On Bound States Concentrating on Spheres for the Maxwell--Schrödinger Equation

We study the semiclassical limit for the following system of Maxwell--Schrödinger equations: \[ -\frac{\hbar^2}{2m}\Delta v + v + \omega\phi v - \gamma v^{p} =0, \;\; -\Delta\phi = 4\pi\omega v^2, \] where $\hbar$, m, $\omega$, $\gamma >0$, v, $\phi: \mathbb{R}^3 \to \mathbb{R}$, $1 < p < \frac{11}{7}$. This system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field: the unknowns v and $\phi$ represent the \emph{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 $(v_\hbar, \phi_\hbar)$ such that $v_\hbar$ concentrates around a sphere $\{|x| = s_0\}$ when $\hbar \to 0$.