Multiple clustered layer solutions for semilinear Neumann problems on a ball

A. Malchiodi; Wei Ming Ni; Juncheng Wei

doi:10.1016/j.anihpc.2004.05.003

Multiple clustered layer solutions for semilinear Neumann problems on a ball

A. Malchiodi, Wei Ming Ni, Juncheng Wei

Mathematics

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55 Scopus citations

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=1^N{|x|=r_j ^ε}, where 1>r₁^ε>r ₂^ε>⋯>r_N^ε are such that 1-r₁^ε∼εlog1/ε,r _j-1^ε-r_j^ε∼εlog1/ ε,j=2,...,N.

Original language	English (US)
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	https://doi.org/10.1016/j.anihpc.2004.05.003
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

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title = "Multiple clustered layer solutions for semilinear Neumann problems on a ball",

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 {\=Ω}. 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.",

keywords = "Multiple clustered layers, Singularly perturbed Neumann problem",

author = "A. Malchiodi and Ni, {Wei Ming} and Juncheng Wei",

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.",

year = "2005",

doi = "10.1016/j.anihpc.2004.05.003",

language = "English (US)",

volume = "22",

pages = "143--163",

journal = "Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire",

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TY - JOUR

T1 - Multiple clustered layer solutions for semilinear Neumann problems on a ball

AU - Malchiodi, A.

AU - Ni, Wei Ming

AU - Wei, Juncheng

N1 - 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.

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

KW - Multiple clustered layers

KW - Singularly perturbed Neumann problem

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VL - 22

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EP - 163

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 2

ER -

Multiple clustered layer solutions for semilinear Neumann problems on a ball

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Keywords

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