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

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.