , which reflects some relationship between b > 0 and b = 0 .