A classification of positive solutions of some integral systems. (English) Zbl 1227.45005

The authors examine the following system of integral equations:
\[ \begin{aligned} & u(x)=\int_{\mathbb R^n}|x-y|^{\alpha-n}e^{v(y)}dy,\\ & v(x)=\int_{\mathbb R^n}|x-y|^{\beta-n}e^{u(y)}dy, \end{aligned} \]
where \(0<\alpha,\beta<n\). The main result of this paper states that if \(\{u,v\}\) is a pair of positive solutions to the above system such that \(e^{u(y)}\in L^r(\mathbb R^n)\) and \(e^{v(y)}\in L^s(\mathbb R^n)\), where \(r,s>1\), \(\frac{n}{s}+\frac{n}{r}=\alpha+\beta\), then \(u\) and \(v\) are radially symmetric and monotone decreasing about some point \(x_0\in\mathbb R^n\). The proofs are based on the method of moving planes.


45G15 Systems of nonlinear integral equations
45M20 Positive solutions of integral equations
45G05 Singular nonlinear integral equations
Full Text: DOI


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