Abstract: In this paper, we employ Li-Yau's method and delicate analysis techniques to provide a unified and simple approach to the gradient estimate of the positive solution to the nonlinear elliptic equation $\Delta u + au^{p+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ where $a > 0$ and $ p < 4/n $ or $a < 0$ and $p >0$ are two constants. For the case $a>0$, we extend the range of $p$ and improve some results in \cite{J-L, MHL} and supplement the results for the case $\dim(M)= 2$. For the case $a<0$ and $p>0$, we improve or perfect the previous results due to Ma, Huang and Luo \cite{MHL} since one does not need to suppose the positive solutions are bounded. When the Ricci curvature of $(M,g)$ is nonnegative, we also obtain a Liouville-type theorem for the above equation.

Key words: Gradient estimate, Nonlinear elliptic equation, Liouville theorem