Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian

Ran  Zhuo; Wenxiong Chen; Xuewei  Cui; Zixia  Yuan

doi:10.3934/dcds.2016.36.1125

Article Contents

2016, Volume 36, Issue 2: 1125-1141. Doi: 10.3934/dcds.2016.36.1125

This issue Previous Article Exact controllability for first order quasilinear hyperbolic systems with internal controls Next Article

Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian

1.
Department of Mathematical Sciences, Yeshiva University, New York, NY, 10033
2.
Department of Mathematics, Yeshiva University, New York, NY 10033
3.
Department of Applied Mathematics, Northwestern Polytechnical University, Xian 710072, Shanxi
4.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan

Received: June 30, 2014

Revised: January 31, 2015

Published: May 2017

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Abstract

In this paper, we consider the following system of pseudo-differential nonlinear equations in $R^n$ \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u_i (x)= f_i( u_1(x), \cdots u_m(x)), & i=1, \cdots, m, \\ u_i \geq 0 , & i=1, \cdots, m, （1） \end{array} \right. \label{b1} \end{equation} where $\alpha$ is any real number between $0$ and $2$.
We obtain radial symmetry in the critical case and non-existence in the subcritical case for positive solutions.
To this end, we first establish the equivalence between (1) and the corresponding integral system $$ \left\{\begin{array}{ll} u_i(x) = \int_{R^n} \frac{c_n}{|x-y|^{n-\alpha}} f_i( u_1(y), \cdots, u_m(y)), & i=1, \cdots, m, \\ u_i(x) \geq 0, & i=1, \cdots, m. \end{array} \right. $$ A new idea is introduced in the proof, which may hopefully be applied to many other problems. Combining this equivalence with the existing results on the integral system, we obtained much more general results on the qualitative properties of the solutions for (1).

Keywords:

Fractional Laplacian,
pseudo-differential nonlinear system,
equivalence,
integral equations,
method of moving planes in integral forms,
radial symmetry,
non-existence of solutions.

Mathematics Subject Classification: Primary: 31A10, 35B45; Secondary: 35B53, 35J91.

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