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Further study of a fourth-order elliptic equation with negative exponent

Published online by Cambridge University Press:  03 June 2011

Zongming Guo  and
Zhongyuan Liu
Zongming Guo
Affiliation:
Department of Mathematics, Henan Normal University, Xinxiang 453007, People's Republic of China (guozm@public.xxptt.ha.cn)
Zhongyuan Liu
Affiliation:
Department of Mathematics, Henan Normal University, Xinxiang 453007, People's Republic of China (liuzhongyuan1984@yahoo.com.cn)
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Abstract

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 141 , Issue 3 , June 2011 , pp. 537 - 549
Copyright
Copyright © Royal Society of Edinburgh 2011

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