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

Abstract

In this paper we prove some integral estimates on the minimal growth of the positive 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 $u_{+}\not\equiv 0$. Here $A$ satisfies the structural assumption $|A(x,u,\nabla u){|}^{p/(p-1)}\le k⟨A(x,u,\nabla u),\nabla u⟩$ 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.

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