## Existence and symmetry of positive solutions of an integral equation system.(English)Zbl 1206.45007

The main result of this paper is the following existence and uniqueness theorem. Assume $$(u,v)$$ is a pair of positive $${C}^1$$ solutions of the system of integral equations
$u(x) = \int_{\mathbb{R}^n} |x-y|^{\alpha -n} v^p(y) dy,\quad v(x) =\int_{\mathbb{R}^n} |x-y|^{\beta -n} u^q(y) dy,$
where $$0<\alpha, \beta < n,$$ $$1<p\leq \frac{n+\alpha}{n-\beta},$$ $$1<q < \frac{n+\beta}{n-\alpha},$$ then this system of integral equations has no positive solution unless $$p=\frac{n+\alpha}{n-\beta}$$ and $$q=\frac{n+\beta}{n-\alpha}.$$ Furthermore, when $$p=\frac{n+\alpha}{n-\beta}$$ and $$q=\frac{n+\beta}{n-\alpha},$$ $$u(x), v(x)$$ are radially symmetric and monotonic about some points with the form
$u(x) = \biggl( \frac{a_1}{d_1+|x-x_0|^2} \biggr )^{\frac{n-\alpha}{2}},\quad v(x) = \biggl( \frac{a_2}{d_2+|x-x_0|^2} \biggr )^{\frac{n-\beta}{2}},$
where $$a_1, d_1, a_2, d_2 >0$$ are constants, and $$x_0\in\mathbb{R}^n$$.

### MSC:

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

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