Multiple clustered layer solutions for semilinear Neumann problems on a ball

A. Malchiodi, Wei Ming Ni, Juncheng Wei

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55 Scopus citations


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 languageEnglish (US)
Pages (from-to)143-163
Number of pages21
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Issue number2
StatePublished - 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.


  • Multiple clustered layers
  • Singularly perturbed Neumann problem


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