Multiple clustered layer solutions for semilinear Neumann problems on a ball

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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|=rjε}, where 1>r1ε>r 2ε>⋯>rNε are such that 1-r1ε∼εlog1/ε,r j-1ε-rjε∼εlog1/ ε,j=2,...,N. © 2004 Elsevier SAS. All rights reserved.

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Malchiodi, A., Ni, W. M., & Wei, J. (2005). Multiple clustered layer solutions for semilinear Neumann problems on a ball. Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire, 22(2), 143–163. https://doi.org/10.1016/j.anihpc.2004.05.003

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