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Liouville theorem for fractional Hénon–Lane–Emden systems on a half space

Part of: Elliptic equations and systems Miscellaneous topics - Partial differential equations Partial differential equations

Published online by Cambridge University Press:  17 September 2019

Phuong Le
Phuong Le*
Affiliation:
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam and Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam ([email protected])
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Abstract

This paper is concerned with the fractional system

\begin{cases} (-\Delta)^{\frac{\alpha}{2}} u(x) = \vert x \vert ^a v^p(x), &x\in\mathbb{R}^n_+,\\ (-\Delta)^{\frac{\beta}{2}} v(x) = \vert x \vert ^b u^q(x), &x\in\mathbb{R}^n_+,\\ u(x)=v(x)=0, &x\in\mathbb{R}^n{\setminus}\mathbb{R}^n_+, \end{cases}
where n ⩾ 2, 0 < α, β < 2, a > −α, b > −β and p, q ⩾ 1. By exploiting a direct method of scaling spheres for fractional systems, we prove that if $p \leqslant \frac {n+\alpha +2a}{n-\beta }$, $q \leqslant \frac {n+\beta +2b}{n-\alpha }$, $p+q<\frac {n+\alpha +2a}{n-\beta }+\frac {n+\beta +2b}{n-\alpha }$ and (u, v) is a nonnegative strong solution of the system, then u ≡ v ≡ 0.

Keywords

Fractional LaplacianHénon–Lane–Emden systemsLiouville theoremsmethod of scaling sphereshalf spaces

MSC classification

Primary: 35R11: Fractional partial differential equations
Secondary: 35J47: Second-order elliptic systems 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35B09: Positive solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3060 - 3073
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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