We deal with the nonlinear Kirchhoff problem

$$-\left(a+b{\displaystyle {\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}x}\right)\mathrm{\Delta }u+V(x)u=f(u),x\in {\mathbb{R}}^3,\left(\mathcal{P}\right)$$

where $a$ is a positive constant, $b>0$ is a parameter, the potential function $V$ is allowed to change its sign, and the nonlinearity $f\in C(\mathbb{R},\mathbb{R})$ exhibits subcritical growth. Under some suitable conditions on $V$, we first prove that the problem has a positive ground state solution for all $b>0$. Then, by using a more general global compactness lemma and a sign-changing Nehari manifold, combined with the method of constructing a sign-changing $(PS{)}_c$ sequence, we show the existence of a least energy sign-changing solution for $b>0$ that is sufficiently small. Moreover, the asymptotic behavior $b\searrow 0$ is established.

In this paper, we consider the following Schrödinger-Poisson system

$$\{\begin{array}{ll}-\mathrm{\Delta }u+V(x)u+\varphi u=|u{|}^{p-2}u+\lambda K(x)|u{|}^{q-2}u& \mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi =u^2& \mathrm{i}\mathrm{n}{\mathbb{R}}^3.\end{array}$$

Under the weakly coercive assumption on $V$ and an appropriate condition on $K$, we investigate the cases when the nonlinearities are of concave-convex type, that is, $1<q<2$ and $4<p<6$. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that $\lambda \in (-\mathrm{\infty },{\lambda }_{\ast })$, where ${\lambda }_{\ast }>0$ is a constant.