Symmetry for an integral system with general nonlinearity

Yingshu Lü; Chunqin Zhou

doi:10.3934/dcds.2018121

Article Contents

2019, Volume 39, Issue 3: 1533-1543. Doi: 10.3934/dcds.2018121

This issue Previous Article A priori bounds and existence result of positive solutions for fractional Laplacian systems Next Article Liouville's theorem for a fractional elliptic system

Symmetry for an integral system with general nonlinearity

Yingshu Lü^, and
Chunqin Zhou

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Yingshu Lü
* Corresponding author: Yingshu Lü

Received: June 30, 2017

Revised: November 30, 2017

Published: December 2018

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Abstract

In this paper, we study the radial symmetry of the solution to the following system of integral form:

$\left\{ {\begin{array}{*{20}{l}}{{u_i}(x) = \int_{{{\bf{R}}^n}} {\frac{1}{{|x - y{|^{n - \alpha }}}}} {f_i}(u(y))dy,\;\;x \in {{\bf{R}}^n},\;\;i = 1, \cdots ,m,}\\{0 < \alpha < n,\;{\rm{ and }}\;u(x) = ({u_1}(x),{u_2}(x), \cdots ,{u_m}(x)).}\end{array}} \right.\;\;\;\;\;\;\left( 1 \right)$

Here $f_i(s)∈ C^1(\mathbf{R^m_+})\bigcap$$ C^0(\mathbf{\overline{R^m_+}})$$(i = 1,2,···,m)$ are real-valued functions, nonnegative and monotone nondecreasing with respect to the variables $s_1$, $s_2$, $···$, $s_m$. We show that the nonnegative solution $u = (u_1,u_2,···,u_m)$ is radially symmetric in the general condition that $f_i$ satisfies monotonicity condition which contains the critical and subcritical homogeneous degree as special cases. The main technique we use is the method of moving planes in an integral form. Due to our condition here is more general, the more subtle method is needed to deal with this difficulty.

Keywords:

Integral system,
general nonlinearity,
method of moving planes in an integral form,
radial symmetry.

Mathematics Subject Classification: Primary: 45G05, 45G15, 45M99.

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