Abstract
We consider the following singularly perturbed Neumann problem {ε2Δu-u+f(u)=0 in Ω; u>0 in Ω and ∂u/∂ν=0 on ∂Ω, where Ω=B1(0) is the unit ball in Rn, ε>0 is a small parameter and f is superlinear. It is known that this problem has multiple solutions (spikes) concentrating at some points of Ω̄. In this paper, we prove the existence of radial solutions which concentrate at N spheres ∪j=1N{|x|=rj ε}, where 1>r1ε>r 2ε>⋯>rNε are such that 1-r1ε∼εlog1/ε,r j-1ε-rjε∼εlog1/ ε,j=2,...,N.
Original language | English (US) |
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Pages (from-to) | 143-163 |
Number of pages | 21 |
Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |
Volume | 22 |
Issue number | 2 |
DOIs | |
State | Published - 2005 |
Bibliographical note
Funding Information:The first author is supported by MURST, under the project Variational Methods and Nonlinear Differential Equations and by NSF under agreement No. DMS-9729992. He is also grateful to ETH for the kind hospitality. The second author is partially supported by the National Science foundation. The research of the third author is supported by an Earmarked Grant from RGC of Hong Kong.
Keywords
- Multiple clustered layers
- Singularly perturbed Neumann problem