Multiple clustered layer solutions for semilinear Neumann problems on a ball

Abstract
We consider the following singularly perturbed Neumann problem \begin{cases} ɛ^{2}\mathrm{\Delta }u−u + f(u) = 0\:& \text{in}\:\Omega \text{;} \\ u > 0\:\text{in}\:\Omega \:\text{and}\:\frac{\partial u}{\partial \nu } = 0\:& \text{on}\:\partial \Omega \text{,} \end{cases} where \Omega = B_{1}(0) is the unit ball in \mathbb{R}^{n} , ɛ > 0 is a small parameter and f is superlinear. It is known that this problem has multiple solutions (spikes) concentrating at some points of \bar \Omega . In this paper, we prove the existence of radial solutions which concentrate at N spheres \bigcup _{j = 1}^{N}\{|x| = r_{j}^{ɛ}\} , where 1 > r_{1}^{ɛ} > r_{2}^{ɛ} > ⋯ > r_{N}^{ɛ} are such that 1−r_{1}^{ɛ} \sim ɛ\mathrm{\log }\frac{1}{ɛ}\text{,}r_{j−1}^{ɛ}−r_{j}^{ɛ} \sim ɛ\mathrm{\log }\frac{1}{ɛ}\text{,}\:j = 2\text{,}…\text{,}N . Résumé: On considère le problème de Neumann singulièrement perturbé suivant \begin{cases} ɛ^{2}\mathrm{\Delta }u−u + f(u) = 0\:&\text{dans}\:\Omega \text{;} \\ u > 0\:\text{dans}\:\Omega \:\text{et}\:\frac{\partial u}{\partial \nu } = 0\:&\text{sur}\:\partial \Omega \text{,} \end{cases} où \Omega = B_{1}(0) est la boule unité de \mathbb{R}^{n} , ɛ > 0 est un paramètre petit et f est surlinéaire. Il est bien connu que ce problème possède plusieurs solutions se concentrant en certains points de \bar \Omega . Dans cet article nous prouvons l'existence de solutions radiales qui se concentrent en N sphères \bigcup _{j = 1}^{N}\{|x| = r_{j}^{ɛ}\} , où 1 > r_{1}^{ɛ} > r_{2}^{ɛ} > ⋯ > r_{N}^{ɛ} sont tels que 1−r_{1}^{ɛ} \sim ɛ\mathrm{\log }\frac{1}{ɛ}\text{,}r_{j−1}^{ɛ}−r_{j}^{ɛ} \sim ɛ\mathrm{\log }\frac{1}{ɛ}\text{,}\:j = 2\text{,}…\text{,}N .

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