Growth of Subsolutions of $$\Delta _p u = V|u|^{p-2}u$$ and of a General Class of Quasilinear Equations [Growth of Subsolutions of Δₚ u = V │u │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 divA(x,u,∇u)=V|u|p-2u on complete Riemannian manifolds M, in the non-trivial case u+≢ 0 . Here A satisfies the structural assumption | A(x, u, ∇ u) | p/(p-1)≤ k⟨ A(x, u, ∇ u) , ∇ 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.