Growth of Subsolutions of $$\Delta _p u = V|u|^{p-2}u$$ and of a General Class of Quasilinear Equations

Abstract

AbstractIn this paper we prove some integral estimates on the minimal growth of the positive part $$u_+$$ u + of subsolutions of quasilinear equations $$\begin{aligned} \textrm{div}A(x,u,\nabla u) = V|u|^{p-2}u \end{aligned}$$ div A ( x , u , ∇ u ) = V | u | p - 2 u on complete Riemannian manifolds M, in the non-trivial case $$u_+\not \equiv 0$$ u + ≢ 0 . Here A satisfies the structural assumption $$|A(x,u,\nabla u)|^{p/(p-1)} \le k \langle A(x,u,\nabla u),\nabla u\rangle $$ | A ( x , u , ∇ u ) | p / ( p - 1 ) ≤ k ⟨ A ( x , u , ∇ u ) , ∇ u ⟩ for some constant $$k>0$$ k > 0 and for $$p>1$$ 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.