Necessary and sufficient conditions on the existence of solutions for the exterior Dirichlet problem of Hessian equations

In this paper, we consider the exterior Dirichlet problem of Hessian equations σk(λ(D2u))=g(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma _{k}(\lambda (D^{2}u))=g(x)$\end{document} with g being a perturbation of a general positive function at infinity. By estimating the eigenvalues of the solution, we obtain the necessary and sufficient conditions of existence of radial symmetric solutions with asymptotic behavior at infinity.

The exterior Dirichlet problem of Monge-Ampère equations is closely related to the classical theorem of Jörgens [18] (n = 2), Calabi [8] (n ≤ 5), and Pogorelov [26] (n ≥ 2) which states that any classical convex solution of det D 2 u = 1 in R n must be a quadratic polynomial. Cheng and Yau [10], Caffarelli [6], Jost and Xin [19], and Trudinger and Wang [27] also gave related results with the Jörgens-Calabi-Pogorelov theorem. The cases of det D 2 u = f in R n with f being a periodic function can be referred to Li and Lu [25] and the references therein.
In 2003, Caffarelli and Li [7] extended the Jörgens-Calabi-Pogorelov theorem to exterior domains and also investigated the existence of solutions to the exterior Dirichlet problem They got that if is a smooth, bounded, strictly convex open subset and φ ∈ C 2 (∂ ), then for any given b ∈ R n and any given n × n real symmetric positive definite matrix A with det A = 1, there exists some constant c * depending only on n, , φ, b, and A, such that for every c > c * there exists a unique function u ∈ C ∞ (R n \ ) ∩ C 0 (R n \ ) which satisfies (1.3) and Since then, many results of the exterior problem for the fully nonlinear elliptic equations have been obtained. For instance, in 2011, the first author and Bao [13], the first author [11] studied the Dirichlet problem of Hessian equation and got the existence and uniqueness of viscosity solutions with the asymptotic behavior where α = n or k, c ∈ R and In 2013, Wang and Bao [28] studied the necessary and sufficient conditions on the existence of radially symmetric solutions for the Dirichlet problem outside a unit ball with the asymptotic behavior and where c, d ∈ R. Recently, Li and Lu [24] characterized the existence and nonexistence of solutions for exterior problem of Monge-Ampère equations Bao, Li, and Li [2] and Cao and Bao [9] studied the solutions with the generalized asymptotic behavior for exterior Dirichlet problem of Hessian equation (1.1). The results of the exterior Dirichlet problem for Monge-Ampère equations can also be referred to [1, 3-5, 17, 20] and the references therein. However, for the Hessian quotient equations where 0 ≤ l < k ≤ n, n ≥ 3, and σ 0 (λ) = 1, one can refer to [12,[21][22][23]. Note that if l = 0, the Hessian quotient equation is the Hessian equation. Moreover, for n = 2, the exterior Dirichlet problem of Monge-Ampère equations can be referred to the earlier works by Ferrer, Martínez, and Milán [15,16] using the complex variable methods. One can also refer to Delanoë [14].
To work in the realm of elliptic equations, we restrict the class of functions. Let We shall discuss the necessary and sufficient conditions of existence for radially symmetric solutions to the exterior Dirichlet problem of Hessian equation.
Let ω 0 ∈ C 0 (R n ) be positive and radially symmetric in x, and ω ∈ C 0 (R n \B 1 ) be a radially symmetric function satisfying for β > 2 and Suppose that, for k ≤ m ≤ n, For l = 1, 2, . . . , n, let |u is an l-convex radially symmetric function , and the radially symmetric function andĉ be a constant. Then, for m = k, there exists a unique radially symmetric function u ∈ m satisfying and as |x| → ∞, 2 Proof of Theorem 1.1 We first give several lemmas in order to prove Theorem 1.1.
Then from (2.2) we have that Thus which is equivalent to That is, for anyr > 1, 0 <δ <δ m (r), whereδ m is defined by (2.1). Proof Let

Lemma 2.3 Assume that u
be a radially symmetric solution to (1.9) and (1.10). By a direct computation, we have Then the eigenvalues of the Hessian matrix D 2 u are By Lemma 2.1, we know that δ = u r > 0 forr > 1.
So τ ≥ 0. From (1.9), we have that i.e., which is equivalent to Then the lemma is proved.