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In this paper we prove some integral estimates on the minimal growth of thepositive part $u_+$ of subsolutions of quasilinear equations  \[  \mathrm{div} A(x,u,\nabla u) = V|u|^{p-2}u  \]  on complete Riemannian manifolds $M$, in the non-trivial case $u_+\not\equiv0$. Here $A$ satisfies the structural assumption $|A(x,u,\nabla u)|^{p/(p-1)}\leq k \langle A(x,u,\nabla u),\nabla u\rangle$ for some constant $k>0$ and for$p>1$ the same exponent appearing on the RHS of the equation, and $V$ is acontinuous positive function, possibly decaying at a controlled rate atinfinity. We underline that the equation may be degenerate and that ourarguments do not require any geometric assumption on $M$ beyond completeness ofthe metric. From these results we also deduce a Liouville-type theorem forsufficiently slowly growing solutions.

Growth of subsolutions of $\Delta_p u = V|u|^{p-2}u$ and of a general class of quasilinear equations