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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

Giuseppe Maria Coclite 1, and  Helge Holden 2,

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 $\beta\to 0$.
Keywords: point interaction., Schrödinger-Maxwell system.
Mathematics Subject Classification: Primary: 35D05, 35Q40; Secondary: 35J6.
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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