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Yau’s gradient estimates for a nonlinear elliptic equation

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Abstract

Let \({(M^n,g)}\) be an n-dimensional complete Riemannian manifold. We consider Yau’s gradient estimates for positive solutions to the following nonlinear equation

$$\Delta u + au {\rm log} u=0$$

where a is a constant. As an application, we obtain the Liouville property for this equation in the case of a < 0. In addition, we illustrate, by giving concrete examples, that our results are sharp.

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Authors and Affiliations

  1. Department of Applied Mathematics, Donghua University, No. 2999 North Renmin Rd., Songjiang, Shanghai, 201620, People’s Republic of China

    Bin Qian

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  1. Bin Qian
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Correspondence to Bin Qian.

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Qian, B. Yau’s gradient estimates for a nonlinear elliptic equation. Arch. Math. 108, 427–435 (2017). https://doi.org/10.1007/s00013-016-0983-2

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  • DOI: https://doi.org/10.1007/s00013-016-0983-2

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