Gradient estimates for a nonlinear elliptic equation on a smooth metric measure space

Let (M, g, e −f dv) be a smooth metric measure space. We consider local gradient estimates for positive solutions to the following elliptic equation ∆ f u + au log u + bu = 0 where a, b are two real constants and f be a smooth function defined on M. As an application, we obtain a Liouville type resu...

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Bibliographic Details
Published inCommunications in Mathematics Vol. 32 (2024), Issue 1
Main Authors Wang, Xiaoshan, Cao, Linfen
Format Journal Article
LanguageEnglish
Published University of Ostrava 27.02.2023
Subjects
Mathematics
Liouville type property
m-dimensions Bakry-Émery Ricci curvature
2010 Mathematics Subject Classification. Primary 58J35
2010 Mathematics Subject Classification. Primary 58J35 Secondary 35B45 gradient estimates nonlinear equation m-dimensions Bakry-Émery Ricci curvature Liouville type property
Secondary 35B45 gradient estimates
nonlinear equation
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ISSN2336-1298
1804-1388
2336-1298
DOI10.46298/cm.10951

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Summary:Let (M, g, e −f dv) be a smooth metric measure space. We consider local gradient estimates for positive solutions to the following elliptic equation ∆ f u + au log u + bu = 0 where a, b are two real constants and f be a smooth function defined on M. As an application, we obtain a Liouville type result for such equation in the case a < 0 under the m-dimensions Bakry-Émery Ricci curvature.
ISSN:2336-1298
1804-1388
2336-1298
DOI:10.46298/cm.10951