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