Nodal Solutions with a Prescribed Number of Nodes for Quasilinear Schr\"odinger Equations with a Cubic Term
Received:November 12, 2023  Revised:April 25, 2024
Key Words: quasilinear Schr\"odinger equations   nodal solutions   limit approach   variational method
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.\,12001188) and the Natural Science Foundation of Hunan Province (Grant No.\,2022JJ30235).
 Author Name Affiliation Jing LAI College of Mathematics and Computing Science, Hunan University of Science and Technology, Hunan 411201, P. R. China Na LIU College of Mathematics and Computing Science, Hunan University of Science and Technology, Hunan 411201, P. R. China Tao WANG College of Mathematics and Computing Science, Hunan University of Science and Technology, Hunan 411201, P. R. China
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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.