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A note on positive supersolutions of the fractional Lane–Emden system

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Abstract

In this paper, we are concerned with the classification of positive supersolutions of the fractional Lane–Emden system

$$\begin{aligned} {\left\{ \begin{array}{ll}(- \Delta )^s u= v^p \text{ in } {\mathbb {R}}^N\\ (- \Delta )^s v= u^q \text{ in } {\mathbb {R}}^N\end{array}\right. }, \end{aligned}$$

where \(p,q\in {\mathbb {R}}\) and \(0<s<1\). We prove that this system has no positive supersolution provided that \(p\le 0\) or \(q\le 0\). Consequently, this together with the results in Leite and Montenegro (Differ Integr Equ 30(11–12):947–974, 2017), Biswas (Nonlinearity 32(6):2246–2268, 2019) completes the classification of positive supersolutions of the system in the full range of p, q. On the other hand, we also provide a simple proof of the nonexistence of positive supersolutions of the system in the case \(p>0,q>0\) and \(pq\le 1\) or \(p>0,q>0,pq>1\) and \(\max \left\{ \frac{2s(p+1)}{pq-1},\frac{2s(q+1)}{pq-1}\right\} > N-2s\).

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Acknowledgements

The authors would like to thank the anonymous referee for the careful reading and helpful suggestions which improve the presentation of the paper. This research was supported by Vietnam Ministry of Education and Training under Grant Number B2019-SPH-02.

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Authors and Affiliations

  1. Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam

    Anh Tuan Duong

  2. Faculty of Fundamental Science, Hanoi University of Industry, Hanoi, Vietnam

    Thi Quynh Nguyen

  3. Ha Long High School For Gifted Student, Ha Long, Quang Ninh, Vietnam

    Thi Hien Anh Vu

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Duong, A.T., Nguyen, T.Q. & Vu, T.H.A. A note on positive supersolutions of the fractional Lane–Emden system. J. Pseudo-Differ. Oper. Appl. 11, 1719–1730 (2020). https://doi.org/10.1007/s11868-020-00365-9

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