December  2014, 34(12): 5299-5323. doi: 10.3934/dcds.2014.34.5299

Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received  July 2013 Revised  May 2014 Published  June 2014

We consider the following two coupled Schrödinger systems in a bounded domain $\Omega\subset \mathbb{R}^N(N=2,3)$ with Neumann boundary conditions $$\left\{ \begin{array}{ll} -\epsilon^2 \triangle u + u = \mu_1 u^3+ \beta u v^2,\\ -\epsilon^2 \triangle v + v =\mu_2 v^3+ \beta u^2 v,\\ u>0,    v>0, \\ \partial u/\partial n = 0,\partial v/\partial n = 0, \mbox{on } \partial \Omega. \end{array}\right. $$ Suppose the mean curvature $H(P)$ of the boundary $\partial \Omega$ admits several local maximums( or local minimums), we obtain the existence of segregated solutions $(u_\epsilon,v_\epsilon)$ to the above system such that both of $u_\epsilon$ and $v_\epsilon$ admit more than one local maximums, furthermore as $\epsilon$ goes to zero, the maximum points of $u_\epsilon$ and $v_\epsilon$ concentrate at different local maximum points( or local minimum points) of the mean curvature $H(P)$ respectively.
Citation: Zhongwei Tang. Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5299-5323. doi: 10.3934/dcds.2014.34.5299
References:
[1]

L. A. Maia, E. Nontefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger systems,, J. Diff. Equat., 229 (2006), 743.  doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[2]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var., 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[3]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger systems,, J. Fixed Point Theory Appl., 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[4]

D. Cao and T. Küpper, On the existence of multipeaked solutions to a semilinear Neumann Problem,, Duke Math. J., 97 (1999), 261.  doi: 10.1215/S0012-7094-99-09712-0.  Google Scholar

[5]

N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction,, Tran. Amer. Math. Soci., 361 (2009), 1189.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar

[6]

S. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems,, Arch. Ration. Mech. Anal., 208 (2013), 305.  doi: 10.1007/s00205-012-0598-0.  Google Scholar

[7]

M. Lucia and Z. Tang, Multi-bump bound states for a system of nonlinear Schrödinger equations,, J. Differential Equations, 252 (2012), 3630.  doi: 10.1016/j.jde.2011.11.017.  Google Scholar

[8]

W. M. Ni and I. Takagi, Locating the peaks of the least energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[9]

B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartee-Fock theory for double condensates,, Phys. Rev. Lett., 78 (1997), 3594.   Google Scholar

[10]

C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47.  doi: 10.1016/S0294-1449(99)00104-3.  Google Scholar

[11]

T. Lin and J. Wei, Spike in two coupled of nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403.  doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[12]

T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n,n\leq 3$,, Comm. Math. Phys., 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[13]

T. Lin and J. Wei, Spike in two-component systems of nonlinear Schrödinger equations with trapping potentials,, J. Diff. Equat., 229 (2006), 538.  doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[14]

B. Sirakov, Least energy solitary waves for a system of nonliear Schrödinger equations in $\mathbbR^n$,, Commun. Math. Phys., 271 (2007), 199.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[15]

Z. Tang, Spike-layer solutions to singularly perturbed semilinear systems of coupled schrödinger equations,, J. Math. Anal. Appl., 377 (2011), 336.  doi: 10.1016/j.jmaa.2010.11.001.  Google Scholar

[16]

Z. Tang, Multi-peak solutions to a coupled schrödinger systems with neumann boundary condition,, J. Math. Anal. Appl., 409 (2014), 684.  doi: 10.1016/j.jmaa.2013.07.053.  Google Scholar

[17]

J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations,, Rend. Lincei Mat. Appl., 18 (2007), 279.  doi: 10.4171/RLM/495.  Google Scholar

[18]

J. Wei and T. Weth, Radial solutions and phase sparation in a system of two coupled Schrödinger equations,, Arch. Rat. Mech. Anal., 190 (2008), 83.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[19]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459.  doi: 10.1016/S0294-1449(98)80031-0.  Google Scholar

[20]

W. Yao and J. Wei, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, Commun. Pure. Appl. Anal., 11 (2012), 1003.  doi: 10.3934/cpaa.2012.11.1003.  Google Scholar

show all references

References:
[1]

L. A. Maia, E. Nontefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger systems,, J. Diff. Equat., 229 (2006), 743.  doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[2]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var., 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[3]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger systems,, J. Fixed Point Theory Appl., 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[4]

D. Cao and T. Küpper, On the existence of multipeaked solutions to a semilinear Neumann Problem,, Duke Math. J., 97 (1999), 261.  doi: 10.1215/S0012-7094-99-09712-0.  Google Scholar

[5]

N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction,, Tran. Amer. Math. Soci., 361 (2009), 1189.  doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar

[6]

S. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems,, Arch. Ration. Mech. Anal., 208 (2013), 305.  doi: 10.1007/s00205-012-0598-0.  Google Scholar

[7]

M. Lucia and Z. Tang, Multi-bump bound states for a system of nonlinear Schrödinger equations,, J. Differential Equations, 252 (2012), 3630.  doi: 10.1016/j.jde.2011.11.017.  Google Scholar

[8]

W. M. Ni and I. Takagi, Locating the peaks of the least energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[9]

B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartee-Fock theory for double condensates,, Phys. Rev. Lett., 78 (1997), 3594.   Google Scholar

[10]

C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47.  doi: 10.1016/S0294-1449(99)00104-3.  Google Scholar

[11]

T. Lin and J. Wei, Spike in two coupled of nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403.  doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[12]

T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n,n\leq 3$,, Comm. Math. Phys., 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[13]

T. Lin and J. Wei, Spike in two-component systems of nonlinear Schrödinger equations with trapping potentials,, J. Diff. Equat., 229 (2006), 538.  doi: 10.1016/j.jde.2005.12.011.  Google Scholar

[14]

B. Sirakov, Least energy solitary waves for a system of nonliear Schrödinger equations in $\mathbbR^n$,, Commun. Math. Phys., 271 (2007), 199.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[15]

Z. Tang, Spike-layer solutions to singularly perturbed semilinear systems of coupled schrödinger equations,, J. Math. Anal. Appl., 377 (2011), 336.  doi: 10.1016/j.jmaa.2010.11.001.  Google Scholar

[16]

Z. Tang, Multi-peak solutions to a coupled schrödinger systems with neumann boundary condition,, J. Math. Anal. Appl., 409 (2014), 684.  doi: 10.1016/j.jmaa.2013.07.053.  Google Scholar

[17]

J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations,, Rend. Lincei Mat. Appl., 18 (2007), 279.  doi: 10.4171/RLM/495.  Google Scholar

[18]

J. Wei and T. Weth, Radial solutions and phase sparation in a system of two coupled Schrödinger equations,, Arch. Rat. Mech. Anal., 190 (2008), 83.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[19]

J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459.  doi: 10.1016/S0294-1449(98)80031-0.  Google Scholar

[20]

W. Yao and J. Wei, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, Commun. Pure. Appl. Anal., 11 (2012), 1003.  doi: 10.3934/cpaa.2012.11.1003.  Google Scholar

[1]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[2]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[3]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[4]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[5]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[6]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[7]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[8]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[9]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[10]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[11]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[12]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[13]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[14]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[15]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[16]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[17]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[18]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[19]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[20]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]