February  2010, 27(1): 117-132. doi: 10.3934/dcds.2010.27.117

Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics

1. 

Department of Mathematics, University of Bari, Via E. Orabona 4, I–70125 Bari, Italy

2. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim

Received  January 2009 Revised  December 2009 Published  February 2010

We study a non-relativistic charged quantum particle moving in a bounded open set $\Omega\subset\R^3$ 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 ground state. We show the existence of such solutions for each $\beta>0$ and the compactness as $\beta\to 0$.
Citation: Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117
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