A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 953-969.

We study the set of solutions of the nonlinear elliptic system

{-Δu+λ1u=μ1u3+βv2uinΩ,-Δv+λ2v=μ2v3+βu2vinΩ,u,v>0inΩ,u=v=0onΩ,(P)
in a smooth bounded domain ΩN, N3, with coupling parameter β. This system arises in the study of Bose–Einstein double condensates. We show that the value β=-μ1μ2 is critical for the existence of a priori bounds for solutions of (P). More precisely, we show that for β>-μ1μ2, solutions of (P) are a priori bounded. In contrast, when λ1=λ2, μ1=μ2, (P) admits an unbounded sequence of solutions if β-μ1μ2.

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     title = {A priori bounds versus multiple existence of positive solutions for a nonlinear {Schr\"odinger} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     publisher = {Elsevier},
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Dancer, E.N.; Wei, Juncheng; Weth, Tobias. A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 953-969. doi : 10.1016/j.anihpc.2010.01.009. http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.009/

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  • Yang, Hui; Zou, Wenming Qualitative analysis for an elliptic system in the punctured space, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 71 (2020) no. 2, p. 21 (Id/No 47) | DOI:10.1007/s00033-020-1270-4 | Zbl:1436.35145
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  • Bartsch, Thomas; Soave, Nicola Multiple normalized solutions for a competing system of Schrödinger equations, Calculus of Variations and Partial Differential Equations, Volume 58 (2019) no. 1, p. 24 (Id/No 22) | DOI:10.1007/s00526-018-1476-x | Zbl:1409.35076
  • Tian, Rushun; Wang, Zhi-Qiang; Zhao, Leiga Schrödinger systems with quadratic interactions, Communications in Contemporary Mathematics, Volume 21 (2019) no. 8, p. 20 (Id/No 1850077) | DOI:10.1142/s0219199718500773 | Zbl:1437.35276
  • Peng, Shuangjie; Peng, Yanfang; Wang, Qingfang Sign-changing solutions of an elliptic system with critical exponent in dimension \(N= 5 \), Journal d'Analyse Mathématique, Volume 137 (2019) no. 1, pp. 231-249 | DOI:10.1007/s11854-018-0071-6 | Zbl:1415.35114
  • Song, Hongxue; Chen, Caisheng; Liu, Wei Existence and multiplicity of standing wave solutions for a class of quasilinear Schrödinger systems in \(\mathbb{R}^{N}\), Journal of Dynamical and Control Systems, Volume 25 (2019) no. 1, pp. 79-94 | DOI:10.1007/s10883-018-9399-6 | Zbl:1411.35107
  • Yang, Hui; Zou, Wenming Stable and finite Morse index solutions of a nonlinear elliptic system, Journal of Mathematical Analysis and Applications, Volume 471 (2019) no. 1-2, pp. 147-169 | DOI:10.1016/j.jmaa.2018.10.069 | Zbl:1408.35031
  • Chen, Yan-Hong; Zou, Wenming Vector solutions for two coupled Schrödinger equations on Riemannian manifolds, Journal of Mathematical Physics, Volume 60 (2019) no. 5, p. 051502 | DOI:10.1063/1.5100021 | Zbl:1414.35204
  • Dancer, E. N.; Yang, Hui; Zou, Wenming Liouville-type results for a class of quasilinear elliptic systems and applications, Journal of the London Mathematical Society. Second Series, Volume 99 (2019) no. 2, pp. 273-294 | DOI:10.1112/jlms.12170 | Zbl:1418.35124
  • Yang, Hui; Zou, Wenming Symmetry of components and Liouville-type theorems for semilinear elliptic systems involving the fractional Laplacian, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 180 (2019), pp. 208-224 | DOI:10.1016/j.na.2018.10.006 | Zbl:1414.35015
  • Zhao, Junfang; Liu, Xiangqing; Liu, Jiaquan Infinitely many sign-changing solutions for system of p-Laplace equations in \(\mathbb{R}^N\), Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 182 (2019), pp. 113-142 | DOI:10.1016/j.na.2018.12.005 | Zbl:1418.35204
  • You, Song; Zhao, Peihao; Wang, Qingxuan Positive ground states for coupled nonlinear Choquard equations involving Hardy-Littlewood-Sobolev critical exponent, Nonlinear Analysis. Real World Applications, Volume 48 (2019), pp. 182-211 | DOI:10.1016/j.nonrwa.2019.01.015 | Zbl:1428.35543
  • Noris, Benedetta; Tavares, Hugo; Verzini, Gianmaria Normalized solutions for nonlinear Schrödinger systems on bounded domains, Nonlinearity, Volume 32 (2019) no. 3, pp. 1044-1072 | DOI:10.1088/1361-6544/aaf2e0 | Zbl:1410.35211
  • Byeon, Jaeyoung; Lee, Youngae; Wang, Zhi-Qiang Formation of radial patterns via mixed attractive and repulsive interactions for Schrödinger systems, SIAM Journal on Mathematical Analysis, Volume 51 (2019) no. 2, pp. 1514-1542 | DOI:10.1137/18m1196789 | Zbl:1414.35074
  • You, Song; Wang, Qingxun; Zhao, Peihao Positive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponent, Topological Methods in Nonlinear Analysis, Volume 53 (2019) no. 2, pp. 623-657 | DOI:10.12775/tmna.2019.014 | Zbl:1433.35080
  • Peng, Shuangjie; Wang, Qingfang; Wang, Zhi-Qiang On coupled nonlinear Schrödinger systems with mixed couplings, Transactions of the American Mathematical Society, Volume 371 (2019) no. 11, pp. 7559-7583 | DOI:10.1090/tran/7383 | Zbl:1439.35162
  • Lou, Zhenluo; Weth, Tobias; Zhang, Zhitao Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 70 (2019) no. 1, p. 19 (Id/No 35) | DOI:10.1007/s00033-019-1080-8 | Zbl:1412.35108
  • Clapp, Mónica; Pistoia, Angela Existence and phase separation of entire solutions to a pure critical competitive elliptic system, Calculus of Variations and Partial Differential Equations, Volume 57 (2018) no. 1, p. 20 (Id/No 23) | DOI:10.1007/s00526-017-1283-9 | Zbl:1394.35164
  • Liu, Haidong; Liu, Zhaoli Multiple positive solutions of elliptic systems in exterior domains, Communications in Contemporary Mathematics, Volume 20 (2018) no. 6, p. 32 (Id/No 1750063) | DOI:10.1142/s0219199717500638 | Zbl:1401.35120
  • Ye, Hongyu Positive solutions for critically coupled Schrödinger systems with attractive interactions, Discrete and Continuous Dynamical Systems, Volume 38 (2018) no. 2, pp. 485-507 | DOI:10.3934/dcds.2018022 | Zbl:1374.35165
  • Duong, Anh Tuan; Phan, Quoc Hung A Liouville-type theorem for cooperative parabolic systems, Discrete and Continuous Dynamical Systems, Volume 38 (2018) no. 2, pp. 823-833 | DOI:10.3934/dcds.2018035 | Zbl:1379.35044
  • Liu, Xiangqing; Zhao, Junfang; Liu, Jiaquan On the system of \(p\)-Laplacian equations with critical growth, International Journal of Mathematics, Volume 29 (2018) no. 2, p. 32 (Id/No 1850008) | DOI:10.1142/s0129167x18500088 | Zbl:1392.35170
  • Tang, Zhongwei; Wang, Lushun Segregated vector solutions with multi-scale spikes for nonlinear coupled elliptic systems, Journal of Mathematical Analysis and Applications, Volume 464 (2018) no. 1, pp. 1-31 | DOI:10.1016/j.jmaa.2018.02.019 | Zbl:1390.35012
  • Ghanmi, R. Asymptotics for a class of coupled fourth-order Schrödinger equations, Mediterranean Journal of Mathematics, Volume 15 (2018) no. 3, p. 28 (Id/No 109) | DOI:10.1007/s00009-018-1153-5 | Zbl:1394.35474
  • Guo, Yuxia; Luo, Senping; Zou, Wenming The existence, uniqueness and nonexistence of the ground state to theN-coupled Schrödinger systems in $ \newcommand{\R}{{\mathbb R}} \boldsymbol{\R^n (n\leqslant 4)}$, Nonlinearity, Volume 31 (2018) no. 1, p. 314 | DOI:10.1088/1361-6544/aa8ca9
  • Cao, Xiaofei; Xu, Junxiang; Wang, Jun; Zhang, Fubao Uniqueness of positive solutions for a class of Schrödinger systems with saturable nonlinearity, Rocky Mountain Journal of Mathematics, Volume 48 (2018) no. 6, pp. 1815-1828 | DOI:10.1216/rmj-2018-48-6-1815 | Zbl:1406.35119
  • Lou, Zhenluo Existence of non-radial solutions of an elliptic system, Applied Mathematics Letters, Volume 68 (2017), pp. 157-162 | DOI:10.1016/j.aml.2016.12.020 | Zbl:1365.35026
  • Peng, Yan-Fang; Ye, Hong-Yu Positive solutions for coupled Schrödinger system with critical exponent in \(\mathbb{R}^{N}\) (\(N\geq5\)), Boundary Value Problems, Volume 2017 (2017), p. 18 (Id/No 104) | DOI:10.1186/s13661-017-0834-5 | Zbl:1375.35123
  • Liang, Zhanping; Song, Yuanmin; Li, Fuyi Positive ground state solutions of a quadratically coupled Schrödinger system, Communications on Pure and Applied Analysis, Volume 16 (2017) no. 3, pp. 999-1012 | DOI:10.3934/cpaa.2017048 | Zbl:1359.35051
  • Yang, Jing Segregated vector solutions for nonlinear Schrödinger systems with electromagnetic potentials, Communications on Pure and Applied Analysis, Volume 16 (2017) no. 5, pp. 1785-1805 | DOI:10.3934/cpaa.2017087 | Zbl:1364.35078
  • Zhao, Chunyi; Wang, Liping Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential, Discrete and Continuous Dynamical Systems, Volume 37 (2017) no. 3, pp. 1707-1731 | DOI:10.3934/dcds.2017071 | Zbl:1365.35024
  • Peng, Shuangjie; Shuai, Wei; Wang, Qingfang Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, Journal of Differential Equations, Volume 263 (2017) no. 1, pp. 709-731 | DOI:10.1016/j.jde.2017.02.053 | Zbl:1365.35027
  • Byeon, Jaeyoung; Sato, Yohei; Wang, Zhi-Qiang Pattern formation via mixed interactions for coupled Schrödinger equations under Neumann boundary condition, Journal of Fixed Point Theory and Applications, Volume 19 (2017) no. 1, pp. 559-583 | DOI:10.1007/s11784-016-0365-1 | Zbl:1373.35132
  • Zhang, Zhitao; Wang, Wei Structure of positive solutions to a Schrödinger system, Journal of Fixed Point Theory and Applications, Volume 19 (2017) no. 1, pp. 877-887 | DOI:10.1007/s11784-016-0383-z | Zbl:1387.35144
  • Ye, Hongyu On the existence of positive least energy solutions for a coupled Schrödinger system with critical exponent, Mathematical Methods in the Applied Sciences, Volume 40 (2017) no. 4, pp. 1032-1043 | DOI:10.1002/mma.4034 | Zbl:1368.35128
  • Zheng, Lvzhou Segregated vector solutions for the nonlinear Schrödinger systems in \(\mathbb {R}^{3}\), Mediterranean Journal of Mathematics, Volume 14 (2017) no. 3, p. 21 (Id/No 107) | DOI:10.1007/s00009-017-0909-7 | Zbl:1377.35077
  • Fazly, Mostafa Solutions of multi-component fractional symmetric systems, NoDEA. Nonlinear Differential Equations and Applications, Volume 24 (2017) no. 4, p. 33 (Id/No 44) | DOI:10.1007/s00030-017-0470-x | Zbl:1441.35122
  • He, QiHan; Luo, Xiao A positive solution of a nonlinear Schrödinger system with nonconstant potentials, Science China. Mathematics, Volume 60 (2017) no. 12, pp. 2407-2420 | DOI:10.1007/s11425-016-9131-6 | Zbl:1387.35137
  • Zhang, Zhi-tao; Sun, Yi-min Existence and multiplicity of solutions for nonlocal systems with Kirchhoff type, Acta Mathematicae Applicatae Sinica. English Series, Volume 32 (2016) no. 1, pp. 35-54 | DOI:10.1007/s10255-016-0545-1 | Zbl:1338.35165
  • Bunoiu, Renata; Precup, Radu Vectorial approach to coupled nonlinear Schrödinger systems under nonlocal Cauchy conditions, Applicable Analysis, Volume 95 (2016) no. 4, pp. 731-747 | DOI:10.1080/00036811.2015.1028921 | Zbl:1338.35403
  • Ao, Weiwei; Wang, Liping; Yao, Wei Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials, Communications on Pure and Applied Analysis, Volume 15 (2016) no. 3, pp. 965-989 | DOI:10.3934/cpaa.2016.15.965 | Zbl:1336.35136
  • Nguyen, Nghiem; Wang, Zhi-Qiang Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 2, pp. 1005-1021 | DOI:10.3934/dcds.2016.36.1005 | Zbl:1330.35411
  • Li, Kui; Zhang, Zhitao A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in \(\mathbb{R}^3\), Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 2, pp. 851-860 | DOI:10.3934/dcds.2016.36.851 | Zbl:1330.35130
  • Liu, Haidong; Liu, Zhaoli Positive solutions of a nonlinear Schrödinger system with nonconstant potentials, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 3, pp. 1431-1464 | DOI:10.3934/dcds.2016.36.1431 | Zbl:1352.35038
  • Pi, Huirong; Peng, Shuangjie Spike vector solutions for some coupled nonlinear Schrödinger equations, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 4, pp. 2205-2227 | DOI:10.3934/dcds.2016.36.2205 | Zbl:1327.35038
  • Peng, Yanfang On elliptic systems with Sobolev critical exponent, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 6, pp. 3357-3373 | DOI:10.3934/dcds.2016.36.3357 | Zbl:1335.35052
  • Byeon, Jaeyoung; Sato, Yohei; Wang, Zhi-Qiang Pattern formation via mixed attractive and repulsive interactions for nonlinear Schrödinger systems, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 106 (2016) no. 3, pp. 477-511 | DOI:10.1016/j.matpur.2016.03.001 | Zbl:1345.35032
  • Liu, Jiaquan; Liu, Xiangqing; Wang, Zhi-Qiang Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, Journal of Differential Equations, Volume 261 (2016) no. 12, pp. 7194-7236 | DOI:10.1016/j.jde.2016.09.018 | Zbl:1352.35162
  • Liu, Chuangye; Nguyen, Nghiem V.; Wang, Zhi-Qiang Orbital stability of spatially synchronized solitary waves of an m-coupled nonlinear Schrödinger system, Journal of Mathematical Physics, Volume 57 (2016) no. 10, p. 101501 | DOI:10.1063/1.4964255 | Zbl:1356.35222
  • Li, Kui; Zhang, Zhitao Existence of solutions for a Schrödinger system with linear and nonlinear couplings, Journal of Mathematical Physics, Volume 57 (2016) no. 8, p. 081504 | DOI:10.1063/1.4960046 | Zbl:1344.81075
  • Liu, Weiming Infinitely many solutions for nonlinear Schrödinger systems with magnetic potentials in \(\mathbb{R}^3\), Mathematical Methods in the Applied Sciences, Volume 39 (2016) no. 6, pp. 1452-1479 | DOI:10.1002/mma.3581 | Zbl:1339.35292
  • Guo, Zhenyu; Zou, Wenming On a class of coupled Schrödinger systems with critical Sobolev exponent growth, Mathematical Methods in the Applied Sciences, Volume 39 (2016) no. 7, pp. 1730-1746 | DOI:10.1002/mma.3598 | Zbl:1341.35040
  • Phan, Quoc Hung; Souplet, Philippe A Liouville-type theorem for the 3-dimensional parabolic Gross-Pitaevskii and related systems, Mathematische Annalen, Volume 366 (2016) no. 3-4, pp. 1561-1585 | DOI:10.1007/s00208-016-1368-3 | Zbl:1362.35063
  • He, Qihan; Peng, Shuangjie Synchronized vector solutions to an elliptic system, Proceedings of the American Mathematical Society, Volume 144 (2016) no. 9, pp. 4055-4063 | DOI:10.1090/proc/13160 | Zbl:1342.35097
  • WANG, Chunhua; XIE, Dingyi; ZHAN, Liping; ZHANG, Lipan; ZHAO, Liangpei Segregated vector solutions for nonlinear schrödinger systems in ℝ2, Acta Mathematica Scientia, Volume 35 (2015) no. 2, p. 383 | DOI:10.1016/s0252-9602(15)60010-8
  • Chen, Zhijie; Zou, Wenming Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calculus of Variations and Partial Differential Equations, Volume 52 (2015) no. 1-2, pp. 423-467 | DOI:10.1007/s00526-014-0717-x | Zbl:1312.35158
  • Liu, Jiaquan; Liu, Xiangqing; Wang, Zhi-Qiang Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calculus of Variations and Partial Differential Equations, Volume 52 (2015) no. 3-4, pp. 565-586 | DOI:10.1007/s00526-014-0724-y | Zbl:1311.35291
  • d'Avenia, Pietro; Mederski, Jarosław Positive ground states for a system of Schrödinger equations with critically growing nonlinearities, Calculus of Variations and Partial Differential Equations, Volume 53 (2015) no. 3-4, pp. 879-900 | DOI:10.1007/s00526-014-0770-5 | Zbl:1317.35067
  • Sato, Yohei; Wang, Zhi-Qiang Multiple positive solutions for Schrödinger systems with mixed couplings, Calculus of Variations and Partial Differential Equations, Volume 54 (2015) no. 2, pp. 1373-1392 | DOI:10.1007/s00526-015-0828-z | Zbl:1326.35131
  • Byeon, Jaeyoung Semi-classical standing waves for nonlinear Schrödinger systems, Calculus of Variations and Partial Differential Equations, Volume 54 (2015) no. 2, pp. 2287-2340 | DOI:10.1007/s00526-015-0866-6 | Zbl:1334.35041
  • Zhao, Leiga; Zhao, Fukun; Shi, Junping Higher dimensional solitary waves generated by second-harmonic generation in quadratic media, Calculus of Variations and Partial Differential Equations, Volume 54 (2015) no. 3, pp. 2657-2691 | DOI:10.1007/s00526-015-0879-1 | Zbl:1332.35345
  • Sato, Yohei; Wang, Zhi-Qiang On the least energy sign-changing solutions for a nonlinear elliptic system, Discrete Continuous Dynamical Systems - A, Volume 35 (2015) no. 5, p. 2151 | DOI:10.3934/dcds.2015.35.2151
  • Phan, Quoc Hung Optimal Liouville-type theorems for a parabolic system, Discrete and Continuous Dynamical Systems, Volume 35 (2015) no. 1, pp. 399-409 | DOI:10.3934/dcds.2015.35.399 | Zbl:1304.35161
  • Noris, Benedetta; Tavares, Hugo; Verzini, Gianmaria Stable solitary waves with prescribed \(L^2\)-mass for the cubic Schrödinger system with trapping potentials, Discrete and Continuous Dynamical Systems, Volume 35 (2015) no. 12, pp. 6085-6112 | DOI:10.3934/dcds.2015.35.6085 | Zbl:1336.35321
  • Yue, Xiaorui Positive ground state solutions and multiple nontrivial solutions for coupled critical elliptic systems, Journal of Mathematical Analysis and Applications, Volume 427 (2015) no. 1, p. 88 | DOI:10.1016/j.jmaa.2015.01.073
  • Guo, Zhenyu Positive ground state solutions for a nonlinearly coupled Schrödinger system with critical exponents in \(\mathbb{R}^4\), Journal of Mathematical Analysis and Applications, Volume 430 (2015) no. 2, pp. 950-970 | DOI:10.1016/j.jmaa.2015.05.037 | Zbl:1319.35237
  • Wang, Chunhua; Yang, Jing Infinitely many solutions to a linearly coupled Schrödinger system with non-symmetric potentials, Journal of Mathematical Physics, Volume 56 (2015) no. 5, p. 051505 | DOI:10.1063/1.4921637 | Zbl:1317.35242
  • Long, Wei; Wang, Qingfang Segregated and synchronized vector solutions to linearly coupled systems of Schrödinger equations, Journal of Mathematical Physics, Volume 56 (2015) no. 9, p. 091507 | DOI:10.1063/1.4930189 | Zbl:1329.35283
  • Liu, HaiDong; Liu, ZhaoLi Ground states of a nonlinear Schrödinger system with nonconstant potentials, Science China. Mathematics, Volume 58 (2015) no. 2, pp. 257-278 | DOI:10.1007/s11425-014-4914-z | Zbl:1311.35290
  • Tian, Rushun; Zhang, Zhitao Existence and bifurcation of solutions for a double coupled system of Schrödinger equations, Science China. Mathematics, Volume 58 (2015) no. 8, pp. 1607-1620 | DOI:10.1007/s11425-015-5028-y | Zbl:1326.35133
  • Yue, Xiaorui; Zou, Wenming A perturbation method for \(k\)-mixtures of Bose-Einstein condensates, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 66 (2015) no. 3, pp. 1023-1035 | DOI:10.1007/s00033-014-0468-8 | Zbl:1323.35032
  • Bartsch, Thomas; Tian, Rushun; Wang, Zhi-Qiang Bifurcations for a coupled Schrödinger system with multiple components, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 66 (2015) no. 5, pp. 2109-2123 | DOI:10.1007/s00033-015-0498-x | Zbl:1330.35134
  • Tang, Su Fang; Dou, Jing Bo Liouville type theorems for a system of integral equations on upper half space, Acta Mathematica Sinica. English Series, Volume 30 (2014) no. 2, pp. 261-276 | DOI:10.1007/s10114-014-3071-1 | Zbl:1290.45004
  • Montaru, Alexandre; Sirakov, Boyan; Souplet, Philippe Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems, Archive for Rational Mechanics and Analysis, Volume 213 (2014) no. 1, pp. 129-169 | DOI:10.1007/s00205-013-0719-4 | Zbl:1298.35060
  • Ao, Weiwei; Wei, Juncheng Infinitely many positive solutions for nonlinear equations with non-symmetric potentials, Calculus of Variations and Partial Differential Equations, Volume 51 (2014) no. 3-4, pp. 761-798 | DOI:10.1007/s00526-013-0694-5 | Zbl:1311.35077
  • Chen, Zhijie; Lin, Chang-Shou; Zou, Wenming Sign-changing solutions and phase separation for an elliptic system with critical exponent, Communications in Partial Differential Equations, Volume 39 (2014) no. 10, pp. 1827-1859 | DOI:10.1080/03605302.2014.908391 | Zbl:1308.35084
  • Montaru, Alexandre; Souplet, Philippe Symmetry of components and Liouville theorems for noncooperative elliptic systems on the half-space, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 352 (2014) no. 4, pp. 321-325 | DOI:10.1016/j.crma.2013.10.033 | Zbl:1300.35032
  • Long, Wei; Peng, Shuangjie Segregated vector solutions for a class of Bose-Einstein systems, Journal of Differential Equations, Volume 257 (2014) no. 1, pp. 207-230 | DOI:10.1016/j.jde.2014.03.019 | Zbl:1288.35434
  • D'Ambrosio, Lorenzo; Mitidieri, Enzo Liouville theorems for elliptic systems and applications, Journal of Mathematical Analysis and Applications, Volume 413 (2014) no. 1, pp. 121-138 | DOI:10.1016/j.jmaa.2013.11.052 | Zbl:1312.35075
  • Ye, Hongyu; Peng, Yanfang Positive least energy solutions for a coupled Schrödinger system with critical exponent, Journal of Mathematical Analysis and Applications, Volume 417 (2014) no. 1, pp. 308-326 | DOI:10.1016/j.jmaa.2014.03.028 | Zbl:1312.35086
  • Yue, Xiaorui; Zou, Wenming Infinitely many solutions for the perturbed Bose-Einstein condensates system, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 94 (2014), pp. 171-184 | DOI:10.1016/j.na.2013.08.012 | Zbl:1285.35026
  • Yang, Minbo; Wei, Yuanhong; Ding, Yanheng Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 65 (2014) no. 1, pp. 41-68 | DOI:10.1007/s00033-013-0317-1 | Zbl:1307.35104
  • Sato, Yohei; Wang, Zhi-Qiang On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 30 (2013) no. 1, pp. 1-22 | DOI:10.1016/j.anihpc.2012.05.002 | Zbl:1457.35071
  • Peng, Shuangjie; Wang, Zhi-qiang Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Archive for Rational Mechanics and Analysis, Volume 208 (2013) no. 1, pp. 305-339 | DOI:10.1007/s00205-012-0598-0 | Zbl:1260.35211
  • Kim, Seunghyeok On vector solutions for coupled nonlinear Schrödinger equations with critical exponents, Communications on Pure Applied Analysis, Volume 12 (2013) no. 3, p. 1259 | DOI:10.3934/cpaa.2013.12.1259
  • Tian, Rushun; Wang, Zhi-Qiang Bifurcation results on positive solutions of an indefinite nonlinear elliptic system, Discrete Continuous Dynamical Systems - A, Volume 33 (2013) no. 1, p. 335 | DOI:10.3934/dcds.2013.33.335
  • Noris, Benedetta; Verzini, Gianmaria A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems, Journal of Differential Equations, Volume 254 (2013) no. 3, p. 1529 | DOI:10.1016/j.jde.2012.11.003
  • Chen, Zhijie; Lin, Chang-Shou; Zou, Wenming Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations, Journal of Differential Equations, Volume 255 (2013) no. 11, pp. 4289-4311 | DOI:10.1016/j.jde.2013.08.009 | Zbl:1281.35073
  • Nguyen, Nghiem V.; Tian, Rushun; Deconinck, Bernard; Sheils, Natalie Global existence for a coupled system of Schrödinger equations with power-type nonlinearities, Journal of Mathematical Physics, Volume 54 (2013) no. 1, p. 011503 | DOI:10.1063/1.4774149 | Zbl:1286.35230
  • Chen, Zhijie; Zou, Wenming Standing waves for coupled nonlinear Schrödinger equations with decaying potentials, Journal of Mathematical Physics, Volume 54 (2013) no. 11, p. 111505 | DOI:10.1063/1.4833795 | Zbl:1288.81171
  • Tavares, H.; Weth, T. Existence and symmetry results for competing variational systems, NoDEA. Nonlinear Differential Equations and Applications, Volume 20 (2013) no. 3, pp. 715-740 | DOI:10.1007/s00030-012-0176-z | Zbl:1268.35008
  • Dou, Jingbo; Qu, Changzheng Classification of positive solutions for an elliptic system with a higher-order fractional Laplacian, Pacific Journal of Mathematics, Volume 261 (2013) no. 2, p. 311 | DOI:10.2140/pjm.2013.261.311
  • Zhang, Zhitao Free Boundary Problems, System of Equations for Bose–Einstein Condensate and Competing Species, Variational, Topological, and Partial Order Methods with Their Applications, Volume 29 (2013), p. 285 | DOI:10.1007/978-3-642-30709-6_11
  • Tavares, Hugo; Terracini, Susanna Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 29 (2012) no. 2, pp. 279-300 | DOI:10.1016/j.anihpc.2011.10.006 | Zbl:1241.35046
  • Chen, Zhijie; Zou, Wenming Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Archive for Rational Mechanics and Analysis, Volume 205 (2012) no. 2, pp. 515-551 | DOI:10.1007/s00205-012-0513-8 | Zbl:1256.35132
  • Tavares, Hugo; Terracini, Susanna Regularity of the nodal set of segregated critical configurations under a weak reflection law, Calculus of Variations and Partial Differential Equations, Volume 45 (2012) no. 3-4, pp. 273-317 | DOI:10.1007/s00526-011-0458-z | Zbl:1263.35101
  • Quittner, Pavol; Souplet, Philippe Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications, Communications in Mathematical Physics, Volume 311 (2012) no. 1, pp. 1-19 | DOI:10.1007/s00220-012-1440-0 | Zbl:1254.35077
  • Lucia, Marcello; Tang, Zhongwei Multi-bump bound states for a system of nonlinear Schrödinger equations, Journal of Differential Equations, Volume 252 (2012) no. 5, pp. 3630-3657 | DOI:10.1016/j.jde.2011.11.017 | Zbl:1232.35154
  • Noris, Benedetta; Tavares, Hugo; Terracini, Susanna; Verzini, Gianmaria Convergence of minimax structures and continuation of critical points for singularly perturbed systems, Journal of the European Mathematical Society (JEMS), Volume 14 (2012) no. 4, pp. 1245-1273 | DOI:10.4171/jems/332 | Zbl:1248.35197
  • Souplet, Philippe Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices, Networks Heterogeneous Media, Volume 7 (2012) no. 4, p. 967 | DOI:10.3934/nhm.2012.7.967
  • Quittner, Pavol; Souplet, Philippe Symmetry of Components for Semilinear Elliptic Systems, SIAM Journal on Mathematical Analysis, Volume 44 (2012) no. 4, p. 2545 | DOI:10.1137/11085428x
  • Tavares, Hugo; Terracini, Susanna; Verzini, Gianmaria; Weth, Tobias Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems, Communications in Partial Differential Equations, Volume 36 (2011) no. 10-12, pp. 1988-2010 | DOI:10.1080/03605302.2011.574244 | Zbl:1235.35089
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