Ground State Sign-Changing Solutions for a Kirchhoff Equation with Asymptotically 3-Linear Nonlinearity

Abstract

In this paper, we investigate the following nonlinear Kirchhoff equation

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^{2}dx\right) \triangle u+V(x)u=f(u),&x\in {\mathbb {R}}^3, \end{aligned}$$

where a, b are positive constants, V is a potential and $f\in C({\mathbb {R}}^3\times {\mathbb {R}},{\mathbb {R}})$ is an asymptotically 3-linear nonlinearity. By using sign-changing Nehari manifold, we can not only get the existence of ground state sign-changing solution under the condition of $b>0$. but also can prove that its energy is strictly larger than twice that of ground state solutions. In addition, we obtain a convergence property of $u_{b_n}$ as $b_n\rightarrow 0$. In this article, we present weaker hypotheses than those that can be found (Xie in Adv Differ Equ 121:1–14, 2016).