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Nonexistence of solutions to Δpu+Δqu+us|∇ u|t≤ 0 on geodesically complete noncompact Riemannian manifolds
Abstract
In this paper, we consider the inequality Δpu+Δqu+us|∇ u|t≤ 0 on geodesically complete noncompact Riemannian manifolds. By a test function argument, we establish Liouville-type theorems under the upper bound of volume of geodesic ball. In the Euclidean space ℝn, we obtain new nonexistence results which extend the result of Bhakta-Biswas-Filippucci \cite{BBF}. In particular, we have addressed the influence of the higher order term for s<0. At last, we present some examples to illustrate sharpness in some cases.
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BibTeX
@misc{zhao2026nonexistence,
title = {{Nonexistence of solutions to $Δ_pu+Δ_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds}},
author = {Biqiang Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19790},
eprint = {2609.19790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19790},
}
RIS
TY - UNPB
TI - Nonexistence of solutions to Δ_pu+Δ_qu+u^s|∇ u|^t≤ 0 on geodesically complete noncompact Riemannian manifolds
AU - Biqiang Zhao
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19790
UR - https://arxiv.org/abs/2609.19790
AB - In this paper, we consider the inequality Δ_pu+Δ_qu+u^s|∇ u|^t≤ 0 on geodesically complete noncompact Riemannian manifolds. By a test function argument, we establish Liouville-type theorems under the upper bound of volume of geodesic ball. In the Euclidean space ℝ^n, we obtain new nonexistence results which extend the result of Bhakta-Biswas-Filippucci \cite{BBF}. In particular, we have addressed the influence of the higher order term for s<0. At last, we present some examples to illustrate sharpness in some cases.
ER -