Growth of subsolutions to fully nonlinear equations in halfspaces

Abstract

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.