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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On nonexistence of positive solutions of quasi-linear inequality on Riemannian manifolds
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by Yuhua Sun PDF
Proc. Amer. Math. Soc. 143 (2015), 2969-2984 Request permission

Abstract:

We investigate the nonexistence of a positive solution to the following differential inequality: \begin{equation} div(|\nabla u|^{m-2}\nabla u)+u^{\sigma }\leq 0, \tag {1} \end{equation} on a noncompact complete Riemannian manifold, where $m>1$ and $\sigma >m-1$ are parameters. Our main result is as follows: If the volume of a geodesic ball of radius $r$ with a fixed center $x_0$ is bounded for large enough $r$ by $Cr^{p}\ln ^qr$, where $p=\frac {m\sigma }{\sigma -m+1}, q=\frac {m-1}{\sigma -m+1}$, then (1) has no positive weak solution.

We also show the sharpness of the parameters $p, q$.

References
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Additional Information
  • Yuhua Sun
  • Affiliation: Department of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany
  • Address at time of publication: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
  • Email: sunyuhua@nankai.edu.cn
  • Received by editor(s): February 10, 2014
  • Published electronically: March 18, 2015
  • Communicated by: Joachim Krieger
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 2969-2984
  • MSC (2010): Primary 35J61; Secondary 58J05
  • DOI: https://doi.org/10.1090/S0002-9939-2015-12705-0
  • MathSciNet review: 3336621