Liouville theorem for the inequality ${\Delta }_{m}u+f(u)\le 0$ on Riemannian manifolds

Abstract

In this paper, we study the quasilinear inequality ${\Delta }_{m}u+f(u)\le 0$ on a complete Riemannian manifold, where \begin{align*} m>1,α>m-1 \quad and \quad f(t)> 0,αf(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}.

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