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December  2012, 7(4): 967-988. doi: 10.3934/nhm.2012.7.967

Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices

Philippe Souplet 1,

1. 

Université Paris 13, Sorbonne Paris Cité, Laboratoire Analyse, Géométrie et Applications, CNRS, UMR 7539, 93430 Villetaneuse, France

Received  September 2011 Published  December 2012

We study non-cooperative, multi-component elliptic Schrödinger systems arising in nonlinear optics and Bose-Einstein condensation phenomena. We here reconsider the more delicate case of systems of $m ≥ 3$ components. We prove a Liouville-type nonexistence theorem in space dimensions $n ≥3$ for homogeneous nonlinearities of degree $ p < n/(n-2)$, under the optimal assumption that the associated matrix is strictly copositive. This extends recent work of Tavares, Terracini, Verzini and Weth [22] and of Quittner and the author [17], where the results were limited to $n ≤ 2$ dimensions or to $m=2$ components. The proof of the Liouville theorem is done by combining and improving different arguments from [22] and [17], namely a feedback procedure based on Rellich-Pohozaev type identities and functional analytic inequalities on $S^{n-1}$, and suitable test-function arguments. We also consider a more general class of systems with gradient structure, for which our arguments show the triviality of solutions satisfying a suitable integral bound, a result which may be of independent interest.
Keywords: strictly copositive matrices., nonexistence, Nonlinear elliptic system, Schrödinger system, Liouville-type theorem.
Mathematics Subject Classification: Primary: 35J60, 35J47, 35B53; Secondary: 15B4.
Citation: Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Networks & Heterogeneous Media, 2012, 7 (4) : 967-988. doi: 10.3934/nhm.2012.7.967
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.  Google Scholar

[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.  Google Scholar

[3]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[10]

Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications, Physics Reports, 298 (1998), 81-197. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States," Birkhäuser Advanced Texts, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[10]

Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications, Physics Reports, 298 (1998), 81-197. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States," Birkhäuser Advanced Texts, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-653.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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