# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117
 [1] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 [2] Pingzheng Zhang, Jianhua Sun. Clustered layers for the Schrödinger-Maxwell system on a ball. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 657-688. doi: 10.3934/dcds.2006.16.657 [3] Jaime Angulo Pava, César A. Hernández Melo. On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2093-2116. doi: 10.3934/cpaa.2019094 [4] Kazuhiro Kurata, Yuki Osada. Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4239-4251. doi: 10.3934/cpaa.2021157 [5] Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1511-1547. doi: 10.3934/dcdsb.2021100 [6] Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic and Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005 [7] Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations and Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009 [8] Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060 [9] Jing Cui, Shu-Ming Sun. Nonlinear Schrödinger equations on a finite interval with point dissipation. Mathematical Control and Related Fields, 2019, 9 (2) : 351-384. doi: 10.3934/mcrf.2019017 [10] Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031 [11] Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565 [12] Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292 [13] Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure and Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040 [14] Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100 [15] Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013 [16] Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221 [17] Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure and Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048 [18] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [19] Marcelo Nogueira, Mahendra Panthee. On the Schrödinger-Debye system in compact Riemannian manifolds. Communications on Pure and Applied Analysis, 2020, 19 (1) : 425-453. doi: 10.3934/cpaa.2020022 [20] Silvia Cingolani, Mnica Clapp, Simone Secchi. Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891

2020 Impact Factor: 1.392