Solitary charged waves interacting with the electrostatic field

In this paper we study the semiclassical limit for the following system of Schrödinger-Maxwell equations in the unit ball B1 of R3:−ℏ22mΔv+eϕv=λv,−Δϕ=4πev2 with the boundary conditions u=0, ϕ=g on ∂B1. Here ℏ,m,e,λ>0, v,ϕ:B1→R, g:∂B1→R. This system has been introduced by V. Benci, D. Fortunate in [V. Benci, D. Fortunate, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998) 283-293] as a model describing standing charged waves for the Schrödinger equation in presence of an electrostatic field. We exhibit a family of positive solutions (vℏ,ϕℏ) such that vℏ concentrates (as ℏ→0+) around some points of the boundary ∂B1 which are proved to be minima for g.