Classification of solutions for an integral system with negative exponents on half space

ABSTRACT

In this paper, we classify each pair of positive solutions of the following general integral system on half space ${\mathbb{R}}_{+}^n$ $\begin{array}{rl}u\left(y\right)& ={\int }_{{\mathbb{R}}_{+}^n}|x-y{|}^{\nu }f\left(v\right)\left(x\right)\mathrm{d}x,y\in \mathrm{\partial }{\mathbb{R}}_{+}^n\\ v\left(x\right)& ={\int }_{\mathrm{\partial }{\mathbb{R}}_{+}^n}|x-y{|}^{\nu }g\left(u\right)\left(y\right)\mathrm{d}y,x\in {\mathbb{R}}_{+}^n.\end{array}$ The main technique we used is the method of moving sphere.

Keywords:

• Asymptotic behavior
• method of moving spheres
• regularity

2010 MSC:

• 45G15
• 45M20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China (NNSF of China) (No. 11901031), the second author is supported by the National Natural Science Foundation of China (NNSF of China) [grant numbers 11801237; 11901031].