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Classification of solutions for a system of integral equations. (English) Zbl 1073.45005

Consider the following system of integral equations in \(\mathbb{R}^n\): \[ u(x)= \int_{\mathbb{R}^n}|x-y|^{\alpha- n}v^q(y)\,dy,\quad v(x)= \int_{\mathbb{R}^n}|x-y|^{\alpha-n} u^p(y)\,dy,\tag{1} \] where \(0<\alpha< n\) and \(1/(p+1)+ 1(q+1)= (n- \alpha)/n\). Assume that \(u\in L^{p+1}(\mathbb{R}^n)\) and \(v\in L^{q+1}(\mathbb{R}^n)\). The authors show that if \(u\) and \(v\) are solutions of the system (1), then they are radially symmetric and decreasing about some point \(x_0\).

MSC:

45G15 Systems of nonlinear integral equations
35J60 Nonlinear elliptic equations
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