Classification of positive solutions for a weighted integral system on the half-space

In this article, we study the following weighted integral system: u ( x ) = ∫ R + n + 1 y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n + 1 , v ( x ) = ∫ R + n + 1 y n + 1 β g ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n + 1 . \left\{\begin{array}{l}u\left(x)=\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}_{+}^{n+1}}\frac{{y}_{n+1}^{\beta }f\left(u(y),v(y))}{{| x-y| }^{\lambda }}{\rm{d}}y,\hspace{1em}x\in {{\mathbb{R}}}_{+}^{n+1},\hspace{1.0em}\\ v\left(x)=\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}_{+}^{n+1}}\frac{{y}_{n+1}^{\beta }g\left(u(y),v(y))}{{| x-y| }^{\lambda }}{\rm{d}}y,\hspace{1em}x\in {{\mathbb{R}}}_{+}^{n+1}.\hspace{1.0em}\end{array}\right. Under nature structure conditions on f f and g g , we classify the positive solutions using the method of moving spheres.