Positive and sign-changing solutions for Kirchhoff equations with indefinite potential

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.