Nodal Solutions with a Prescribed Number of Nodes for Quasilinear Schr\"odinger Equations with a Cubic Term

Abstract:

      This paper is concerned with the existence of nodal solutions for the following quasilinear Schr\"odinger equation with a cubic term $$\left\{\begin{array}{l}-\Delta u+V(|x|)u-\frac{1}{2}\Delta(|u|^2)u=\lambda|u|^2u,\ \ \mbox{in}~\mathbb{R}^N,\\u\to 0,~~\mbox{as}\ |x|\to\infty,\end{array}\right.$$ 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 $k\geq 1$, the above equation 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$$\left\{\begin{array}{l}-\Delta u+V(|x|)u=|u|^2u,\ \ \mbox{in}~\mathbb{R}^N,\\ u\to 0,\ \ \mbox{as}\ |x|\to\infty. \end{array}\right.$$ Our results extend the existing ones in the literature from the super-cubic case to the cubic case.