MULTIPLE POSITIVE SOLUTIONS FOR THE GENERALIZED QUASILINEAR SCHRÖDINGER EQUATION IN $\mathbb{R}^N$

doi:10.1007/s10473-025-0512-0

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (5): 2010-2028.doi: 10.1007/s10473-025-0512-0

MULTIPLE POSITIVE SOLUTIONS FOR THE GENERALIZED QUASILINEAR SCHRÖDINGER EQUATION IN $\mathbb{R}^N$

Yongpeng CHEN^1,2, Zhipeng YANG^3,4,*

1. Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming 650500, China;
2. School of Science, Guangxi University of Science and Technology, Liuzhou 545006, China;
3. Department of Mathematics, Yunnan Normal University, Kunming 650500, China;
4. Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming 650500, China

收稿日期:2024-01-04 修回日期:2024-11-19 出版日期:2025-09-25 发布日期:2025-10-14

MULTIPLE POSITIVE SOLUTIONS FOR THE GENERALIZED QUASILINEAR SCHRÖDINGER EQUATION IN $\mathbb{R}^N$

Yongpeng CHEN^1,2, Zhipeng YANG^3,4,*

1. Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming 650500, China;
2. School of Science, Guangxi University of Science and Technology, Liuzhou 545006, China;
3. Department of Mathematics, Yunnan Normal University, Kunming 650500, China;
4. Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming 650500, China

Received:2024-01-04 Revised:2024-11-19 Online:2025-09-25 Published:2025-10-14
Contact: ^*Zhipeng Yang, E-mail: yangzhipeng326@163.com
About author:Yongpeng Chen, E-mail: yongpengchen@mail.bnu.edu.cn
Supported by:
Chen's research was supported by the NSFC (12161007) and the Guangxi Natural Science Foundation Project (2023GXNSFAA026190). Yang's research was supported by the National Natural Science Foundation of China (12301145, 12261107), the Yunnan Fundamental Research Projects (202301AU070144, 202401AU070123).

摘要/Abstract

摘要： In this paper, we investigate the generalized quasilinear Schrödinger equation: $$ -\operatorname{div}\left(g^2(u) \nabla u\right)+g(u) g^{\prime}(u)|\nabla u|^2 +u=P(\varepsilon x) |u|^{\\\alpha p-2}u, \quad x \in \mathbb{R}^N, $$ where $N>3$, $g\!\!:\mathbb{R} \rightarrow \mathbb{R}^{+}$ is a $C^1$ even function, $g(0)=1$, $g^{\prime}(s) \geq 0$ for all $s \geq 0$, $g(s)=\beta|s|^{\alpha-1}+O\left(|s|^{\gamma-1}\right)$ as $s \rightarrow \infty$ for some constants $\alpha \in[1,2]$, $\beta>0$, $\gamma<\alpha$ and $(\alpha-1) g(s) \geq g^{\prime}(s) s$ for all $s \geq 0$, $\varepsilon>0$ is a positive parameter, and $p \in\left(2,2^*\right)$. We will study the impact of the nonlinearity's coefficient $P(x)$ on the quantity of positive solutions.

关键词: variational methods, generalized quasilinear equation

Abstract: In this paper, we investigate the generalized quasilinear Schrödinger equation: $$ -\operatorname{div}\left(g^2(u) \nabla u\right)+g(u) g^{\prime}(u)|\nabla u|^2 +u=P(\varepsilon x) |u|^{\\\alpha p-2}u, \quad x \in \mathbb{R}^N, $$ where $N>3$, $g\!\!:\mathbb{R} \rightarrow \mathbb{R}^{+}$ is a $C^1$ even function, $g(0)=1$, $g^{\prime}(s) \geq 0$ for all $s \geq 0$, $g(s)=\beta|s|^{\alpha-1}+O\left(|s|^{\gamma-1}\right)$ as $s \rightarrow \infty$ for some constants $\alpha \in[1,2]$, $\beta>0$, $\gamma<\alpha$ and $(\alpha-1) g(s) \geq g^{\prime}(s) s$ for all $s \geq 0$, $\varepsilon>0$ is a positive parameter, and $p \in\left(2,2^*\right)$. We will study the impact of the nonlinearity's coefficient $P(x)$ on the quantity of positive solutions.

Key words: variational methods, generalized quasilinear equation

中图分类号:

35J62

Yongpeng CHEN, Zhipeng YANG. MULTIPLE POSITIVE SOLUTIONS FOR THE GENERALIZED QUASILINEAR SCHRÖDINGER EQUATION IN $\mathbb{R}^N$[J]. 数学物理学报(英文版), 2025, 45(5): 2010-2028.

Yongpeng CHEN, Zhipeng YANG. MULTIPLE POSITIVE SOLUTIONS FOR THE GENERALIZED QUASILINEAR SCHRÖDINGER EQUATION IN $\mathbb{R}^N$[J]. Acta mathematica scientia,Series B, 2025, 45(5): 2010-2028.

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MULTIPLE POSITIVE SOLUTIONS FOR THE GENERALIZED QUASILINEAR SCHRÖDINGER EQUATION IN $\mathbb{R}^N$

MULTIPLE POSITIVE SOLUTIONS FOR THE GENERALIZED QUASILINEAR SCHRÖDINGER EQUATION IN $\mathbb{R}^N$

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