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: − ¯h 2 2m v +eφv = λv, − φ = 4πev2 with the boundary conditions u = 0, φ = g on ∂B1. Here ¯h,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 (v¯h ,φ¯h ) such that v¯h concentrates (as ¯h →0+) around some points of the boundary ∂B1 which are proved to be minima for g. © 2005 Elsevier Inc. All rights reserved.