Liouville theorem for elliptic equations involving the sum of the function and its gradient in $\mathbb R^n$

We prove Liouville theorem for the equation $\Delta v + N v^p + M |\nabla v|^{q}= 0$ in $\mathbb R^n$, with $M, N > 0, q = \frac{2p}{p + 1}$ in the critical and subcritical case. The proof is based on a differential identity and Young inequality. We remark that this is the second version for the paper. And we thank Prof. Bidaut-V\'eron and V\'eron for their very useful comments on this paper. Compared with the first one, in this version we correct some errors and adjust the arrangement of the proof so that it can be understood easily.