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

Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions

Zhongwei Tang 1,

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.
Keywords: Neumann boundary condition., Segregated peak solutions, variational methods, coupled Schrödinger systems.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35Q5.
Citation: Zhongwei Tang. Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 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), 743-767. 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), 345-361. doi: 10.1007/s00526-009-0265-y.  Google Scholar

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T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger systems, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.  Google Scholar

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D. Cao and T. Küpper, On the existence of multipeaked solutions to a semilinear Neumann Problem, Duke Math. J., 97 (1999), 261-300. doi: 10.1215/S0012-7094-99-09712-0.  Google Scholar

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N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Tran. Amer. Math. Soci., 361 (2009), 1189-1208. doi: 10.1090/S0002-9947-08-04735-1.  Google Scholar

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S. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339. 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), 3630-3657. 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), 247-281. 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), 3594-3597. 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), 47-82. 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), 403-439. 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), 629-653. 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), 538-569. 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), 199-221. 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), 336-352. 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), 684-704. 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), 279-293. 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), 83-106. 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), 459-492. 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), 1003-1011. 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), 743-767. 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), 345-361. 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), 353-367. 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), 261-300. 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), 1189-1208. 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), 305-339. 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), 3630-3657. 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), 247-281. 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), 3594-3597. 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), 47-82. 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), 403-439. 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), 629-653. 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), 538-569. 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), 199-221. 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), 336-352. 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), 684-704. 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), 279-293. 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), 83-106. 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), 459-492. 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), 1003-1011. doi: 10.3934/cpaa.2012.11.1003.  Google Scholar

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