Growth of Subsolutions of Δpu=V|u|p-2u 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 div A ( x , u , ∇ u ) = V | u | p - 2 u 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.