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Asymptotic behavior for fully nonlinear elliptic equations in exterior domains

Lian Yuanyuan, Zhang Kai

Abstract

In this paper, we obtain the asymptotic behavior at infinity for viscosity solutions of fully nonlinear elliptic equations in exterior domains. We show that if the solution $u$ grows linearly, there exists a linear polynomial $P$ such that $u-P$ is controlled by fundamental solutions of the Pucci's operators. In addition, with proper ellipticity constants, $u(x)-P(x)\to 0$ as $x\to \infty$ (see Theorem 1.11). If $u$ grows quadratically, we obtain similar asymptotic behavior (see Theorem 1.16). In this paper, we don't require any smoothness of the fully nonlinear operator.

Asymptotic behavior for fully nonlinear elliptic equations in exterior domains

Abstract

In this paper, we obtain the asymptotic behavior at infinity for viscosity solutions of fully nonlinear elliptic equations in exterior domains. We show that if the solution grows linearly, there exists a linear polynomial such that is controlled by fundamental solutions of the Pucci's operators. In addition, with proper ellipticity constants, as (see Theorem 1.11). If grows quadratically, we obtain similar asymptotic behavior (see Theorem 1.16). In this paper, we don't require any smoothness of the fully nonlinear operator.
Paper Structure (3 sections, 12 theorems, 137 equations, 1 figure)

This paper contains 3 sections, 12 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic behavior at infinity

Key Result

Lemma 1.2

Suppose that $F$ is positively homogeneous of degree $1$. Then there exists a viscosity solution $\Phi$ of such that: (i) $\Phi$ is bounded below in $B_1$ and bounded above in $B_1^c$; (ii) For any $t>0$ and $x\in \mathbb{R}^n\backslash \{0\}$, where $\alpha^{*}\in (-1,+\infty)\backslash \{0\}$ depends only on $n$ and $F$, and $\alpha^*\Phi>0$. Moreover, any viscosity solution satisfying (i) can

Figures (1)

  • Figure 1: (I): $\alpha^*>0,\tilde{\alpha}^*>0$; (II): $\alpha^*>0,\tilde{\alpha}^*<0$; (III): $\alpha^*<0,\tilde{\alpha}^*>0$; (IV): $\alpha^*<0,\tilde{\alpha}^*<0$.

Theorems & Definitions (26)

  • Definition 1.1
  • Lemma 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Lemma 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 16 more