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). 

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Abstract: 
This paper is concerned with the existence of nodal solutions for the following quasilinear Schr\"odinger equation with a cubic term $$\left\{\begin{array}{l}\Delta u+V(x)u\frac{1}{2}\Delta(u^2)u=\lambdau^2u,\ \ \mbox{in}~\mathbb{R}^N,\\u\to 0,~~\mbox{as}\ x\to\infty,\end{array}\right.$$ 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 supercubic case to the cubic case. 
Citation: 
DOI:10.3770/j.issn:20952651.2024.05.009 
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