A Liouville theorem for the quasilinear elliptic inequality on complete Riemannian manifolds

Abstract:

In this article we establish a Liouville theorem for positive solutions to the differential inequality $\Delta _{p}u+a u^{r}|\nabla u|^{q}\le 0$ on a complete noncompact Riemannian manifold $(M,g)$. The method is based on a suitable change of variable aimed to produce a useful differential inequality, then the use of cut-off functions technique yields some a priori integral estimates useful to reach the claim by removing the condition $q +r-p+1>0$ in previous result (Theorem 1.1 in He, Hu, and Wang, Nash-Moser iteration approach to the logarithmic gradient estimates and Liouville properties of quasilinear elliptic equations on manifolds, Preprint, https://arxiv.org/abs/2311.02568, 2023).