# American Institute of Mathematical Sciences

May  2016, 15(3): 965-989. doi: 10.3934/cpaa.2016.15.965

## Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials

 1 Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2 2 Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241 3 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Chile

Received  October 2015 Revised  November 2015 Published  February 2016

Without any symmetric conditions on potentials, we proved the following nonlinear Schrödinger system \begin{eqnarray} \left\{\begin{array}{ll} \Delta u-P(x)u+\mu_1u^3+\beta uv^2=0, \quad &\mbox{in} \quad R^2\\ \Delta v-Q(x)v+\mu_2v^3+\beta vu^2=0, \quad &\mbox{in} \quad R^2 \end{array} \right. \end{eqnarray} has infinitely many non-radial solutions with suitable decaying rate at infinity of potentials $P(x)$ and $Q(x)$. This is the continued work of [8]. Especially when $P(x)$ and $Q(x)$ are symmetric, this result has been proved in [18].
Citation: Weiwei Ao, Liping Wang, Wei Yao. Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials. Communications on Pure & Applied Analysis, 2016, 15 (3) : 965-989. doi: 10.3934/cpaa.2016.15.965
##### References:
 [1] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris Ser., 1342 (2006), 453-458. doi: 10.1016/j.crma.2006.01.024.  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, Cal. Var. Partial Differential Equations., 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 system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.  Google Scholar [4] J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Memoirs of the American Mathematical Society, 229 (2014), 89pp.  Google Scholar [5] K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations, Phys. Rev. Lett. 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Salter, Special set and solution of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928.  Google Scholar [11] T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$, Comm. Math Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar [12] T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Phy. D, 220 (2006), 99-115. doi: 10.1016/j.physd.2006.07.009.  Google Scholar [13] Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system, Comm. math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.  Google Scholar [14] A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$, Adv. Math., 221 (2009), 1843-1909.  Google Scholar [15] M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-883. doi: 10.1038/43136.  Google Scholar [16] M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953. doi: 10.4171/JEMS/351.  Google Scholar [17] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.  Google Scholar [18] S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0.  Google Scholar [19] E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. Google Scholar [20] S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates, Arch. Rat. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y.  Google Scholar [21] L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615. doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar [22] L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc., 102 (2011), 1099-1126. doi: 10.1112/plms/pdq051.  Google Scholar [23] L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem, Discrete Contin. Dyn. Syst., 34 (2014), 2333-2357.  Google Scholar [24] J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.  Google Scholar [25] J. C. Wei and T. Weth, Radial solutions and phase separation 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 [26] J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1.  Google Scholar [27] J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, CPAA, 11 (2012), 1003-1011.  Google Scholar

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##### References:
 [1] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris Ser., 1342 (2006), 453-458. doi: 10.1016/j.crma.2006.01.024.  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, Cal. Var. Partial Differential Equations., 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 system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.  Google Scholar [4] J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Memoirs of the American Mathematical Society, 229 (2014), 89pp.  Google Scholar [5] K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations, Phys. Rev. Lett. A, 285 (2001), 319-326. doi: 10.1016/S0375-9601(01)00369-3.  Google Scholar [6] M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincar Anal. Non Linaire, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X.  Google Scholar [7] N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincar Anal. Non Linaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar [8] M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Car. Var. PDE., 53 (2015), 473-523. doi: 10.1007/s00526-014-0756-3.  Google Scholar [9] Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., ().   Google Scholar [10] F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928.  Google Scholar [11] T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$, Comm. Math Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar [12] T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Phy. D, 220 (2006), 99-115. doi: 10.1016/j.physd.2006.07.009.  Google Scholar [13] Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system, Comm. math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.  Google Scholar [14] A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$, Adv. Math., 221 (2009), 1843-1909.  Google Scholar [15] M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-883. doi: 10.1038/43136.  Google Scholar [16] M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953. doi: 10.4171/JEMS/351.  Google Scholar [17] B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.  Google Scholar [18] S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational. Mech. Anal., 208 (2013), 305-339. doi: 10.1007/s00205-012-0598-0.  Google Scholar [19] E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. Google Scholar [20] S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates, Arch. Rat. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y.  Google Scholar [21] L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615. doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar [22] L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc., 102 (2011), 1099-1126. doi: 10.1112/plms/pdq051.  Google Scholar [23] L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem, Discrete Contin. Dyn. Syst., 34 (2014), 2333-2357.  Google Scholar [24] J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.  Google Scholar [25] J. C. Wei and T. Weth, Radial solutions and phase separation 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 [26] J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1.  Google Scholar [27] J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, CPAA, 11 (2012), 1003-1011.  Google Scholar
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