Asymptotic behavior of solutions of fully nonlinear equations over exterior domains

Abstract

In this paper, we consider the asymptotic behavior at infinity of solutions of a class of fully nonlinear elliptic equations $F(D^2u)=f(x)$ over exterior domains, where the Hessian matrix $(D^2u)$ tends to some symmetric positive definite matrix at infinity and $f(x)=O(|x|^{-t})$ at infinity with sharp condition $t>2$. Moreover, we also obtain the same result if $(D^2u)$ is only very close to some symmetric positive definite matrix at infinity.