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UNIFORM LOWER BOUND AND LIOUVILLE TYPE THEOREM FOR FRACTIONAL LICHNEROWICZ EQUATIONS

Published online by Cambridge University Press:  21 April 2021

ANH TUAN DUONG
Affiliation:
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Ha Noi, Viet Nam and Thang Long Institute of Mathematics and Applied Sciences, Nghiem Xuan Yem, Hoang Mai, Ha Noi, Viet Nam
VAN HOANG NGUYEN
Affiliation:
Department of Mathematics, FPT University, Ha Noi, Viet Nam e-mail: vanhoang0610@yahoo.com; hoangnv47@fe.edu.vn
THI QUYNH NGUYEN
Affiliation:
Faculty of Fundamental Science, Hanoi University of Industry, Ha Noi, Viet Nam e-mail: nguyen.quynh@haui.edu.vn

Abstract

We study the fractional parabolic Lichnerowicz equation

$$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$

where $p>0$ and $ 0<s<1 $. We establish a Liouville-type theorem for positive solutions in the case $p>1$ and give a uniform lower bound of positive solutions when $0<p\leq 1$. In particular, when v is independent of the time variable, we obtain a similar result for the fractional elliptic Lichnerowicz equation

$$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$

with $p>0$ and $0<s<1$. This extends the result of Brézis [‘Comments on two notes by L. Ma and X. Xu’, C. R. Math. Acad. Sci. Paris 349(5–6) (2011), 269–271] to the fractional Laplacian.

Type
Research Article
Copyright
©2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The research of A. T. Duong is funded by the Vietnam Ministry of Education and Training under grant number B2021-SPH-15.

References

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UNIFORM LOWER BOUND AND LIOUVILLE TYPE THEOREM FOR FRACTIONAL LICHNEROWICZ EQUATIONS
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