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Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices
1. | Université Paris 13, Sorbonne Paris Cité, Laboratoire Analyse, Géométrie et Applications, CNRS, UMR 7539, 93430 Villetaneuse, France |
References:
[1] |
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Sér. I, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[2] |
Th. Bartsch, 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. Partial Differ. Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[3] |
J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51. |
[4] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[5] |
N. Dancer, J.-C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[6] |
N. Dancer and T. Weth, Liouville-type results for noncooperative elliptic systems in a half space, J. London Math. Soc., 86 (2012), 111-128. |
[7] |
D. G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 21 (1994), 387-397. |
[8] |
D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: From theory to experiments, J. Phys. A: Math. Theor., 43 (2010), 213001.
doi: 10.1088/1751-8113/43/21/213001. |
[9] |
D. H. Jacobson, Extensions of linear-quadratic control, optimization and matrix theory. Mathematics in Science and Engineering, 133. Academic Press, London-New York, (1977). |
[10] |
Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications, Physics Reports, 298 (1998), 81-197. |
[11] |
T-C. Lin and J.-C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n$, $n le 3$, Commun. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[12] |
Z. 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] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[14] |
P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[15] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[16] |
P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States," Birkhäuser Advanced Texts, 2007. |
[17] |
P. Quittner and Ph. Souplet, Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications, Comm. Math. Phys., 311 (2012), 1-19.
doi: 10.1007/s00220-012-1440-0. |
[18] |
W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equations, 161 (2000), 219-243.
doi: 10.1006/jdeq.1999.3700. |
[19] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653. |
[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] |
Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[22] |
H. Tavares, S. Terracini, G.-M. Verzini and T. Weth, Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems, Comm. Partial Differ. Eq., 36 (2011), 1988-2010.
doi: 10.1080/03605302.2011.574244. |
[23] |
F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102.
doi: 10.1512/iumj.1980.29.29007. |
show all references
References:
[1] |
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Sér. I, 342 (2006), 453-458.
doi: 10.1016/j.crma.2006.01.024. |
[2] |
Th. Bartsch, 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. Partial Differ. Equations, 37 (2010), 345-361.
doi: 10.1007/s00526-009-0265-y. |
[3] |
J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51. |
[4] |
W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[5] |
N. Dancer, J.-C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.
doi: 10.1016/j.anihpc.2010.01.009. |
[6] |
N. Dancer and T. Weth, Liouville-type results for noncooperative elliptic systems in a half space, J. London Math. Soc., 86 (2012), 111-128. |
[7] |
D. G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 21 (1994), 387-397. |
[8] |
D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: From theory to experiments, J. Phys. A: Math. Theor., 43 (2010), 213001.
doi: 10.1088/1751-8113/43/21/213001. |
[9] |
D. H. Jacobson, Extensions of linear-quadratic control, optimization and matrix theory. Mathematics in Science and Engineering, 133. Academic Press, London-New York, (1977). |
[10] |
Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications, Physics Reports, 298 (1998), 81-197. |
[11] |
T-C. Lin and J.-C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n$, $n le 3$, Commun. Math. Phys., 255 (2005), 629-653.
doi: 10.1007/s00220-005-1313-x. |
[12] |
Z. 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] |
L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.
doi: 10.1016/j.jde.2006.07.002. |
[14] |
P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[15] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[16] |
P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States," Birkhäuser Advanced Texts, 2007. |
[17] |
P. Quittner and Ph. Souplet, Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications, Comm. Math. Phys., 311 (2012), 1-19.
doi: 10.1007/s00220-012-1440-0. |
[18] |
W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equations, 161 (2000), 219-243.
doi: 10.1006/jdeq.1999.3700. |
[19] |
J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653. |
[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] |
Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[22] |
H. Tavares, S. Terracini, G.-M. Verzini and T. Weth, Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems, Comm. Partial Differ. Eq., 36 (2011), 1988-2010.
doi: 10.1080/03605302.2011.574244. |
[23] |
F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102.
doi: 10.1512/iumj.1980.29.29007. |
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