# American Institute of Mathematical Sciences

March  2019, 39(3): 1533-1543. doi: 10.3934/dcds.2018121

## Symmetry for an integral system with general nonlinearity

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

* Corresponding author: Yingshu Lü

Received  July 2017 Revised  December 2017 Published  April 2018

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.
Citation: Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121
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