Nonexistence of solutions to $Δ_pu+Δ_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds

Abstract: In this paper, we consider the inequality $$Δ_pu+Δ_qu+u^s|\nabla u|^t\leq 0$$ 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 $\mathbb{R}<sup>n$, 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 $s&lt;0$. At last, we present some examples to illustrate sharpness in some cases.