We study the set of solutions of the nonlinear elliptic system
@article{AIHPC_2010__27_3_953_0, author = {Dancer, E.N. and Wei, Juncheng and Weth, Tobias}, 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}, pages = {953--969}, publisher = {Elsevier}, volume = {27}, number = {3}, year = {2010}, doi = {10.1016/j.anihpc.2010.01.009}, mrnumber = {2629888}, zbl = {1191.35121}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.009/} }
TY - JOUR AU - Dancer, E.N. AU - Wei, Juncheng AU - Weth, Tobias TI - A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 953 EP - 969 VL - 27 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.009/ DO - 10.1016/j.anihpc.2010.01.009 LA - en ID - AIHPC_2010__27_3_953_0 ER -
%0 Journal Article %A Dancer, E.N. %A Wei, Juncheng %A Weth, Tobias %T A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 953-969 %V 27 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.009/ %R 10.1016/j.anihpc.2010.01.009 %G en %F AIHPC_2010__27_3_953_0
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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- 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
- 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
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- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Symmetry of Components for Semilinear Elliptic Systems, SIAM Journal on Mathematical Analysis, Volume 44 (2012) no. 4, p. 2545 | DOI:10.1137/11085428x
- 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
- A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calculus of Variations and Partial Differential Equations, Volume 37 (2010) no. 3-4, pp. 345-361 | DOI:10.1007/s00526-009-0265-y | Zbl:1189.35074
- Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Communications on Pure and Applied Mathematics, Volume 63 (2010) no. 3, pp. 267-302 | DOI:10.1002/cpa.20309 | Zbl:1189.35314
- Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proceedings of the American Mathematical Society, Volume 138 (2010) no. 5, pp. 1681-1692 | DOI:10.1090/s0002-9939-10-10231-7 | Zbl:1189.35086
- Multipulse phases in k-mixtures of Bose-Einstein condensates, Archive for Rational Mechanics and Analysis, Volume 194 (2009) no. 3, pp. 717-741 | DOI:10.1007/s00205-008-0172-y | Zbl:1181.35069
- A priori estimates and existence for quasi-linear elliptic equations, Calculus of Variations and Partial Differential Equations, Volume 33 (2008) no. 4, pp. 417-437 | DOI:10.1007/s00526-008-0168-3 | Zbl:1169.35336
- Multiple bound states of nonlinear Schrödinger systems, Communications in Mathematical Physics, Volume 282 (2008) no. 3, pp. 721-731 | DOI:10.1007/s00220-008-0546-x | Zbl:1156.35093
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