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Ground states of the Schrödinger-Maxwell system with Dirac mass: existence and asymptotics. (English) Zbl 1190.35189

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 \(\beta\to 0\).

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
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