# American Institute of Mathematical Sciences

March  2019, 39(3): 1595-1611. doi: 10.3934/dcds.2019071

## Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian

 1 School of mathematics and statistics, Huanghuai University, Zhumadian, Henan 463000, China 2 Department of Mathematics, Baylor University, Waco, TX 76798, USA

* Corresponding author: Ran Zhuo

Received  January 2018 Revised  April 2018 Published  December 2018

Fund Project: The first author is supported by NSFC grant 11701207 and Education Department of Henan Province grant 18B110011.

In this paper, we consider the following Schrödinger systems involving pseudo-differential operator in
 $R^n$
 $\left\{ {\begin{array}{*{20}{l}}{{{( - \Delta )}^{\frac{\alpha }{2}}}u(x) = {u^{{\beta _1}}}(x){v^{{\tau _1}}}(x),}&{{\rm{in}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {R^n},}\\{{{( - \Delta )}^{\frac{\gamma }{2}}}v(x) = {u^{{\beta _2}}}(x){v^{{\tau _2}}}(x),}& \ \ \ {{\rm{ in}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {R^n},}\end{array}} \right.\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$
where
 $α$
and
 $γ$
are any number between 0 and 2,
 $α$
does not identically equal to
 $γ$
.
We employ a direct method of moving planes to partial differential equations (PDEs) (1). Instead of using the Caffarelli-Silvestre's extension method and the method of moving planes in integral forms, we directly apply the method of moving planes to the nonlocal fractional order pseudo-differential system. We obtained radial symmetry in the critical case and non-existence in the subcritical case for positive solutions.
In the proof, combining a new approach and the integral definition of the fractional Laplacian, we derive the key tools, which are needed in the method of moving planes, such as, narrow region principle, decay at infinity. The new idea may hopefully be applied to many other problems.
Citation: Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
##### References:

show all references

##### References:
 [1] Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051 [2] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [3] Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 [4] Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810 [5] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [6] Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 [7] Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 [8] De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431 [9] Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273 [10] Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 [11] Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141 [12] Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065 [13] Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068 [14] Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393 [15] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154 [16] Hongyu Ye. Positive solutions for critically coupled Schrödinger systems with attractive interactions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 485-507. doi: 10.3934/dcds.2018022 [17] Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138 [18] Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166 [19] Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499 [20] Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168

2018 Impact Factor: 1.143