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

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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, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [3] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3821-3836. doi: 10.3934/dcdss.2020436 [4] 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 [5] 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 [6] 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 [7] Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 [8] Guofa Li, Yisheng Huang. Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021214 [9] Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 [10] Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079 [11] Trad Alotaibi, D. D. Hai, R. Shivaji. Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4655-4666. doi: 10.3934/cpaa.2020131 [12] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1857-1870. doi: 10.3934/dcdss.2020461 [13] 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 [14] 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 [15] Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141 [16] Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065 [17] 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 [18] Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068 [19] 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 [20] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154

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