Abstract :
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.
