American Institute of Mathematical Sciences

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May  2015, 35(5): 2151-2164. doi: 10.3934/dcds.2015.35.2151

On the least energy sign-changing solutions for a nonlinear elliptic system

Yohei Sato 1, and  Zhi-Qiang Wang 2,

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

Received  August 2013 Revised  September 2014 Published  December 2014

In this paper, as bound state solutions we consider least energy sign-changing solutions to a nonlinear elliptic system which consists of N-equations defined on a bounded domain $\Omega$. For any subset $K\subset \{1,\cdots, N\}$, we show the existence of sign-changing solution $\vec{u}=(u_1,\cdots,u_n)$ such that, for $i\in K$, $u_i$ are sign-changing functions that change sign exactly once in $\Omega$, and, for $i\notin K$, $u_i$ are one sign functions. We give a variational characterization of such solutions on modified Nehari type constrained sets.
Keywords: sign-changing solutions, least energy solution, multiple existence of solutions, Nonlinear Schrödinger system.
Mathematics Subject Classification: Primary: 35J50; Secondary: 47J3.
Citation: Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151
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.  Google Scholar

[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.  Google Scholar

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T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. Mitchell and M. Segev, Self-trapping of inconherentwhite light, Nature, 387 (1997), 880-882. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[19]

Ch. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl$_3$, Nature, 423 (2003), 62-65. Google Scholar

[20]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Meth. Non. Anal., 37 (2011), 203-223.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. Mitchell and M. Segev, Self-trapping of inconherentwhite light, Nature, 387 (1997), 880-882. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Ch. Rüegg et al, Bose-Einstein condensation of the triple states in the magnetic insulator TlCuCl$_3$, Nature, 423 (2003), 62-65. Google Scholar

[20]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Meth. Non. Anal., 37 (2011), 203-223.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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