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

Abstract

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.