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The regularity and stability of solutions to semilinear fourth-order elliptic problems with negative exponents

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  07 January 2016

Baishun Lai
Baishun Lai*
Affiliation:
Institute of Contemporary Mathematics, Henan University, Kaifeng 475004, People's Republic of China (laibaishun@henu.edu.cn)
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Extract

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem

on a general bounded domain Ω in ℝN, with Navier boundary condition u = Δu on ∂Ω. Firstly, we prove the extremal solution is smooth for any p > 1 and N ⩽ 4, which improves the result of Guo and Wei (Discrete Contin. Dynam. Syst. A 34 (2014), 2561–2580). Secondly, if p = 3, N = 3, we prove that any radial weak solution of this nonlinear eigenvalue problem is smooth in the case Ω = 𝔹, which completes the result of Dávila et al. (Math. Annalen348 (2009), 143–193). Finally, we also consider the stability of the entire solution of Δ2u = –l/up in ℝN with u > 0.

Keywords

biharmonic equationentire solutionregularitystability

MSC classification

Secondary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B08: Entire solutions 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 1 , February 2016 , pp. 195 - 212
Copyright
Copyright © Royal Society of Edinburgh 2016 

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