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

In this paper, we consider the inequality ${\mathrm{\Delta }}_{p}u+{\mathrm{\Delta }}_{q}u+u^s\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^t\le 0$ 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.