On the exterior Dirichlet problem for Hessian type fully nonlinear elliptic equations

Abstract

We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the formwith prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli-Nirenberg-Spruck [8], Trudinger [35] and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on the Perron's method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations by assuming f to satisfy certain structure conditions as in [8,35], which may embrace the well-known Monge-Ampère equations, Hessian equations and Hessian quotient equations as special cases but do not require the concavity.