# American Institute of Mathematical Sciences

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

## 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

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.
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, 342 (2006), 453. 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. 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. Google Scholar [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. 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. 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. 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. Google Scholar [8] D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: From theory to experiments,, J. Phys. A: Math. Theor., 43 (2010). 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, 133 (1977). Google Scholar [10] Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications,, Physics Reports, 298 (1998), 81. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1512/iumj.1980.29.29007. Google Scholar

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##### References:
 [1] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, C. R. Acad. Sci. Paris, 342 (2006), 453. 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. 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. Google Scholar [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. 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. 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. 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. Google Scholar [8] D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: From theory to experiments,, J. Phys. A: Math. Theor., 43 (2010). 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, 133 (1977). Google Scholar [10] Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications,, Physics Reports, 298 (1998), 81. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1512/iumj.1980.29.29007. Google Scholar
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