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 and 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.

[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.

[1]

Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399

[2]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317

[3]

Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134

[4]

Yuxia Guo, Ting Liu. Liouville-type theorem for high order degenerate Lane-Emden system. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2073-2100. doi: 10.3934/dcds.2021184

[5]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855

[6]

Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035

[7]

Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887

[8]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure and Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807

[9]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067

[10]

Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236

[11]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869

[12]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947

[13]

Ran Zhuo, Fengquan Li, Boqiang Lv. Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space. Communications on Pure and Applied Analysis, 2014, 13 (3) : 977-990. doi: 10.3934/cpaa.2014.13.977

[14]

Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206

[15]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[16]

Congcong Li, Chunqiu Li, Jintao Wang. Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021311

[17]

Foued Mtiri. Liouville type theorems for stable solutions of elliptic system involving the Grushin operator. Communications on Pure and Applied Analysis, 2022, 21 (2) : 541-553. doi: 10.3934/cpaa.2021187

[18]

Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565

[19]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3821-3836. doi: 10.3934/dcdss.2020436

[20]

Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058

2020 Impact Factor: 1.213

Metrics

  • PDF downloads (104)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]