Asymptotic behavior for fully nonlinear elliptic equations in exterior domains

Lian Yuanyuan, Zhang Kai

Published 2024-01-11Version 1

In this paper, we obtain the asymptotic behavior at infinity for viscosity solutions of fully nonlinear elliptic equations in exterior domains. We show that if the solution $u$ grows linearly, there exists a linear polynomial $P$ such that $u-P$ is controlled by fundamental solutions of the Pucci's operators. In addition, with proper ellipticity constants, $u(x)-P(x)\to 0$ as $x\to \infty$ (see Theorem 1.11). If $u$ grows quadratically, we obtain similar asymptotic behavior (see Theorem 1.16). In this paper, we don't require any smoothness of the fully nonlinear operator.