Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	B. Sirakov, Least energy solitary waves for a system of nonliear Schrödinger equations in $\mathbb{R}^{N}$, Commun. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	B. Sirakov, Least energy solitary waves for a system of nonliear Schrödinger equations in $\mathbb{R}^{N}$, Commun. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.
[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.
[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.
[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.
[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.
[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.
[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.

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