# 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
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