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Solvability of Hessian quotient equations in exterior domains

Part of: Partial differential equations Representations of solutions Parabolic equations and systems

Published online by Cambridge University Press:  14 December 2023

Limei Dai ,
Jiguang Bao Open the ORCID record for Jiguang Bao [Opens in a new window]  and
Bo Wang
Limei Dai
Affiliation:
School of Mathematics and Information Science, Weifang University, Weifang 261061, P. R. China e-mail: lmdai@wfu.edu.cn
Jiguang Bao
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China e-mail: jgbao@bnu.edu.cn
Bo Wang*
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China
*
e-mail: wangbo89630@bit.edu.cn
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Abstract

In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\equiv \mbox {const.}$, we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

Keywords

Hessian quotient equationsexterior Dirichlet problemradially symmetric solutionsasymptotic behaviornecessary and sufficient conditions

MSC classification

Primary: 35K55: Nonlinear parabolic equations 35D40: Viscosity solutions 35B40: Asymptotic behavior of solutions 35B07: Axially symmetric solutions
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 31
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

Dai is supported by the Shandong Provincial Natural Science Foundation (Grant No. ZR2021MA054). Bao is supported by the Beijing Natural Science Foundation (Grant No. 1222017). Wang is supported by the National Natural Science Foundation of China (Grant Nos. 11971061 and 12271028), the Beijing Natural Science Foundation (Grant No. 1222017), and the Fundamental Research Funds for the Central Universities.

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