Gradient estimates for a nonlinear equation $\Delta_fu+cu^{-\alpha}=0$ on complete noncompact manifolds

Zhang, Jing ; Ma, Bingqing

Communications in Mathematics, Tome 19 (2011) no. 1, pp. 73-84.

Résumé

Let $(M,g)$ be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation $\Delta_fu+cu^{-\alpha}=0$ in $M$, where $\alpha$, $c$ are two real constants and $\alpha>0$, $f$ is a smooth real valued function on $M$ and $\Delta_f=\Delta-\nabla f\nabla$. When $N$ is finite and the $N$-Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that $\infty$-Bakry-Emery Ricci tensor is bounded from below and $|\nabla f|$ is bounded from above, we also obtain a gradient estimate for positive solutions of the above equation. It extends the results of Yang [Yang, Y.Y. Gradient estimates for the equation $\Delta u+cu^{-\alpha}=0$ on Riemannian manifolds Acta. Math. Sin. 26(B) 2010 1177–1182].

Détail

Classification : 35J60, 58J05
Keywords: gradient estimates; positive solution; Bakry-Emery Ricci tensor