Liouville theorem for one kind of elliptic equations on complete Riemannian manifold

Abstract

We use maximum principle to prove the Liouville theorem of the equation $\Delta U+b\cdot \nabla U+h{U}^{\alpha }=0,U\ge 0,0<\alpha <\frac{n+2}{n-2}$ on the complete Riemannian manifold with non-negative Ricci tensor, which improve the result of Gidas-Spruck and Catino-Monticelli. We remark that this is the second version and all of the results come from the first version. Two months after we posted version 1 of this preprint on arXiv, we found Zhihao Lu has already posted a paper arXiv:2308.14764 before us and part of his result coincides with ours. So after deleting these parts and adding more reference and details, we post this second version on arXiv.