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

Published online by Cambridge University Press:  07 January 2016

Baishun Lai*
Affiliation:
Institute of Contemporary Mathematics, Henan University, Kaifeng 475004, People's Republic of China ([email protected])

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2016 

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