Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems inR3

Abstract

In this paper, we study the existence and asymptotic behavior of nodal solutions to the following Kirchhoff problem − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( | x | ) u = f ( | x | , u ) , in R 3 , u ∈ H 1 ( R 3 ) , where V ( x ) is a smooth function, a , b are positive constants. Because the so-called nonlocal term ( ∫ R 3 | ∇ u | 2 d x ) Δ u 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 u k b , which changes signs exactly k times. Moreover, the energy of u k b is strictly increasing in k , and for any sequence { b n } → 0 + ( n → + ∞ ) , there is a subsequence { b n s } , such that u k b n s converges in H 1 ( R 3 ) to w k as s → ∞ , where w k also changes signs exactly k times and solves the following equation − a Δ u + V ( | x | ) u = f ( | x | , u ) , in R 3 , u ∈ H 1 ( R 3 ) .