Existence and nonexistence of subsolutions for augmented Hessian equations

Limei Dai

doi:10.3934/dcds.2020023

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2020, Volume 40, Issue 1: 579-596. Doi: 10.3934/dcds.2020023

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Existence and nonexistence of subsolutions for augmented Hessian equations

Limei Dai

School of Mathematics and Information Science, Weifang University, Weifang 261061, China

Received: March 31, 2019

Published: October 2019

The author is supported by Shandong Provincial Natural Science Foundation, China (ZR2018LA006)

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In this paper, we consider the augmented Hessian equations $ S_k^{\frac{1}{k}}[D^2u+\sigma(x)I] = f(u) $ in $ \mathbb{R}^{n} $ or $ \mathbb{R}^{n}_+ $. We first give the necessary and sufficient condition of the existence of classical subsolutions to the equations in $ \mathbb{R}^{n} $ for $ \sigma(x) = \alpha $, which is an extended Keller-Osserman condition. Then we obtain the nonexistence of positive viscosity subsolutions of the equations in $ \mathbb{R}^{n} $ or $ \mathbb{R}^{n}_+ $ for $ f(u) = u^p $ with $ p>1 $.

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Mathematics Subject Classification: 35J60, 35D40.

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