Liouville theorem for elliptic equations with a source reaction term involving the product of the function and its gradient in ${\mathbb{R}}^n$

We improve the Liouville theorem for the equation $-\mathrm{\Delta }v=v^p\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^q$ in ${\mathbb{R}}^n$, which was studied by Bidaut-Véron, García-Huidobro, and Véron. The proof is based on a differential identity and Young inequality. We remark that this is the second version for this paper and the first one was submitted one year ago. We thank Prof. Bidaut-Véron and Véron for their very useful comments on this paper. Compared with the first version, we correct some errors and provide more details for the proof.