Liouville Theorems and Sharp Homogeneity Defects for Weighted Potential Systems

In this paper, we consider \[ \begin{cases} u=K*(|x|^αa_0(x)v^p)&\quad\hbox{in }\mathbb R^n,\\ v=K*(|x|^βb_0(x)u^q)&\quad\hbox{in }\mathbb R^n, \end{cases} \] where $n\ge2$, $00$, and $α,β>-ρ$. The radial kernel is nonincreasing and satisfies $K(r)\asymp r^{ρ-n}$. The coefficients satisf