Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3R3

In this paper, we study the existence and asymptotic behavior of nodal solutions to the following Kirchhoff problem−(a+b∫R3|∇u|2dx)Δu+V(|x|)u=f(|x|,u),inR3,u∈H1(R3), where V(x)V(x) is a smooth function, a,ba,b are positive constants. Because the so-called nonlocal term (∫R3|∇u|2dx)Δu(∫R3|∇u|2dx)Δu is involved in the equation, the variational functional of the equation has totally different properties from the case of b=0b=0. Under suitable construction conditions, we prove that, for any positive integer k  , the problem has a sign-changing solution ukb, which changes signs exactly k   times. Moreover, the energy of ukb is strictly increasing in k  , and for any sequence {bn}→0+(n→+∞), there is a subsequence {bns}{bns}, such that ukbns converges in H1(R3)H1(R3) to wkwk as s→∞s→∞, where wkwk also changes signs exactly k times and solves the following equation−aΔu+V(|x|)u=f(|x|,u),inR3,u∈H1(R3).