Abstract
We study positive solutions to the following higher order Schrödinger system with Dirichlet boundary conditions on a half space:
where α is any even number between 0 and n. This PDE system is closely related to the integral system
where G is the corresponding Green’s function on the half space. More precisely, we show that every solution to (0.2) satisfies (0.1), and we believe that the converse is also true. We establish a Liouville type theorem — the non-existence of positive solutions to (0.2) under a very weak condition that u and v are only locally integrable. Some new ideas are involved in the proof, which can be applied to a system of more equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Berestycki H, Nirenberg L. On the method of moving planes and the sliding method. Bol Soc Brazs Mat, 1991, 1: 1–37
Caffarelli L, Gidas B, Spruck J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm Pure Appl Math, 1989, XLII: 271–297
Chen D Z, Ma L. A Liouville type theorem for an integral system. Comm Pure Appl Anal, 2006, 5: 855–859
Chen W X, Li C M. A priori estimates for prescribing scalar curvature equations. Ann of Math, 1997, 145: 547–564
Chen W X, Li C M. Regularity of solutions for a system of integral equations. Comm Pure Appl Anal, 2005, 4: 1–8
Chen W X, Li C M. The best constant in a weighted Hardy-Littlewood-Sobolev inequality. Proc Amer Math Soc, 2008, 136: 955–962
Chen W X, Li C M. An integral system and the Lane-Emden conjecture. Discrete Contin Dyn Syst, 2009, 4: 1167–1184
Chen W X, Li C M. Methods on Nonlinear Elliptic Equations. New York: American Institute of Mathematical Sciences, 2010
Chen W X, Li C M, Ou B. Qualitative properties of solutions for an integral equation. Discrete Contin Dyn Syst, 2005, 12: 347–354
Chen W X, Li C M, Ou B. Classification of solutions for a system of integral equations. Comm PDEs, 2005, 30: 59–65
Chen W X, Li C M, Ou B. Classification of solutions for an integral equation. Comm Pure Appl Math, 2006, 59: 330–343
Chen W X, Zhu J Y. Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems. J Math Anal Appl, 2011, 377: 744–753
Fang Y Q, Chen W X. A Liouville type theorem for poly-harmonic Dirichlet problem in a half space. Adv Math, 2012, 229: 2835–2867
Fang Y Q, Zhang J H. Nonexistence of positive solution for an integral equation on a half-space R n+ . Comm Pure Appl Anal, 2013, 12: 663–678
Gidas B, Ni W M, Nirenberg L. Symmetry of Positive Solutions of Nonlinear Elliptic Equations in R n. New York: Academic Press, 1981
Jin C, Li C M. Symmetry of solutions to some systems of integral equations. Proc Amer Math Soc, 2006, 134: 1661–1670
Kanna T, Lakshmanan M. Exact soliton solutions, shape changing collisions and partially coherent solitons in coupled nonlinear Schröedinger equations. Phys Rev Lett, 2001, 86: 5043–5046
Li C M, Lim J. The singularity analysis of solutions to some integral equations. Comm Pure Appl Anal, 2007, 6: 453–464
Li C M, Ma L. Uniqueness of positive bound states to Schrödinger systems with critical exponents. SIAM J Math Anal, 2008, 40: 1049–1057
Li D Y, Zhuo R. An integral equation on half space. Proc Amer Math Soc, 2010, 138: 2779–2791
Lieb E. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann of Math, 1983, 118: 349–374
Lin T C, Wei J C. Spikes in two coupled nonlinear Schrödinger equations. Ann Inst H Poincaré Anal Non Linéaire, 2005, 22: 403–439
Ma C, Chen WX, Li C M. Regularity of solutions for an integral system ofWolff type. Adv Math, 2011, 226: 2676–2699
Ma L, Chen D Z. Radial symmetry and monotonicity for an integral equation. J Math Anal Appl, 2008, 342: 943–949
Ma L, Zhao L. Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system. J Math Phys, 2008, doi: 10.1063/1.2939238
Ou B. A remark on a singular integral equation. Houston J Math, 1999, 25: 181–184
Reichel W, Weth T. A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems. Math Z, 2009, 261: 805–827
Serrin J. A symmetry problem in potential theory. Arch Rational Mech Anal, 1971, 43: 304–318
Wei J C, Xu X W. Classification of solutions of higher order conformally invariant equations. Math Ann, 1999, 313: 207–228
Zhuo R, Li D Y. A system of integral equations on half space. J Math Anal Appl, 2011, 381: 392–401
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhuo, R., Li, F. Liouville type theorems for Schrödinger systems. Sci. China Math. 58, 179–196 (2015). https://doi.org/10.1007/s11425-014-4925-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-014-4925-9
Keywords
- Schrödinger systems
- poly-harmonic operators
- Dirichlet boundary conditions
- method of moving planes in integral forms
- Kelvin transforms
- monotonicity
- rotational symmetry
- non-existence