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

Giuseppe Maria Coclite; Helge Holden

doi:10.3934/dcds.2010.27.117

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2010, Volume 27, Issue 1: 117-132. Doi: 10.3934/dcds.2010.27.117

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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
2.
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim

Received: December 31, 2008

Revised: November 30, 2009

Published: February 2010

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Abstract

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:

Schrödinger-Maxwell system,
point interaction.

Mathematics Subject Classification: Primary: 35D05, 35Q40; Secondary: 35J65.

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

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