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In this paper we consider the gradient estimates on positive solutions to thefollowing 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 thedimension of $M$ and $\kappa$ is a nonnegative constant), we employ theNash-Moser iteration technique to derive a Cheng-Yau's type gradient estimatefor positive solution to the above equation under some suitable geometric andanalysis conditions. Moreover, it is shown that when the Ricci curvature of $M$is nonnegative, this elliptic equation does not admit any positive solutionexcept for $u\equiv 1$ if $r<s$ and $$1<r<\frac{n+3}{n-1}\quad\quad~~\mbox{or}~~\quad 1<s<\frac{n+3}{n-1}.$$

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