Coclite, Giuseppe Maria; Holden, Helge Ground states of the Schrödinger-Maxwell system with Dirac mass: existence and asymptotics. (English) Zbl 1190.35189 Discrete Contin. Dyn. Syst. 27, No. 1, 117-132 (2010). Summary: We study a non-relativistic charged quantum particle moving in a bounded open set \(\Omega\subset\mathbb R^3\) with smooth boundary under the action of a zero-range potential. In the electrostatic case the standing wave solutions take the form \(\psi(t,x)=u(x)e^{-i\omega t}\) where \(u\) formally satisfies \(-\Delta u+\alpha\varphi u-\beta\delta_{x_0} u=\omega u\) and the electric potential \(\varphi\) is given by \(-\Delta\varphi = u^2\). We introduce the definition of a ground state. We show the existence of such solutions for each \(\beta>0\) and the compactness as \(\beta\to 0\). Cited in 1 Document MSC: 35Q40 PDEs in connection with quantum mechanics 35D30 Weak solutions to PDEs 35J57 Boundary value problems for second-order elliptic systems 35B40 Asymptotic behavior of solutions to PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics Keywords:Schrödinger-Maxwell system; point interaction; ground state PDFBibTeX XMLCite \textit{G. M. Coclite} and \textit{H. Holden}, Discrete Contin. Dyn. Syst. 27, No. 1, 117--132 (2010; Zbl 1190.35189) Full Text: DOI