| This paper is concerned with the existence of the following quasilinear Schr\"odinger equation with a cubic term
\begin{equation}\label{eq1}\left\{\begin{aligned}
&-\Delta u+V(|x|)u-\frac{1}{2}\Delta(|u|^2)u=\lambda|u|^2u,\quad\mbox{in }\mathbb{R}^N,\&u\to 0, \qquad as\ |x|\to\infty,\\end{aligned}\right.\end{equation}
where $N\geq 3,\lambda>0$, the function $V(|x|)$ is a radially symmetric and positive potential. By using the variational method and energy comparison method, for any given integer $k\geq 1$, equation \eqref{eq1} admits a radial nodal solution $U_{k,4}^\lambda$ having exactly $k$ nodes via a limit approach. Furthermore, the energy of $U_{k,4}^\lambda$ is monotonically increasing in $k$ and for any sequence $\{\lambda_n\}$, up to a subsequence, $\lambda_n^{\frac{1}{2}}U_{k,4}^{\lambda_n}$ converges strongly to some $\bar{U}_{k,4}^0$ as $\lambda_n\to +\infty,$ which is a radial nodal solution with exactly $k$ nodes of the classical Schr\"odinger equation
\begin{equation*}\left\{\begin{aligned}
&-\Delta u+V(|x|)u=|u|^2u\quad\mbox{in }\mathbb{R}^N,\&u\to 0 \qquad as\ |x|\to\infty.
\end{aligned}\right.\end{equation*}
Our results extend the existing ones in the literature from the super-cubic case to the cubic case. |
