Electron. J. Differential Equations, Vol. 2017 (2017), No. 187, pp. 1-18.

Second-order boundary estimate for the solution to infinity Laplace equations

Ling Mi

In this article, we establish a second-order estimate for the solutions to the infinity Laplace equation
 -\Delta_{\infty} u=b(x)g(u), \quad u>0, \quad x \in \Omega,\;
 u|_{\partial \Omega}=0,
where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $g\in C^1((0,\infty),(0,\infty))$, $g$ is decreasing on $(0,\infty)$ with $\lim_{s \to 0^+}g(s)=\infty$ and g is normalized regularly varying at zero with index $-\gamma$ ( $\gamma>1$), $b \in C({\bar{\Omega}})$ is positive in $\Omega$, may be vanishing on the boundary. Our analysis is based on Karamata regular variation theory.

Submitted December 18, 2016. Published July 24, 2017.
Math Subject Classifications: 35J55, 35J60, 35J65.
Key Words: Infinity Laplace equation; second order estimate; Karamata regular variation theory; comparison functions.

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Ling Mi
College of Mathematics and Statistics
Linyi University
Linyi, Shang Dong 276005, China
email: mi-ling@163.com, miling@lyu.edu.cn

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