Abstract
In this paper, we consider Liouville type theorems for positive solutions to the following nonlinear elliptic equation:
where a is a nonzero real constant. By using gradient estimates, we obtain upper bounds of \(|\nabla u|\) with respect to \(\sup u\) and the lower bound of Bakry-Emery Ricci curvature. In particular, for complete noncompact manifolds with \(a<0\), we prove that any positive solution must be \(u\equiv 1\) under a suitable condition for a with respect to the lower bound of Bakry-Emery Ricci curvature. It generalizes a classical result of Yau.
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The research of authors is supported by NSFC (Nos. 11371018, 11401179, 11671121).
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Huang, G., Li, Z. Liouville type theorems of a nonlinear elliptic equation for the V-Laplacian. Anal.Math.Phys. 8, 123–134 (2018). https://doi.org/10.1007/s13324-017-0168-6
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DOI: https://doi.org/10.1007/s13324-017-0168-6