Core-periphery detection aims to separate the nodes of a complex network into two subsets: a core that is densely connected to the entire network and a periphery that is densely connected to the core but sparsely connected internally. The definition of core-periphery structure in multiplex networks that record different types of interactions between the same set of nodes on different layers is nontrivial since a node may belong to the core in some layers and to the periphery in others. We propose a nonlinear spectral method for multiplex networks that simultaneously optimises a node and a layer coreness vector by maximising a suitable nonconvex homogeneous objective function by a provably convergent alternating fixed point iteration. We derive a quantitative measure for the quality of a given multiplex core-periphery structure that allows the determination of the optimal core size. Numerical experiments on synthetic and real-world networks illustrate that our approach is robust against noisy layers and significantly outperforms baseline methods while improving the latter with our novel optimised layer coreness weights. As the runtime of our method depends linearly on the number of edges of the network it is scalable to large-scale multiplex networks.