Liouville Theorems and Sharp Homogeneity Defects for Weighted Potential Systems

Abstract

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 $n\ge 2$, $0<\rho <n$, $p,q>0$, and $\alpha ,\beta >-\rho$. The radial kernel is nonincreasing and satisfies $K(r)\asymp r^{\rho -n}$. The coefficients satisfy $0<c_0\le a_0(x),b_0(x)\le C_0$ and $|x|(|\nabla \log a_0(x)|+|\nabla \log b_0(x)|)\le L$. We prove nonexistence if \[ \frac{n+α-\varepsilon_a}{p+1} +\frac{n+β-\varepsilon_b}{q+1}>n-ρ+\varepsilon_K, \] where the defects bound the radial logarithmic derivatives. No symmetry, finite-energy condition, or prescribed decay is assumed. The proof combines a localized virial identity with a source-layer pressure estimate whose constants are uniform under centered dilations, including the low-order range $0<\rho \le 1$. We also identify the necessary scaling range for existence. For radial weighted Newton systems with unequal powers, we prove that the combined coefficient defect has a sharp, unattained infimum equal to the subcritical gap. A fixed bound on the coefficient ratios forces a strictly larger defect, whereas suitable coefficient--solution pairs approach the threshold with excess at most $C/\log M$. In the scalar Newton case, the threshold is explicit and the approximating family converges, after normalization, to an explicit critical weighted profile. The sharpness results concern the Newton kernel; no corresponding construction is asserted for general fractional kernels.

arXiv comment36 pages