Classification of Anti-symmetric Solutions to Fractional Lane-Emden System in Borderline Case

We investigate the existence and non-existence of anti-symmetric solutions to the fractional Lane--Emden system in the half-space $\mathbb{R}^n_+$: \begin{equation*}\begin{cases} (-Δ)^s u = v^p & \text{in } \mathbb{R}^n_+, (-Δ)^s v = u^q & \text{in } \mathbb{R}^n_+, \end{cases}\end{equation*} where $00,2s+p=1$, with $u,v$ nonnegative in $\mathbb{R}^n_+$ and anti-symmetric with respect to $\partial\mathbb{R}^n_+$. Li and Zhuo [2023, Sci. China Math.] previously proved non-existence in $L_{2s}$, as well as non-existence in the extended space $L_{2s+1}$ when $1-2s