# American Institute of Mathematical Sciences

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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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. 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. 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. 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. 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. 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. 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. 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. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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