Multiple Periodic Solutions of a Class of Fractional Laplacian Equations-Reference-Cited by-同舟云学术

Multiple Periodic Solutions of a Class of Fractional Laplacian Equations

Published:2020-11-20 Issue:1 Volume:21 Page:41-56
ISSN:2169-0375
Container-title:Advanced Nonlinear Studies
language:en
Short-container-title:

Author:

Cui Ying-Xin¹,Wang Zhi-Qiang²

Affiliation:

1. Center for Applied Mathematics , Tianjin University , Tianjin 300072 , P. R. China

2. College of Mathematics and Informatics , Fujian Normal University , Fuzhou 350117 , P. R. China ; and Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

Abstract

Abstract In this paper, we study the existence of multiple periodic solutions for the following fractional equation:

( - Δ ) s ⁢ u + F ′ ⁢ ( u ) = 0 , u ⁢ ( x ) = u ⁢ ( x + T ) x ∈ ℝ .

(-\Delta)^{s}u+F^{\prime}(u)=0,\qquad u(x)=u(x+T)\quad x\in\mathbb{R}.

For an even double-well potential, we establish more and more periodic solutions for a large period T. Without the evenness of F we give the existence of two periodic solutions of the problem. We make use of variational arguments, in particular Clark’s theorem and Morse theory.

Funder

National Natural Science Foundation of China

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics,Statistical and Nonlinear Physics

Link

https://www.degruyter.com/document/doi/10.1515/ans-2020-2113/pdf

Reference23 articles.

1. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

2. V. Ambrosio, On the existence of periodic solutions for a fractional Schrödinger equation, Proc. Amer. Math. Soc. 146 (2018), no. 9, 3767–3775.

3. V. Ambrosio, Periodic solutions for critical fractional problems, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 45.

4. V. Ambrosio, J. Mawhin and G. M. Bisci, (Super)Critical nonlocal equations with periodic boundary conditions, Selecta Math. (N. S.) 24 (2018), no. 4, 3723–3751.

5. X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 1, 23–53.

Cited by 2 articles. 订阅此论文施引文献订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献

1. Periodic solutions of a fractional Schrödinger equation;Applicable Analysis;2023-09-10

2. Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit;Topological Methods in Nonlinear Analysis;2022-12-10