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

Abstract

In this article, we study the following weighted integral system: u(x)=∫R+n+1yn+1βf(u(y),v(y))∣x−y∣λdy,x∈R+n+1,v(x)=∫R+n+1yn+1βg(u(y),v(y))∣x−y∣λdy,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 ff and gg, we classify the positive solutions using the method of moving spheres.

Keywords

• method of moving spheres
• weighted integral system
• 45g15
• 45m20