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On the least energy sign-changing solutions for a nonlinear elliptic system
1. | Osaka City University Advanced Mathematical Institute, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Smiyoshi-ku, Osaka 558-8585, Japan |
2. | Chern Institute Mathematics and LPMC, Nankai University, Tianjin 300071, China |
References:
[1] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[2] |
A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. PDEs, 30 (2007), 85-112.
doi: 10.1007/s00526-006-0079-0. |
[3] |
T. Bartsch, E. 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. PDEs, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[4] |
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207. |
[5] |
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. |
[6] |
A. Castro, J. Cossio and J. Neuberger, A sign changing solutions for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053.
doi: 10.1216/rmjm/1181071858. |
[7] |
S. Chang, C. S. Lin, T. C. Lin and W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361.
doi: 10.1016/j.physd.2004.06.002. |
[8] |
M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species system, Ann. I. H. Poincaré, 19 (2002), 871-888.
doi: 10.1016/S0294-1449(02)00104-X. |
[9] |
E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. I. H. Poincaré, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[10] |
T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n, n\leq3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[11] |
T.-C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire., 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[12] |
Z. L. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[13] |
Z. L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193. |
[14] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[15] |
M. Mitchell and M. Segev, Self-trapping of inconherentwhite light, Nature, 387 (1997), 880-882. |
[16] |
E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 41-71.
doi: 10.4171/JEMS/103. |
[17] |
B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proceedings of the AMS, 138 (2010), 1681-1692.
doi: 10.1090/S0002-9939-10-10231-7. |
[18] |
B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure and Appl. Math., 63 (2010), 267-302. |
[19] |
Ch. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl$_3$, Nature, 423 (2003), 62-65. |
[20] |
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[21] |
Y. Sato and Z.-Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincare Anal. Non Lineaire , 30 (2013), 1-22.
doi: 10.1016/j.anihpc.2012.05.002. |
[22] |
H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. H. Poincare Anal. Non Lineaire, 29 (2012), 279-300.
doi: 10.1016/j.anihpc.2011.10.006. |
[23] |
S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condensates, Arch. Rat. Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[24] |
R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Meth. Non. Anal., 37 (2011), 203-223. |
[25] |
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. |
[26] |
J. 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. |
show all references
References:
[1] |
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.
doi: 10.1112/jlms/jdl020. |
[2] |
A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. PDEs, 30 (2007), 85-112.
doi: 10.1007/s00526-006-0079-0. |
[3] |
T. Bartsch, E. 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. PDEs, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[4] |
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207. |
[5] |
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. |
[6] |
A. Castro, J. Cossio and J. Neuberger, A sign changing solutions for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053.
doi: 10.1216/rmjm/1181071858. |
[7] |
S. Chang, C. S. Lin, T. C. Lin and W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361.
doi: 10.1016/j.physd.2004.06.002. |
[8] |
M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species system, Ann. I. H. Poincaré, 19 (2002), 871-888.
doi: 10.1016/S0294-1449(02)00104-X. |
[9] |
E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. I. H. Poincaré, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[10] |
T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n, n\leq3$, Comm. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[11] |
T.-C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire., 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[12] |
Z. L. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.
doi: 10.1007/s00220-008-0546-x. |
[13] |
Z. L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193. |
[14] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[15] |
M. Mitchell and M. Segev, Self-trapping of inconherentwhite light, Nature, 387 (1997), 880-882. |
[16] |
E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 41-71.
doi: 10.4171/JEMS/103. |
[17] |
B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proceedings of the AMS, 138 (2010), 1681-1692.
doi: 10.1090/S0002-9939-10-10231-7. |
[18] |
B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure and Appl. Math., 63 (2010), 267-302. |
[19] |
Ch. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl$_3$, Nature, 423 (2003), 62-65. |
[20] |
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221.
doi: 10.1007/s00220-006-0179-x. |
[21] |
Y. Sato and Z.-Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincare Anal. Non Lineaire , 30 (2013), 1-22.
doi: 10.1016/j.anihpc.2012.05.002. |
[22] |
H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. H. Poincare Anal. Non Lineaire, 29 (2012), 279-300.
doi: 10.1016/j.anihpc.2011.10.006. |
[23] |
S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condensates, Arch. Rat. Mech. Anal., 194 (2009), 717-741.
doi: 10.1007/s00205-008-0172-y. |
[24] |
R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Meth. Non. Anal., 37 (2011), 203-223. |
[25] |
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. |
[26] |
J. 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. |
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