Solitary charged waves interacting with the electrostatic field (Q820038)

The Schrödinger-Maxwell equation is solved for the case of a charged particle interacting with its own electromagnetic field. The semiclassical limit \(\hbar\to{0^{+}}\) with unknown potential \(\Phi\) and magnetic potential \(A=0\) is considered for the system \[ -\hbar^{2}\Delta{v}+e\Phi{v}=\lambda{v}, \qquad -\Delta\Phi=4\pi{e}{v}^{2}. \] in the unit ball \(B_{1}\). The main theorem is a statement that this system for every \(\hbar\) has a solution \((v_{\hbar},\Phi_{\hbar},\lambda_{\hbar})\) which belongs to the Sobolov space \(H^{1}(B_{1})\). Each \((v_{\hbar},\Phi_{\hbar})\) is obtained as a stationary point of a suitable functional having \(\lambda=\lambda_{\hbar}\) as a Lagrangian multiplier for the constraint \(\| v\| _{L_{2}}=1\).