Clustered solutions around harmonic centers to a coupled elliptic system

We study the following system of Schrödinger–Maxwell equations ε^2Δv−v−ωvφ+f(v)=0, Δφ+γv2=0 in Ω, v,φ > 0 in Ω, v,φ = 0 on ∂Ω, where Ω is a smooth and bounded domain of R3 . We prove that for any integer k the system has a family of solutions (vε , φε ) such that the form of vε consists of k spikes concentrating at a harmonic center of Ω as ε → 0+. Furthermore we show that the spikes approach the vertexes of a configuration which maximizes a suitable geometrical problem.