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

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

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

Weiwei Ao ^1, , Liping Wang ^2, and Wei Yao ^3,

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

Abstract
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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].

Keywords: intermediate reduction method., non-symmetric potentials, infinitely many solutions, Schrödinger system.

Mathematics Subject Classification: Primary: 35B35, 35J25; Secondary: 35B40, 92D2.

Citation: Weiwei Ao, Liping Wang, Wei Yao. Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials. Communications on Pure and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order, preprint.
[10]	F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928.
[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.
[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.
[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.
[14]	A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\mathbb{R}^{N}$, Adv. Math., 221 (2009), 1843-1909.
[15]	M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-883. doi: 10.1038/43136.
[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.
[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.
[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.
[19]	E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, CPAA, 11 (2012), 1003-1011.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order, preprint.
[10]	F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928.
[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.
[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.
[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.
[14]	A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\mathbb{R}^{N}$, Adv. Math., 221 (2009), 1843-1909.
[15]	M. Mitchell and M. Segev, Self-trapping of inconherent white light, Nature, 387 (1997), 880-883. doi: 10.1038/43136.
[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.
[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.
[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.
[19]	E. Timmermans, Phase seperation of Bose Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, CPAA, 11 (2012), 1003-1011.

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