Abstract
Let (M n, g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation:
where a is a nonzero constant. In particular, for a < 0, we prove that any bounded positive solution of the above equation with a suitable condition for a with respect to the lower bound of Ricci curvature must be \({u\equiv 1}\). This generalizes a classical result of Yau.
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The research of the G. Huang is supported by NSFC (Nos. 11371018, 11171091), Henan Provincial Key Teacher (No. 2013GGJS-057) and IRTSTHN (14IRTSTHN023). The research of the B. Ma is supported by NSFC (No. 11401179) and Henan Provincial Education department (No. 14B110017).
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Huang, G., Ma, B. Gradient estimates and Liouville type theorems for a nonlinear elliptic equation. Arch. Math. 105, 491–499 (2015). https://doi.org/10.1007/s00013-015-0820-z
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DOI: https://doi.org/10.1007/s00013-015-0820-z