Summary: We consider a quasilinear Neumann problem with exponent \(p\in ]1,+\infty[\), in a multidomain of \({\mathbb{R}}^N\), \(N\geq 2\), 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 \(1<p\leq N-1\).