Neumann condition in the Schrödinger-Maxwell system
Lorenzo Pisani, Gaetano Siciliano · Project Euclid (Cornell University) · 2007
We study a system of (nonlinear) Schrödinger and Maxwell equation in a bounded domain, with a Dirichelet boundary condition for the wave function $\\psi$ and a nonhomogeneous Neumann datum for the electric potential $\\phi$. Under a suitable compatibility condition, we establish the existence of infinitely many static solutions $\\psi=u(x)$ in equilibrium with a purely electrostatic field ${\\mathbf{E}}=-\ abla\\phi$. Due to the Neumann condition, the same electric field is in equilibrium with stationary solutions $\\psi=e^{-i\\omega t}u(x)$ of arbitrary frequency $\\omega$.