  We use maximum principle to give a new proof for the Liouville theorem of the
equation $\Delta u + u^{\alpha} = 0, u \geq 0 , 0 < \alpha < \frac{n + 2}{n -
2}$ in $\mathbb R^n$. Besides we can use this method to prove the Liouville
theorem of the equation $\Delta U + b\cdot \nabla U + h U^{\alpha} = 0, U \geq
0, 0 < \alpha < \frac{n + 2}{n - 2}$ on the complete Riemannian manifold.
