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<p>This paper is concerned with the existence and optimal boundary behavior of large solutions to the Monge-Ampère type equations det <i>D</i><sup>2</sup><i>u</i>(<i>x</i>) = <i>λu</i><sup><i>n</i></sup>(<i>x</i>)+<i>b</i>(<i>x</i>)<i>g</i>(∣∇<i>u</i>(<i>x</i>)∣), <i>x</i> ∈ Ω, where Ω is a uniformly convex, bounded smooth domain in ℝ<sup><i>n</i></sup> with <i>n</i> ≥ 2, <i>b</i> ∈ <i>C</i><sup>∞</sup>(Ω) is positive in Ω, <i>g</i> ∈ <i>C</i>[0, ∞) ∩ <i>C</i><sup>1</sup> (0, ∞), <i>g</i>(0) = 0 and <i>g</i> is increasing on [0, ∞). The author finds new structure conditions on <i>g</i> which play a crucial role in boundary behavior of such solutions.</p>

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Boundary Behavior of Large Solutions for Equations of Monge-Ampère Type

  • 期刊论文

  • 发表日期：

    2025年11月7日

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  • Zhijun Zhang

摘要

This paper is concerned with the existence and optimal boundary behavior of large solutions to the Monge-Ampère type equations det D2u(x) = λun(x)+b(x)g(∣∇u(x)∣), x ∈ Ω, where Ω is a uniformly convex, bounded smooth domain in ℝn with n ≥ 2, b ∈ C∞(Ω) is positive in Ω, g ∈ C[0, ∞) ∩ C1 (0, ∞), g(0) = 0 and g is increasing on [0, ∞). The author finds new structure conditions on g which play a crucial role in boundary behavior of such solutions.