Nonexistence of solutions to ${\Delta }_{p}u+{\Delta }_{q}u+u^s|\nabla u{|}^t\le 0$ on geodesically complete noncompact Riemannian manifolds

Abstract

In this paper, we consider the inequality ${\Delta }_{p}u+{\Delta }_{q}u+u^s|\nabla u{|}^t\le 0$ on geodesically complete noncompact Riemannian manifolds. By a test function argument, we establish Liouville-type theorems under the upper bound of volume of geodesic ball. In the Euclidean space ${\mathbb{R}}^n$, we obtain new nonexistence results which extend the result of Bhakta-Biswas-Filippucci \cite{BBF}. In particular, we have addressed the influence of the higher order term for $s<0$. At last, we present some examples to illustrate sharpness in some cases.

arXiv comment28 pages, all comments welcome!