  By employing a novel perturbation approach and the method of invariant sets
of descending flow, this manuscript investigates the existence and multiplicity
of sign-changing solutions to a class of semilinear Kirchhoff equations in the
following form $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle
{u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ are constants, $V\in C(\R^3,\R)$,
$f\in C(\R,\R)$. The methodology proposed in the current paper is robust, in
the sense that, the monotonicity condition for the nonlinearity $f$ and the
coercivity condition of $V$ are not required. Our result improves the study
made by Y. Deng, S. Peng and W. Shuai ({\it J. Functional Analysis},
3500-3527(2015)), in the sense that, in the present paper, the nonlinearities
include the power-type case $f(u)=|u|^{p-2}u$ for $p\in(2,4)$, in which case,
it remains open in the existing literature that whether there exist infinitely
many sign-changing solutions to the problem above without the coercivity
condition of $V$. Moreover, {\it energy doubling} is established, i.e., the
energy of sign-changing solutions is strictly large than two times that of the
ground state solutions for small $b>0$.
