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Radial symmetry results for systems of integral equations on \(\Omega \subset {\mathbb{R}}^n\). (English) Zbl 1246.45013

The authors investigate positive solutions of the integral equations with weight Riesz potential on the bounded domain \(\Omega\subseteq\mathbb{R}^n\): \[ \begin{gathered} u(x)= \int_\Omega|x|^{-\alpha} |x-y|^{-\mu}|y|^{-\beta} u^a(y) v^b(y)\,dy,\\ v(x)= \int_\Omega |x|^{-\gamma} |x-y|^{-\nu} |y|^{-k} u^c(y) v^d(y)\,dy,\\ u(x)> 0,\quad u(x)\equiv A,\end{gathered}\tag{1} \] where \[ \begin{gathered} a\geq 0,\;b\geq 0,\;c\geq 0,\;d\geq 0,\;A> 0,\;B> 0,\;0\leq\alpha< n,\;0\leq\beta< n,\\ 0\leq\gamma< n,\;0\leq k< n,\;0<\mu< n,\;0<\nu< n.\end{gathered}\tag{2} \] The authors prove that if \(\Omega\) is a \(C^1\) domain, \(0\in\Omega\), the system (1), with conditions (2), have positive solutions \((u,v)\in L^p(\Omega)\times L^q(\Omega)\), such that \(p> 1\), \(q> 1\), \[ \begin{aligned} (a- 1)p^{-1}+ bq^{-1} &= (n-\mu-\alpha-\beta) n^{-1},\\ cp^{-1}+ (d- 1)q^{-1} &= (n-\nu-\gamma- k)n^{-1},\end{aligned} \] then \(\Omega\) must be ball, \(u\) and \(v\) must be radially symmetric and monotonic with the radius of the ball.
The authors, also investigate the symmetry for system of integral equation weight Riesz potential on an exterior domain.

MSC:

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