Asymptotic behavior for fully nonlinear elliptic equations in exterior domains

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 $x\to \mathrm{\infty }$ (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.