Hessian equations on exterior domain

In the paper, we consider the Hessian equation σk(λ(D2u))=f(x) where f is a positive function outside a bounded domain of Rn, n≥3 and f(x)=1+O(|x|−β) for some β>2 at infinity. Using the Perron's method we prove the existence and uniqueness for viscosity solutions of exterior Dirichlet problem with prescribed asymptotic behavior at infinity. There are examples to show that the result is optimal. This is an extension of the theorems given by Bao-Li-Li in [2] for f≡1 and Bao-Li-Zhang in [3] for k=n.