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

Sun, Yuhua

doi:10.1090/S0002-9939-2015-12705-0

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

Proc. Amer. Math. Soc. 143 (2015), 2969-2984

DOI: https://doi.org/10.1090/S0002-9939-2015-12705-0

Published electronically: March 18, 2015

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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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