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Classification of positive solutions to a Lane-Emden type integral system with negative exponents

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  • In this paper, we classify the positive solutions to the following Lane-Emden type integral system with negative exponents \begin{equation*} \begin{cases} u(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau} u^{-p}(y)v^{-q}(y) \, dy, ~x\in \mathbb{R}^{n}, \\ v(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau}u^{-r}(y)v^{-s}(y) \, dy,~ x\in \mathbb{R}^{n}, \end{cases} \end{equation*}where $n \geq 1$ is an integer and $ \tau, p,q,r,s>0.$ Particularly, using an integral form of the method of moving spheres, we classify the positive solutions to the integral system whenever $$p+q=r+s=1 + 2n/\tau.$$ We also establish the non-existence of positive solutions under the condition $$\max\{p+q,r+s\} \leq 1 + 2n/\tau \,\text{ and }\, p + q + r + s < 2(1 + 2n/\tau).$$
    Mathematics Subject Classification: Primary: 45G15; Secondary: 35B65, 35B53.


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