  In this paper, we consider the following nonlinear Kirchhoff type problem: \[
\left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla
u|^2\right)\Delta u+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in
H^1(\mathbb{R}^3), \end{array}\right. \] where $a,b>0$ are constants, the
nonlinearity $f$ is superlinear at infinity with subcritical growth and $V$ is
continuous and coercive. For the case when $f$ is odd in $u$ we obtain
infinitely many sign-changing solutions for the above problem by using a
combination of invariant sets method and the Ljusternik-Schnirelman type
minimax method. To the best of our knowledge, there are only few existence
results for this problem. It is worth mentioning that the nonlinear term may
not be 4-superlinear at infinity, in particular, it includes the power-type
nonlinearity $|u|^{p-2}u$ with $p\in(2,4]$.
