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Symmetry for an integral system with general nonlinearity

  • * Corresponding author: Yingshu Lü

    * Corresponding author: Yingshu Lü 
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  • 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.

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


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