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

In this paper, we study the existence and asymptotic behavior of nodal solutions to the following Kirchhoff problem -(a+b integral(3)(R)vertical bar del u vertical bar(2)dx)Delta u + v(vertical bar x vertical bar)u = f(vertical bar x vertical bar, u), in R-3 , u is an element of H-1(R-3), where V(x) is a smooth function, a, b are positive constants. Because the so-called nonlocal term (integral(3)(R)vertical bar del u vertical bar(2)dx)Delta u)Au 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 -> + infinity), there is a subsequence {b(ns)}, such that u(k)(bs) converges in H-1 (R-3) to wk as s -> infinity, where wk also changes signs exactly k times and solves the following equation -a Delta u +V (|x|), u) = f(|x|, u), in R-3, u is an element of H-1 (R-3). (C) 2015 Elsevier Inc. All rights reserved.