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Liouville type theorems for general integral system with negative exponents. (English) Zbl 1401.45007

Summary: In this paper, we establish a Liouville type theorem for the following integral system with negative exponents \[ \begin{cases} u(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} f(u,v)(y)dy,\quad & x \in \mathbb{R}^n, \\ v(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} g(u,v)(y) dy, & x \in \mathbb{R}^n, \end{cases} \] where \(n \geq 1\), \(\nu > 0\), and \(f\), \(g\) are continuous functions defined on \(\mathbb{R}_{+} \times \mathbb{R}_{+}\). Under nature structure conditions on \(f\) and \(g\), we classify each pair of positive solutions for above integral system by using the method of moving sphere in integral forms. Moreover, some other Liouville theorems are established for similar integral systems.

MSC:

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

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