On bound states concentrating on spheres for the Maxwell-Schrodinger equation

We study the semiclassical limit for the following system of Maxwell - Schrodinger equations: - (h) over bar (2)/ 2m Delta v + v + omega phi v -gamma v(p) = 0, -Delta phi = 4 pi omega v(2), where (h) over bar, m, omega, gamma > 0, v, phi : R-3 --> R, 1 < p < 11/7. This system describes standing waves for the nonlinear Schrodinger equation interacting with the electrostatic field: the unknowns v and phi represent the wave function associated to the particle and the electric potential, respectively. By using localized energy method, we construct a family of positive radially symmetric bound states (v((h) over bar), phi((h) over bar)) such that v((h) over bar) concentrates around a sphere {| x| = s(0)} when (h) over bar --> 0.