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IMA Journal of Applied Mathematics Advance Access originally published online on December 5, 2008
IMA Journal of Applied Mathematics 2009 74(3):468-480; doi:10.1093/imamat/hxn035
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Asymptotic behaviour of ground state solutions for the Hénon equation

Daomin Cao{dagger}

Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China

Shuangjie Peng{ddagger}

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People's Republic of China

Shusen Yan

School of Mathematics, Statistics and Computer Science, The University of New England, Armidale NSW 2351, Australia

{dagger} Email: dmcao{at}amt.ac.cn

{ddagger} Corresponding author. Email: sjpeng{at}mail.ccnu.edu.cn

Let B1(0) sub RN be the unit ball centred at the origin, N ≥ 3. In this paper, we analyse the profile of the ground state solution of the Hénon equation – {Delta}u = |x|{alpha}up–1 in B1(0), u = 0 on{partial}B1(0). We prove that for fixed p isin (2, 2*), (2* = 2N/(N – 2)), the maximum point x{alpha} of the ground state solution u{alpha} satisfies {alpha}(1 – |x{alpha}|) -> l isin (0, +{infty}) as {alpha} -> {infty}. We also obtain the asymptotic behaviour of u{alpha}, which shows that the ground state solution is non-radial. Moreover, we prove the existence of multi-peaked solutions and give their asymptotic behaviour.

Keywords: Hénon equation; ground state solutions; asymptotic behaviour.


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