  In this paper, we consider the quasilinear Schr\"{o}dinger equation
\begin{equation*} -\Delta u+V(x)u-u\Delta(u^2)=g(u),\ \ x\in \mathbb{R}^{3},
\end{equation*} where $V$ and $g$ are continuous functions. Without the
coercive condition on $V$ or the monotonicity condition on $g$, we show that
the problem above has a least energy sign-changing solution and infinitely many
sign-changing solutions. Our results especially solve the problem above in the
case where $g(u)=|u|^{p-2}u$ ($2<p<4$) and complete some recent related works
on sign-changing solutions, in the sense that, in the literature only the case
$g(u)=|u|^{p-2}u$ ($p\geq4$) was considered. The main results in the present
paper are obtained by a new perturbation approach and the method of invariant
sets of descending flow. In addition, in some cases where the functional merely
satisfies the Cerami condition, a deformation lemma under the Cerami condition
is developed.
