# American Institute of Mathematical Sciences

September  2006, 16(3): 657-688. doi: 10.3934/dcds.2006.16.657

## Clustered layers for the Schrödinger-Maxwell system on a ball

 1 Department of Mathematics, Nanjing University, Nanjing 210093, China, China

Received  January 2006 Revised  August 2006 Published  August 2006

We consider the following singularly perturbed Schrödinger-Maxwell system with Dirichlet boundary condition

$-\varepsilon^2\Delta v+v+\omega\phi v- \varepsilon^{\frac{p-1}{2}} v^p=0 \quad \text{ in}\ B_1,$
$-\Delta \phi=4\pi \omega v^2 \quad \text{in}\ B_1$,
$v,\ \phi>0 \ \text{in}\ B_1 \quad \text{and}\quad v=\phi=0 \quad \text{on}\ \partial B_1$,

where $B_1$ is the unit ball in $\mathbb{R}^3,\ \omega>0$ and $\ \frac{7}{3}$<$p\leq 5$ are constants, and $\varepsilon$>$0$ is a small parameter. Using the localized energy method, we prove that for every sufficiently large integer $N$, the system has a family of radial solutions $(v_\varepsilon, \phi_\varepsilon)$ such that $v_\varepsilon$ has $N$ sharp spheres concentrating on a sphere $\{|x|=r_N\}$ as $\varepsilon\to 0$.

Citation: Pingzheng Zhang, Jianhua Sun. Clustered layers for the Schrödinger-Maxwell system on a ball. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 657-688. doi: 10.3934/dcds.2006.16.657
 [1] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 [2] Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117 [3] Pascal Bégout, Jesús Ildefonso Díaz. A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3371-3382. doi: 10.3934/dcds.2014.34.3371 [4] Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971 [5] Olivier Bourget, Matias Courdurier, Claudio Fernández. Construction of solutions for some localized nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 841-862. doi: 10.3934/dcds.2019035 [6] Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337 [7] Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040 [8] Nan Lu. Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3533-3567. doi: 10.3934/dcds.2015.35.3533 [9] Dachun Yang, Dongyong Yang, Yuan Zhou. Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators. Communications on Pure & Applied Analysis, 2010, 9 (3) : 779-812. doi: 10.3934/cpaa.2010.9.779 [10] Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303 [11] Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253 [12] Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239 [13] Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565 [14] Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193 [15] Emil Minchev, Mitsuharu Ôtani. $L^∞$-energy method for a parabolic system with convection and hysteresis effect. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1613-1632. doi: 10.3934/cpaa.2018077 [16] Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [19] Silvia Cingolani, Mnica Clapp, Simone Secchi. Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891 [20] Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005

2018 Impact Factor: 1.143