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Integral Volume Criteria for Semilinear and Quasilinear Liouville Theorems on Complete Manifolds
Abstract
Let M be a connected complete noncompact Riemannian manifold without boundary, and let Vo(R) denote its ball volume. We prove Liouville theorems without any curvature assumption. For p>1 and q>p−1, divergence of ∫∞ Rpq/(p−1)−1Vo(R)−(q−p+1)/(p−1)dd R forces every nonnegative Sobolev weak supersolution of −Δpu≥ uq to vanish. For q>max{1,p−1}, divergence of ∫∞ Rp−1Vo(R)−(q−p+1)dd R forces every nonnegative global distributional energy supersolution of ut−Δpu≥ uq to vanish. Consequently, the corresponding Cauchy problem has no global solution in this class with a nonzero nonnegative local L1 initial trace.
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BibTeX
@misc{qin2026integral,
title = {{Integral Volume Criteria for Semilinear and Quasilinear Liouville Theorems on Complete Manifolds}},
author = {Yongheng Qin and Yuhua Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22803},
eprint = {2610.22803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22803},
}
RIS
TY - UNPB
TI - Integral Volume Criteria for Semilinear and Quasilinear Liouville Theorems on Complete Manifolds
AU - Yongheng Qin
AU - Yuhua Sun
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22803
UR - https://arxiv.org/abs/2610.22803
AB - Let M be a connected complete noncompact Riemannian manifold without boundary, and let V_o(R) denote its ball volume. We prove Liouville theorems without any curvature assumption. For p>1 and q>p−1, divergence of ∫^∞ R^pq/(p−1)−1V_o(R)^−(q−p+1)/(p−1)dd R forces every nonnegative Sobolev weak supersolution of −Δ_pu≥ u^q to vanish. For q>max{1,p−1}, divergence of ∫^∞ R^p−1V_o(R)^−(q−p+1)dd R forces every nonnegative global distributional energy supersolution of u_t−Δ_pu≥ u^q to vanish. Consequently, the corresponding Cauchy problem has no global solution in this class with a nonzero nonnegative local L^1 initial trace.
ER -