INFINITELY MANY SOLUTIONS FOR FRACTIONAL SCHRÖDINGER-MAXWELL EQUATIONS

Jiafa Xu; Zhongli Wei; Donal O'Regan; Yujun Cui

doi:10.11948/2156-907X.20190022

2019 Volume 9 Issue 3

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Jiafa Xu, Zhongli Wei, Donal O'Regan, Yujun Cui. INFINITELY MANY SOLUTIONS FOR FRACTIONAL SCHRÖDINGER-MAXWELL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 1165-1182. doi: 10.11948/2156-907X.20190022

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Jiafa Xu, Zhongli Wei, Donal O'Regan, Yujun Cui. INFINITELY MANY SOLUTIONS FOR FRACTIONAL SCHRÖDINGER-MAXWELL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 1165-1182. doi: 10.11948/2156-907X.20190022

INFINITELY MANY SOLUTIONS FOR FRACTIONAL SCHRÖDINGER-MAXWELL EQUATIONS

1.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
2.
Department of Mathematics, Shandong Jianzhu University, Jinan 250101, China
3.
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
4.
State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Author Bio: Email address: jnwzl32@163.com(Z. Wei); Email address: donal.oregan@nuigalway.ie(D. O'Regan); Email address: cyj720201@163.com(Y. Cui)
Corresponding author: Email address: xujiafa292@sina.com(J. Xu)
Fund Project: The authors were supported by Talent Project of Chongqing Normal University (No. 02030307-0040), the National Natural Science Foundation of China(No. 11601048), Natural Science Foundation of Chongqing (No. cstc2016jcyjA0181), Natural Science Foundation of Chongqing Normal University (No. 16XYY24)

Abstract

Abstract

In this paper using fountain theorems we study the existence of infinitely many solutions for fractional Schrödinger-Maxwell equations $ \begin{cases} (-\Delta)^\alpha u+\lambda V(x)u+\phi u = f(x,u)-\mu g(x)|u|^{q-2}u, \text{ in } \mathbb R^3,\\ (-\Delta)^\alpha \phi = K_\alpha u^2, \text{ in } \mathbb R^3, \end{cases} $ where $ \lambda,\mu >0 $ are two parameters, $ \alpha\in (0,1] $, $ K_\alpha = \frac{\pi^{-\alpha}\Gamma(\alpha)}{\pi^{-(3-2\alpha)/2}\Gamma((3-2\alpha)/2)} $ and $ (-\Delta)^\alpha $ is the fractional Laplacian. Under appropriate assumptions on $ f $ and $ g $ we obtain an existence theorem for this system.
- Fractional Laplacian /
- Schrödinger-Maxwell equations /
- infinitely many solutions
MSC: 35J20, 35J60

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