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Properties of the extremal solution for a fourth-order elliptic problem

Published online by Cambridge University Press:  20 September 2012

Baishun Lai  and
Zhuoran Du
Baishun Lai
Affiliation:
Institute of Contemporary Mathematics, and School of Mathematics and Information Science, Henan University, Kaifeng 475004, People's Republic of China (laibaishun@henu.edu.cn)
Zhuoran Du
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha 410082, People's Republic of China
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Abstract

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Let λ* > 0 denote the largest possible value of λ such that the system

has a solution, where is the unit ball in ℝn centred at the origin, p > 1 and n is the exterior unit normal vector. We show that for λ = λ* this problem possesses a unique weak solution u*, called the extremal solution. We prove that u* is singular when n ≥ 13 for p large enough and actually solve part of the open problem which Dávila et al. left unsolved.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 142 , Issue 5 , October 2012 , pp. 1051 - 1069
Copyright
Copyright © Royal Society of Edinburgh 2012