Liouville theorem for the inequality $\Delta_m u+f(u)\leq 0$ on Riemannian manifolds

Abstract

In this paper, we study the quasilinear inequality $ \Delta_m u+f(u)\leq 0$ on a complete Riemannian manifold, where \begin{align*} m>1,\alpha>m-1 \quad and \quad f(t)> 0,\alpha f(t)-tf^{'}(t)\geq 0, \forall t>0. \end{align*} If for some point $x_0$ and large enough $r$, \begin{align*} vol B_r(x_0)\leq C r^p ln^q r, \end{align*} where $p=\frac{m\alpha}{\alpha-(m-1)},q=\frac{m-1}{\alpha-(m-1)}$ and $B_r(x_0) $ is a geodesic ball of radius $r$ centered at $x_0$, then the inequality possesses no positive weak solution. This generalizes the result in \cite{AS,Sun}.