We consider a quasilinear Neumann problem with exponent p ∈]1, +∞[, in a multidomain of R N , N ≥ 2, consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the other one with small height and given cross section. Assuming that the volumes of the two cylinders tend to zero with same rate, we prove that the limit problem is well posed in the union of the limit domains, with respective dimension 1 and N − 1. Moreover, this limit problem is coupled if p > N − 1 and uncoupled if 1 < p ≤ N − 1. IntroductionLet N ≥ 2, let ω ⊂ R N −1 be a bounded open connected set with a smooth boundary such that the origin in R N −1 , denoted by 0 , belongs to ω, and let {r n } n∈N , {h n } n∈N be two sequences of positive numbers converging to 0. For every n ∈ N, consider the thin multidomain Ω n = Ω 1 n ∪ Ω 2 n , the union of two vertical cylinders with small volumes: Ω 1 n = r n ω × [0, 1[ with small cross section r n ω and constant height, Ω 2 n = ω×] − h n , 0[ with small height h n and constant cross section (see figure next page). This paper arises from the desire of studying the asymptotic behaviour, as n → +∞, of the following model problem:

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