Liouville Theorems and Sharp Homogeneity Defects for Weighted Potential Systems

In this paper, we consider $$\{\begin{array}{ll}u=K\ast (\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{a}_0(x)v^p)& \text{in }{\mathbb{R}}^n,\\ v=K\ast (\mathrm{\mid }x{\mathrm{\mid }}^{\beta }{b}_0(x)u^q)& \text{in }{\mathbb{R}}^n,\end{array}$$ 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 $\mathrm{\mid }x\mathrm{\mid }(\mathrm{\mid }\mathrm{\nabla }\log a_0(x)\mathrm{\mid }+\mathrm{\mid }\mathrm{\nabla }\log b_0(x)\mathrm{\mid })\le L$. We prove nonexistence if $$\frac{n+\alpha -{\epsilon }_a}{p+1}+\frac{n+\beta -{\epsilon }_b}{q+1}>n-\rho +{\epsilon }_K,$$ 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 $C\mathrm{/}\log M$. 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.