Liouville Theorems and Sharp Homogeneity Defects for Weighted Potential Systems
Abstract
In this paper, we consider u=K*(|x|αa0(x)vp)& in ℝn, v=K*(|x|βb0(x)uq)& in ℝn, where n≥2, 0<ρ<n, p,q>0, and α,β>−ρ. The radial kernel is nonincreasing and satisfies K(r)asymp rρ−n. The coefficients satisfy 0<c0≤ a0(x),b0(x)≤ C0 and |x|(|∇log a0(x)|+|∇log b0(x)|)≤ L. We prove nonexistence if (n+α−εa)/(p+1) +(n+β−εb)/(q+1)>n−ρ+εK, 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<ρ≤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.
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BibTeX
@misc{deng2026liouville,
title = {{Liouville Theorems and Sharp Homogeneity Defects for Weighted Potential Systems}},
author = {Shengbing Deng and Bo Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31167},
eprint = {2610.31167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31167},
}
RIS
TY - UNPB
TI - Liouville Theorems and Sharp Homogeneity Defects for Weighted Potential Systems
AU - Shengbing Deng
AU - Bo Zhang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31167
UR - https://arxiv.org/abs/2610.31167
AB - In this paper, we consider u=K*(|x|^αa_0(x)v^p)& in ℝ^n, v=K*(|x|^βb_0(x)u^q)& in ℝ^n, where n≥2, 0<ρ<n, p,q>0, and α,β>−ρ. The radial kernel is nonincreasing and satisfies K(r)asymp r^ρ−n. The coefficients satisfy 0<c_0≤ a_0(x),b_0(x)≤ C_0 and |x|(|∇log a_0(x)|+|∇log b_0(x)|)≤ L. We prove nonexistence if (n+α−ε_a)/(p+1) +(n+β−ε_b)/(q+1)>n−ρ+ε_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<ρ≤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.
ER -