Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3

• Published in: Journal of functional analysis Vol. 269; no. 11; pp. 3500 - 3527
• Main Authors: Deng, Yinbin, Peng, Shuangjie, Shuai, Wei
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2015
• Subjects: Kirchhoff-type equations; Nodal solutions; Nonlocal term; Nodal solutions; Nonlocal term; Kirchhoff-type equations
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2015.09.012

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) is a smooth function, a,b are positive constants. Because the so-called nonlocal term (∫R3|∇u|2dx)Δu is involved in the equation, the variational functional of the equation has totally different properties from the case of b=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}, such that ukbns converges in H1(R3) to wk as s→∞, where wk also changes signs exactly k times and solves the following equation−aΔu+V(|x|)u=f(|x|,u),inR3,u∈H1(R3).