Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation
Abstract We 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...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-05-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-018-1705-z |
| _version_ | 1818887711188058112 |
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| author | Bingqing Ma Yongli Dong |
| author_facet | Bingqing Ma Yongli Dong |
| author_sort | Bingqing Ma |
| collection | DOAJ |
| description | Abstract We 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. |
| first_indexed | 2024-12-19T16:41:34Z |
| format | Article |
| id | doaj.art-8e6e6a02a6664c5884b7589080c5036c |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-19T16:41:34Z |
| publishDate | 2018-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-8e6e6a02a6664c5884b7589080c5036c2022-12-21T20:13:46ZengSpringerOpenJournal of Inequalities and Applications1029-242X2018-05-012018111010.1186/s13660-018-1705-zGradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equationBingqing Ma0Yongli Dong1College of Physics and Materials Science, Henan Normal UniversityDepartment of Mathematics, Henan Normal UniversityAbstract We 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.http://link.springer.com/article/10.1186/s13660-018-1705-zGradient estimateNonlinear elliptic equationLiouville-type theorem |
| spellingShingle | Bingqing Ma Yongli Dong Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation Journal of Inequalities and Applications Gradient estimate Nonlinear elliptic equation Liouville-type theorem |
| title | Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation |
| title_full | Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation |
| title_fullStr | Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation |
| title_full_unstemmed | Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation |
| title_short | Gradient estimates and Liouville-type theorems for a weighted nonlinear elliptic equation |
| title_sort | gradient estimates and liouville type theorems for a weighted nonlinear elliptic equation |
| topic | Gradient estimate Nonlinear elliptic equation Liouville-type theorem |
| url | http://link.springer.com/article/10.1186/s13660-018-1705-z |
| work_keys_str_mv | AT bingqingma gradientestimatesandliouvilletypetheoremsforaweightednonlinearellipticequation AT yonglidong gradientestimatesandliouvilletypetheoremsforaweightednonlinearellipticequation |