A Liouville theorem for the Stokes system in a half-space

2013 ◽  
Vol 195 (1) ◽  
pp. 13-19 ◽  
Author(s):  
H. Jia ◽  
G. Seregin ◽  
V. Sverak
Keyword(s):  
Half Space ◽  
Stokes System ◽  
Liouville Theorem

scholarly journals A Liouville theorem of parabolic Monge-AmpÈre equations in half-space

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Ziwei Zhou ◽  
◽  
Jiguang Bao ◽  
Bo Wang ◽  
Keyword(s):  
Half Space ◽  
Liouville Theorem ◽  
Monge Ampere

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

2019 ◽  
Vol 150 (6) ◽  
pp. 3060-3073
Author(s):  
Phuong Le
Keyword(s):  
Strong Solution ◽  
Half Space ◽  
Liouville Theorem ◽  
Direct Method ◽  
Fractional Systems ◽  
Fractional System ◽  
Beta 2 ◽  
Image Position ◽  
A Direct Method

AbstractThis 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.

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