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

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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

Abstract
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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 and 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.
[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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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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