Radial symmetry results for systems of integral equations on \(\Omega \subset {\mathbb{R}}^n\)
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.
scientific article; zbMATH DE number 6006487
scientific article