On nonnegative solutions of the inequality \(\Delta u+u^{\sigma }\leq 0\) on Riemannian manifolds. (English) Zbl 1296.58011
This paper is concerned with the study of nonnegative solutions of the differential inequality
\[
\Delta u+u^\sigma\leq 0,
\]
on a geodesically complete connected Riemannian manifold \(M\), where \(\sigma >1\). Let \(p=2\sigma/(\sigma -1)\), \(q=1/(\sigma -1)\) and let \(B(x,r)\) denote the geodesic ball centered at \(x\in M\) and of radius \(r>0\). The main Liouville-type result of the present paper establishes that if, for some \(x_0\in M\) and all large enough \(r\), we have
\[
vol\, B(x_0,r)\leq Cr^p\ln^qr,
\]
then the only nonnegative solution of the above differential inequality is the trivial solution.
Reviewer: Vicenţiu D. Rădulescu (Craiova)
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