Gradient Estimate for Solutions of $\Delta v+v^r-v^s= 0$ on A Complete Riemannian Manifold

In this paper we consider the gradient estimates on positive solutions to the following elliptic equation defined on a complete Riemannian manifold $(M,\,g)$: $$\Delta v+v^r-v^s= 0,$$ where $r$ and $s$ are two real constants. When$(M,\,g)$ satisfies $Ric \geq -(n-1)\kappa$ (where $n\geq2$ is the dimension of $M$ and $\kappa$ is a nonnegative constant), we employ the Nash-Moser iteration technique to derive a Cheng-Yau's type gradient estimate for positive solution to the above equation under some suitable geometric and analysis conditions. Moreover, it is shown that when the Ricci curvature of $M$ is nonnegative, this elliptic equation does not admit any positive solution except for $u\equiv 1$ if $r<s$ and $$1<r<\frac{n+3}{n-1}\quad\quad ~~\mbox{or}~~\quad 1<s<\frac{n+3}{n-1}.$$