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Remarks on Liouville-type theorems on complete noncompact Finsler manifolds
Volume 59, no. 2 (2018), pp. 255–264

DOI: https://doi.org/10.33044/revuma.v59n2a03

Abstract

We give a gradient estimate of the positive solution to the equation $\Delta u=- \lambda^2u, \quad \lambda \geq 0$ on a complete noncompact Finsler manifold. Then we obtain the corresponding Liouville-type theorem and Harnack inequality for the solution. Moreover, on a complete noncompact Finsler manifold we also prove a Liouville-type theorem for a $C^2$-nonnegative function $f$ satisfying $\Delta f \geq cf^d, \quad c>0, \; d>1,$ which improves a result obtained by Yin and He.