An eigenvalue semiclassical problem for the Schrödinger operator with an electrostatic field

Abstract

We consider the following system of Schrödinger-Maxwell equations in the unit ball $B_1$ of ${\mathbb R}^3$ $$ -\frac{\hbar^2}{2m}\Delta v+ e\phi v=\omega v, \quad -\Delta\phi=4\pi e v^2 $$ with the boundary conditions $ u=0$, $ \phi=g$ on $\partial B_1$, where $\hbar, m, e, \omega > 0$, $v$, $\phi\colon B_1\rightarrow {\mathbb R}$, $g\colon \partial B_1\to {\mathbb R}$. Such system describes the interaction of a particle constrained to move in $B_1$ with its own electrostatic field. We exhibit a family of positive solutions $(v_\hbar, \phi_\hbar)$ corresponding to eigenvalues $\omega_\hbar$ such that $v_\hbar$ concentrates around some points of the boundary $\partial B_1$ which are minima for $g$ when $\hbar\rightarrow 0$.