## A classification of positive solutions of some integral systems.(English)Zbl 1227.45005

The authors examine the following system of integral equations:
\begin{aligned} & u(x)=\int_{\mathbb R^n}|x-y|^{\alpha-n}e^{v(y)}dy,\\ & v(x)=\int_{\mathbb R^n}|x-y|^{\beta-n}e^{u(y)}dy, \end{aligned}
where $$0<\alpha,\beta<n$$. The main result of this paper states that if $$\{u,v\}$$ is a pair of positive solutions to the above system such that $$e^{u(y)}\in L^r(\mathbb R^n)$$ and $$e^{v(y)}\in L^s(\mathbb R^n)$$, where $$r,s>1$$, $$\frac{n}{s}+\frac{n}{r}=\alpha+\beta$$, then $$u$$ and $$v$$ are radially symmetric and monotone decreasing about some point $$x_0\in\mathbb R^n$$. The proofs are based on the method of moving planes.

### MSC:

 45G15 Systems of nonlinear integral equations 45M20 Positive solutions of integral equations 45G05 Singular nonlinear integral equations
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### References:

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