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June, 2018 Liouville Type Theorems for General Integral System with Negative Exponents
Zhao Liu, Lu Chen, Xumin Wang
Taiwanese J. Math. 22(3): 661-675 (June, 2018). DOI: 10.11650/tjm/170810


In this paper, we establish a Liouville type theorem for the following integral system with negative exponents \[ \begin{cases} u(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} f(u,v)(y) \, dy, & x \in \mathbb{R}^n, \\ v(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} g(u,v)(y) \, dy, & x \in \mathbb{R}^n, \end{cases} \] where $n \geq 1$, $\nu \gt 0$, 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.


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Zhao Liu. Lu Chen. Xumin Wang. "Liouville Type Theorems for General Integral System with Negative Exponents." Taiwanese J. Math. 22 (3) 661 - 675, June, 2018.


Received: 24 April 2017; Revised: 20 July 2017; Accepted: 21 August 2017; Published: June, 2018
First available in Project Euclid: 4 October 2017

zbMATH: 06965391
MathSciNet: MR3807331
Digital Object Identifier: 10.11650/tjm/170810

Primary: 45G15 , 45M20

Keywords: integral system , Liouville type theorems , method of moving spheres

Rights: Copyright © 2018 The Mathematical Society of the Republic of China


Vol.22 • No. 3 • June, 2018
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