American Institute of Mathematical Sciences

Advanced
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
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show all references

References:
[1]

J. Lond. Math. Soc., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.  Google Scholar

[2]

Calc. Var. PDEs, 30 (2007), 85-112. doi: 10.1007/s00526-006-0079-0.  Google Scholar

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Calc. Var. PDEs, 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.  Google Scholar

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J. Part. Diff. Equ., 19 (2006), 200-207.  Google Scholar

[5]

J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.  Google Scholar

[6]

Rocky Mountain J. Math., 27 (1997), 1041-1053. doi: 10.1216/rmjm/1181071858.  Google Scholar

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Phys. D, 196 (2004), 341-361. doi: 10.1016/j.physd.2004.06.002.  Google Scholar

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Ann. I. H. Poincaré, 19 (2002), 871-888. doi: 10.1016/S0294-1449(02)00104-X.  Google Scholar

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Ann. I. H. Poincaré, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

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Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar

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Ann. Inst. H. Poincaré Anal. Non Linéaire., 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

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Comm. Math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.  Google Scholar

[13]

Advanced Nonlinear Studies, 10 (2010), 175-193.  Google Scholar

[14]

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[15]

Nature, 387 (1997), 880-882. Google Scholar

[16]

J. Eur. Math. Soc., 10 (2008), 41-71. doi: 10.4171/JEMS/103.  Google Scholar

[17]

Proceedings of the AMS, 138 (2010), 1681-1692. doi: 10.1090/S0002-9939-10-10231-7.  Google Scholar

[18]

Comm. Pure and Appl. Math., 63 (2010), 267-302.  Google Scholar

[19]

Nature, 423 (2003), 62-65. Google Scholar

[20]

Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[21]

Ann. Inst. H. Poincare Anal. Non Lineaire , 30 (2013), 1-22. doi: 10.1016/j.anihpc.2012.05.002.  Google Scholar

[22]

Ann. Inst. H. Poincare Anal. Non Lineaire, 29 (2012), 279-300. doi: 10.1016/j.anihpc.2011.10.006.  Google Scholar

[23]

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[24]

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Rend. Lincei Mat. Appl., 18 (2007), 279-293. doi: 10.4171/RLM/495.  Google Scholar

[26]

Arch. Rat. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.  Google Scholar

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