MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A FRACTIONAL $\frac{N}{S}$-LAPLACIAN PROBLEM WITH EXPONENTIAL CRITICAL GROWTH AND POTENTIALS COMPETITION

doi:10.1007/s10473-025-0309-1

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (3): 885-918.doi: 10.1007/s10473-025-0309-1

MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A FRACTIONAL $\frac{N}{S}$-LAPLACIAN PROBLEM WITH EXPONENTIAL CRITICAL GROWTH AND POTENTIALS COMPETITION

Wei CHEN¹, Chao JI², Nguyen Van THIN^3,†

1. Chongqing University of Posts and Telecommunications, School of Science, Chongqing 400065, China;
2. School of Mathematics, East China University of Science and Technology, Shanghai 200237, China;
3. Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen street, Thai Nguyen city, Thai Nguyen, Viet Nam; Thang Long Institute of Mathematics and Applied Sciences, Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi, Viet Nam

收稿日期:2023-09-25 修回日期:2024-05-23 出版日期:2025-05-25 发布日期:2025-09-30

MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A FRACTIONAL $\frac{N}{S}$-LAPLACIAN PROBLEM WITH EXPONENTIAL CRITICAL GROWTH AND POTENTIALS COMPETITION

Wei CHEN¹, Chao JI², Nguyen Van THIN^3,†

1. Chongqing University of Posts and Telecommunications, School of Science, Chongqing 400065, China;
2. School of Mathematics, East China University of Science and Technology, Shanghai 200237, China;
3. Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen street, Thai Nguyen city, Thai Nguyen, Viet Nam; Thang Long Institute of Mathematics and Applied Sciences, Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi, Viet Nam

Received:2023-09-25 Revised:2024-05-23 Online:2025-05-25 Published:2025-09-30
Contact: ^†Nguyen Van THIN, E-mail: thinmath@gmail.com and thinnv@tnue.edu.vn
About author:Wei CHEN, E-mail: weichensdu@126.com; Chao JI, E-mail: jichao@ecust.edu.cn
Supported by:
National Natural Science Foundation of China (No. 12171152). The third author was supported by Thang Long University under project with the name "Nevanlinna theory and Kirchhoff-Schrödinger-Hardy type problems for the fractional p-Laplacian" and grant number: 01/2020/STS01.

摘要/Abstract

摘要： By using the Ljusternik-Schnirelmann category and variational method, we study the existence, multiplicity and concentration of solutions to the fractional Schrödinger equation with potentials competition as follows, $$ \varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)|u|^{\frac{N}{s}-2}u=Q(x)h(u)\,\,in\,\, \mathbb R^{N},$$ where $\varepsilon>0$ is a parameter, $s\in (0,1)$, $2\le p<+\infty$ and $N=ps$. The nonlinear term $h$ is a differentiable function with exponential critical growth, the absorption potential $V$ and the reaction potential $Q$ are continuous functions.

关键词: exponential critical growth, fractional $p$-Laplace, Ljusternik-Schnirelmann theory, mountain pass theorem, Trudinger-Moser inequality, variational methods

Abstract: By using the Ljusternik-Schnirelmann category and variational method, we study the existence, multiplicity and concentration of solutions to the fractional Schrödinger equation with potentials competition as follows, $$ \varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)|u|^{\frac{N}{s}-2}u=Q(x)h(u)\,\,in\,\, \mathbb R^{N},$$ where $\varepsilon>0$ is a parameter, $s\in (0,1)$, $2\le p<+\infty$ and $N=ps$. The nonlinear term $h$ is a differentiable function with exponential critical growth, the absorption potential $V$ and the reaction potential $Q$ are continuous functions.

Key words: exponential critical growth, fractional $p$-Laplace, Ljusternik-Schnirelmann theory, mountain pass theorem, Trudinger-Moser inequality, variational methods

中图分类号:

35A15

Wei CHEN, Chao JI, Nguyen Van THIN. MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A FRACTIONAL $\frac{N}{S}$-LAPLACIAN PROBLEM WITH EXPONENTIAL CRITICAL GROWTH AND POTENTIALS COMPETITION[J]. 数学物理学报(英文版), 2025, 45(3): 885-918.

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MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A FRACTIONAL $\frac{N}{S}$-LAPLACIAN PROBLEM WITH EXPONENTIAL CRITICAL GROWTH AND POTENTIALS COMPETITION

MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A FRACTIONAL $\frac{N}{S}$-LAPLACIAN PROBLEM WITH EXPONENTIAL CRITICAL GROWTH AND POTENTIALS COMPETITION

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