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We characterize lower growth estimates for subsolutions in halfspaces offully nonlinear partial differential equations on the form $$ F(x,u,Du,D^2u) =0 $$ in terms of solutions to ordinary differential equations built solely upona growth assumption on $F$. Using this characterization we derive several sharpPhragmen-Lindel\"of-type theorems for certain classes of well known PDEs. Theequation need not be uniformly elliptic nor homogeneous and we obtain resultsboth in case the subsolution is bounded or unbounded. Among our results weretrieve classical estimates in the halfspace for $p$-subharmonic functions andextend those to more general equations; we prove sharp growth estimates, interms of $k$ and the asymptotic behaviour of $\int_{0}^{R} C(s) ds$, forsubsolutions of equations allowing for sublinear growth in the gradient of theform $C(|x|)|Du|^k$ with $k\geq 1$; we establish a Phragmen-Lindel\"of theoremfor weak subsolutions of the variable exponent $p$-Laplace equation inhalfspaces, $1 < p(x) < \infty$, $p(x) \in C^1$, of which we conclude sharpnessby finding the "slowest growing" $p(x)$-harmonic function together with itscorresponding family of $p(x)$-exponents. The paper ends with a discussion ofour results from the point of view of a spatially dependent diffusion problem.

Growth of subsolutions to fully nonlinear equations in halfspaces