Optimal Liouville-type Theorems for Noncooperative Elliptic Schrödinger Systems and Applications

Communications in Mathematical Physics

Volumn 311, Issue 1, 2012, Pages 1-19

  Quittner, Pavol ^a   Souplet, Philippe ^b  

^a Physics and Informatics   (Slovakia)

^b UNIVERSITÉ PARIS 13   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84857351089     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-012-1440-0     Document Type: Article

References (33)

1. Ambrosetti A., Colorado E.: Bound and ground states of coupled nonlinear Schrödinger equations. C. R. Acad. Sci. Paris, Sér. I 342, 453-458 (2006).
2. Bartsch Th., Dancer N., Wang Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Par. Diff. Eqs. 37, 345-361 (2010).
3. Bidaut-Véron M.-F.: Local behaviour of the solutions of a class of nonlinear elliptic systems. Adv. Diff. Eqs. 5, 147-192 (2000).
4. Bidaut-Véron M.-F., Giacomini H.: A new dynamical approach of Emden-Fowler equations and systems. Adv. Diff. Eqs. 15, 1033-1082 (2010).
5. Bidaut-Véron M.-F., Raoux Th.: Asymptotics of solutions of some nonlinear elliptic systems. Comm. Part. Diff. Eqs. 21, 1035-1086 (1996).
6. Busca J., Manásevich R.: A Liouville-type theorem for Lane-Emden system. Indiana Univ. Math. J. 51, 37-51 (2002).
7. Chang J., Liu Z.: Ground states of nonlinear Schrödinger systems. Proc. Amer. Math. Soc. 138, 687-693 (2010).
8. Chen J., Guo B.: Blow-up profile to the solutions of two-coupled Schrödinger equations. J. Math. Phys. 50, 023505 (2009).
9. Chen W., Li C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615-622 (1991).
10. Dancer N., Wei J.-C., Weth T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré Anal. Non-Linéaire 27, 953-969 (2010).
11. Dancer, N., Weth, T.: Liouville type results for noncooperative elliptic systems in a half space. J. London. Math. Soc (to appear).
12. de Figueiredo D. G., Felmer P.: A Liouville-type theorem for elliptic systems. Ann. Sc. Norm. Super. Pisa Cl. Sci. 21(4), 387-397 (1994).
13. Frantzeskakis D. J.: Dark solitons in atomic Bose-Einstein condensates: from theory to experiments. J. Phys. A: Math. Theor. 43, 213001 (2010).
14. Gidas, B.: Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations. In: Nonlinear partial differential equations in engineering and applied science. Lecture Notes in Pure and Appl. Math. 54, New York: Dekker, 1980, pp. 255-273.
15. Gidas B., Spruck J.: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Part. Diff. Eqs. 6, 883-901 (1981).
16. Hirano N.: Multiple existence of nonradial positive solutions for a coupled nonlinear Schrödinger system. NoDEA Nonlin. Diff. Eqs. Appl. 16, 159-188 (2009).
17. Kivshar Yu. S., Luther-Davies B.: Dark optical solitons: physics and applications. Phys. Rep. 298, 81-197 (1998).
18. n, n ≤ 3. Commun. Math. Phys. 255, 629-653 (2005).
19. Liu Z., Wang Z.-Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282, 721-731 (2008).
20. Liu Z., Wang Z-Q.: Ground states and bound states of a nonlinear Schrödinger system. Adv. Nonlinear Stud. 10, 175-193 (2010).
21. 2-critical nonlinear Schrödinger system in the plane. Milan J. Math. 78(2), 591-601 (2010).
22. Ma L., Zhao L.: Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system J. Math. Phys. 49, 062103 (2008).
23. Maia L. A., Montefusco E., Pellacci B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Diff. Eqs. 229, 743-767 (2006).
24. Poláčik P., Quittner P., Souplet Ph.: Singularity and decay estimates in superlinear prooblems via Liouville-type theorems, I: elliptic equations and systems. Duke Math. J. 139, 555-579 (2007).
25. Pucci P., Serrin J.: A general variational identity. Indiana Univ. Math. J. 35, 681-703 (1986).
26. Quittner, P., Souplet, Ph.: Superlinear parabolic problems. Blow-up, global existence and steady states. Birkhäuser Advanced Texts, Basel-Boston: Birkhäuser, 2007.
27. Reichel W., Zou H.: Non-existence results for semilinear cooperative elliptic systems via moving spheres. J. Diff. Eqs. 161, 219-243 (2000).
28. Serrin J., Zou H.: Non-existence of positive solutions of Lane-Emden systems. Diff. Int. Eqs. 9, 635-653 (1996).
29. n. Commun. Math. Phys. 271, 199-221 (2007).
30. Souplet Ph.: The proof of the Lane-Emden conjecture in four space dimensions. Adv. Math. 221, 1409-1427 (2009).
31. Souto M. A. S.: A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems. Diff. Int. Eqs. 8, 1245-1258 (1995).
32. Tavares H., Terracini S., Verzini G.-M., Weth T.: Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems. Comm. Part. Diff. Eqs. 36(11), 1988-2010 (2011).
33. Wei J.-C., Weth T.: Nonradial symmetric bound states for a system of coupled Schrödinger equations. Rend. Lincei Mat. Appl. 18, 279-294 (2007).