Abstract
Let \({(M^n,g)}\) be an n-dimensional complete Riemannian manifold. We consider Yau’s gradient estimates for positive solutions to the following nonlinear equation $$\Delta u + au {\rm log} u=0$$ where a is a constant. As an application, we obtain the Liouville property for this equation in the case of a < 0. In addition, we illustrate, by giving concrete examples, that our results are sharp.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.