Gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces and applications

Heliyon

Sulyman O. Salawu¹  Abimbola Abolarinwa²  Clement A. Onate³ 

[1] Department of Physical Sciences, Landmark University, Omu-Aran, Kwara State, Nigeria;Corresponding author.;Department of Physical Sciences, Landmark University, Omu-Aran, Kwara State, Nigeria;

关键词: Mathematics;    Riemannian manifolds;    Elliptic equations;    Liouville theorem;    Gradient estimates;    Yamabe problem;   

DOI  :  

来源: DOAJ

【 摘 要 】

In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equationΔfu+p(x)u+q(x)uα=0, on complete smooth metric measure spaces (MN,g,e−fdv) with ∞-Bakry-Émery Ricci tensor bounded from below, where α is an arbitrary real constant, p(x) and q(x) are smooth functions. As an application, Liouville-type theorems for various special cases of the equation are recovered. Furthermore, we discuss nonexistence of smooth solution to Yamabe type problem on (MN,g,e−fdv) with nonpositive weighted scalar curvature.

【 授权许可】

Unknown