Growth of subsolutions of ${\mathrm{\Delta }}_{p}u=V\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$ and of a general class of quasilinear equations

In this paper we prove some integral estimates on the minimal growth of the positive part $u_{+}$ of subsolutions of quasilinear equations $$\mathrm{d}\mathrm{i}\mathrm{v}A(x,u,\mathrm{\nabla }u)=V\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$$ on complete Riemannian manifolds $M$, in the non-trivial case $u_{+}\equiv ̸0$. Here $A$ satisfies the structural assumption $\mathrm{\mid }A(x,u,\mathrm{\nabla }u){\mathrm{\mid }}^{p\mathrm{/}(p-1)}\le k⟨A(x,u,\mathrm{\nabla }u),\mathrm{\nabla }u⟩$ for some constant $k>0$ and for $p>1$ the same exponent appearing on the RHS of the equation, and $V$ is a continuous positive function, possibly decaying at a controlled rate at infinity. We underline that the equation may be degenerate and that our arguments do not require any geometric assumption on $M$ beyond completeness of the metric. From these results we also deduce a Liouville-type theorem for sufficiently slowly growing solutions.