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Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  17 September 2020

Zhijun Zhang
Zhijun Zhang*
Affiliation:
School of Mathematics and Information Sciences, Yantai University, Yantai264005, Shandong, People's Republic of China ([email protected])
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Abstract

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝn, g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$, b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

Keywords

Semi-linear elliptic equationsa singular nonlinearityboundary value problemsthe unique solutionglobal asymptotic behaviour

MSC classification

Primary: 35J75: Singular elliptic equations
Secondary: 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 3 , June 2021 , pp. 1116 - 1134
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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