Asymptotic behavior of solutions of fully nonlinear equations over exterior domains

Abstract

In this paper, we consider the asymptotic behavior at infinity of solutions of a class of fully nonlinear elliptic equations F (D 2 u) = f (x) over exterior domains, where the Hessian matrix (D 2 u) tends to some symmetric positive definite matrix at infinity and f (x) = O(|x| −t ) at infinity with sharp condition t > 2. Moreover, we also obtain the same result if (D 2 u) is only very close to some symmetric positive definite matrix at infinity.

Asymptotic behavior of solutions to the Monge--Ampère equations with slow convergence rate at infinity

“…In this section, we prove Theorem 1.3. By Theorem 3.1 and Remark 3.3 in [18] (see also Corollary 2.1 in [16] or Theorem 2.2 in [11]), we have the following result on linear elliptic equations.…”

“…By constructing barrier functions (see for instance [16,11]), there exists C > 0 such that for all x ∈ R n ,…”

Existence of entire solutions to the Lagrangian mean curvature equations in supercritical phase

“…Remark 4.5. In the original statement of Theorem 1.1 in [19], the asymptotic behavior results were stated as below…”

“…Eventually, we would like to mention that the condition (1.6) origins from the asymptotic behavior results of solutions on entire R n or exterior domain. There are generous results on this topic, see for instance Caffarelli-Li [8] and Bao-Li-Zhang [2] for the Monge-Ampère equations, Li-Li-Yuan [22] for the special Lagrangian equations, Liu-Bao [26,27,28,29] for a family of mean curvature equations of gradient graphs and Jia [19] for a family of general fully nonlinear elliptic equations under asymptotic assumptions of the Hessian matrix.…”

Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity

“…We would also like to mention that for the Monge-Ampère type equations, there are also classification results and asymptotic behavior analysis for f (x) − C 0 being a periodic function by Caffarelli-Li [10] or asymptotically periodic at infinity by Teixeira-Zhang [36], etc. Under additional assumptions on D 2 u at infinity, the asymptotic behavior results were obtained for general fully nonlinear elliptic equations by Jia [25].…”