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On nonexistence of positive solutions of quasi-linear inequality on Riemannian manifolds


Author: Yuhua Sun
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
Published electronically: March 18, 2015
MathSciNet review: 3336621
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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 $p, q$.


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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

Keywords: Quasi-linear inequalities, critical exponent, Riemannian manifold, volume growth
Received by editor(s): February 10, 2014
Published electronically: March 18, 2015
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society