Article ID Journal Published Year Pages File Type
4589705 Journal of Functional Analysis 2015 28 Pages PDF
Abstract

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).

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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