In this paper, we consider the exterior Dirichlet problem of Hessian equations $$σ_k (λ(D^2u)) = g(x)$$ with $g$ being a perturbation of a general positive function at infinity. The existence of the viscosity solutions with generalized asymptotic behavior at infinity is established by the Perron’s method which extends the previous results for Hessian equations. By the solutions of Bernoulli ordinary differential equations, the viscosity subsolutions and supersolutions are constructed.