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

Abstract

We improve the Liouville theorem for the equation $-\Delta v = v^p |\nabla v|^q$ in $\mathbb R^n$, which was studied by Bidaut-V\'eron, Garc\'ia-Huidobro, and V\'eron. 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-V\'eron and V\'eron for their very useful comments on this paper. Compared with the first version, we correct some errors and provide more details for the proof.