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

TL;DR

This paper establishes a Liouville theorem for a class of elliptic equations involving the function and its gradient in Euclidean space, using differential identities and inequalities, with improvements over previous versions.

Contribution

It provides a new Liouville theorem for elliptic equations with gradient terms, including corrections and clearer proofs compared to earlier versions.

Findings

Liouville theorem proven for the equation in critical and subcritical cases

Uses differential identities and Young inequality in the proof

Improved clarity and corrected errors from previous version

Abstract

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.