  In this paper, we consider the following Kirchhoff type equation $$ -\left(a+
b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where
$a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive,
bounded and satisfies suitable decay assumptions. By using a new perturbation
approach together with a new version of global compactness lemma of Kirchhoff
type, we prove the existence and multiplicity of bound state solutions for the
above problem with a general nonlinearity. We especially point out that neither
the corresponding Ambrosetti-Rabinowitz condition nor any monotonicity
assumption is required for $f$. Moreover, the potential $V$ may not be radially
symmetry or coercive. As a prototype, the nonlinear term involves the
power-type nonlinearity $f(u) = |u|^{p-2}u$ for $p\in (2, 6)$. In particular,
our results generalize and improve the results by Li and Ye (J.Differential
Equations, 257(2014): 566-600), in the sense that the case $p\in(2,3]$ is left
open there.
