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 form [Formula: see text] with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli et al. [The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301], Trudinger [On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] 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 [Formula: see text] is a concave function. In this paper, based on Perron’s method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming [Formula: see text] to satisfy certain structure conditions as in [L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301; N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] but without requiring the concavity of [Formula: see text]. The equations in our setting may embrace the well-known Monge–Ampère equations, Hessian equations and Hessian quotient equations as special cases.