Abstract
In this paper, we study an integral system \[ \begin{cases} u_{i}(x) = K_{i}(x) (|x|^{\alpha-n} \ast u^{p_{i+1}}_{i+1})(x), &i = 1,2,\ldots,m-1, \\ u_{m}(x) = K_{m}(x) (|x|^{\alpha-n} \ast u^{p_{1}}_{1})(x). \end{cases} \] When $\alpha \in (0,n)$, $p_{i} \gt 0$ ($i = 1,2,\ldots,m$), the Serrin-type condition is critical for existence of positive solutions for some double bounded functions $K_{i}(x)$ ($i = 1,2,\ldots,m$). When $\alpha \in (0,n)$, $p_{i} \lt 0$ ($i = 1,2,\ldots,m$), the system has no positive solution for any double bounded $K_{i}(x)$ ($i = 1,2,\ldots,m$). When $\alpha \gt n$, $p_{i} \lt 0$ ($i = 1,2,\ldots,m$), and $\max_{i} \{-p_{i}\} \gt \alpha/(\alpha-n)$, then the system exists positive solutions increasing with the rate $\alpha-n$.
Citation
Xiaoqian Liu. Yutian Lei. "On the Existence for an Integral System Including $m$ Equations." Taiwanese J. Math. 24 (2) 421 - 437, April, 2020. https://doi.org/10.11650/tjm/190406
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