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Regularity of solutions for a system of integral equations. (English) Zbl 1073.45004

The authors study boundedness properties of positive solutions of system of integral equations of the form \[ \begin{aligned} u(x)&=\int_{\mathbb R^n}| x-y|^{\alpha-n} v^q(y)dy\\ v(x)&=\int_{\mathbb R^n}| x-y|^{\alpha-n} u^p(y)dy\end{aligned}\tag{\(*\)} \] with \(\frac{1}{q+1}+\frac{1}{p+1}=\frac{n-\alpha}{n}\). The following results are established:
Theorem1. Let \((u(x), v(x))\) be a pair of positive solutions of (\(*\)) and \(u\in L^{p+1}(\mathbb R^n)\) and \(v\in L^{q+1}(\mathbb R^n)\). Then \(u(x)\) and \(v(x)\) are uniformly bounded in \(\mathbb R^n\).
Theorem 2. If \(p = q\) in (\(*\)) then \(u = v\) and they both assume the form \[ c\left(\frac{t}{t^2+| x-x_0|^2}\right)^{(n-\alpha)/2} \] with some constant \(c = c(n,\alpha)\) and for some \(t>0\) and \(x_0\in\mathbb R^n\).

MSC:

45G15 Systems of nonlinear integral equations
45M20 Positive solutions of integral equations
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