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BOUNDARY BLOW-UP SOLUTIONS TO EQUATIONS INVOLVING THE INFINITY LAPLACIAN

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  23 August 2022

CUICUI LI ,
FANG LIU  and
PEIBIAO ZHAO
CUICUI LI
Affiliation:
Department of Mathematics, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China e-mail: licui1121@njust.edu.cn
FANG LIU*
Affiliation:
Department of Mathematics, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China
PEIBIAO ZHAO
Affiliation:
Department of Mathematics, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China e-mail: pbzhao@njust.edu.cn
*
e-mail: sdqdlf78@126.com, liufang78@njust.edu.cn
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Abstract

In this paper, we study the boundary blow-up problem related to the infinity Laplacian

$$ \begin{align*}\begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u=\infty &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\Delta _{\infty }^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When $q>h>1$ , we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q \leq h.$

Keywords

infinity Laplacianboundary blow-up solutioncomparison principleboundary asymptotic estimate

MSC classification

Primary: 35J60: Nonlinear elliptic equations 35J70: Degenerate elliptic equations 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 114 , Issue 3 , June 2023 , pp. 337 - 358
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Florica Cirstea

This work was supported by National Natural Science Foundation of China (Nos. 11501292, 11871275 and 12141104).

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