Abstract: In this paper, we consider the nonlinear Kirchhoff type equation with a steep potential well \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,{\rm d}x\Big)\Delta u+\lambda V(x)u=f(u) \qquad {\rm in}\ \mathbb{R}^{3}, \end{equation*} where $a,b>0$ are constants, $\lambda$ is a positive parameter, $V\in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well and the nonlinearity $f\in C(\mathbb{R}, \mathbb{R})$ satisfies certain assumptions. By applying a sign-changing Nehari manifold combined with the method of constructing a sign-changing $(PS)_{C}$ sequence, we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when $\lambda$ is large enough, and find that its energy is strictly larger than twice that of the ground state solutions. In addition, we also prove the concentration of ground state sign-changing solutions.

Key words: Kirchhoff-type equation, ground state sign-changing solutions, steep potential well