On the growth of positive entire solutions of elliptic PDEs and their gradients

Abstract

We investigate the growth of entire positive functions $u(x)$ and their gradients $Du$ in Sobolev spaces when a polynomial growth is assumed for their image $Lu$ through a linear second-order uniform elliptic operator $L$. In particular, under suitable assumptions on the coefficients, we show that if $Lu$ is bounded, then $u(x)$ may grow at most quadratically at infinity. We also discuss, by counterexamples, the optimality of the assumptions and extend the results to viscosity solutions of fully nonlinear equations.