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AbstractWe consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space (M,g,e−fdv)$(M, g,e^{-f}\,dv)$: Δfu+aulogu+bu=0,$$\Delta_{f} u+au\log u+bu=0, $$ where a, b are two… Click to show full abstract
AbstractWe consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space (M,g,e−fdv)$(M, g,e^{-f}\,dv)$: Δfu+aulogu+bu=0,$$\Delta_{f} u+au\log u+bu=0, $$ where a, b are two real constants. When the ∞-Bakry–Émery Ricci curvature is bounded from below, we obtain a global gradient estimate which is not dependent on |∇f|$|\nabla f|$. In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions.
Journal Title: Journal of Inequalities and Applications
Year Published: 2018
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