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On decay of solutions to nonlinear Schrödinger equations

Author: Alexander Pankov
Journal: Proc. Amer. Math. Soc. 136 (2008), 2565-2570
MSC (2000): Primary 35J60, 35B40
Published electronically: March 14, 2008
MathSciNet review: 2390527
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Abstract: We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. Under certain natural assumptions we show that any such solution is continuous and vanishes at infinity. This allows us to interpret the solution as a finite multiplicity eigenfunction of a certain linear Schrödinger operator and, hence, apply well-known results on the decay of eigenfunctions.

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  • 1. T. Bartsch, A. Pankov, Zh.-Q. Wang, Nonlinear Schrödinger equation with a steep potential well, Commun. Contemp. Math. 3 (2001), 549-569. MR 1869104 (2002k:35079)
  • 2. F. A. Berezin, M. A. Shubin, The Schrödinger Equation, Kluwer, Dordrecht, 1991. MR 1186643 (93i:81001)
  • 3. H. Berestycki, P.-L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rat. Mech. Anal. 82 (1983), 313-345. MR 695535 (84h:35054a)
  • 4. R. Fukuizumi, T. Ozawa, Exponential decay of solutions to nonlinear elliptic equations with potentials, Z. Angew. Math. Phys. 56 (2005), 1000-1011. MR 2187002 (2006i:35089)
  • 5. P. D. Hislop, Exponential decay of two-body eigenfunctions: A review, Math. Phys. Quant. Field Theory, Electr. J. Differ. Equat. Conf. 4, 2000, pp. 265-288. MR 1785381 (2001j:81247)
  • 6. P. D. Hislop, I. M. Sigal, Introduction to Spectral Theory, with Applications to Schrödinger Operators, Springer-Verlag, New York, 1996. MR 1361167 (98h:47003)
  • 7. V. Kondrat'ev, M. Shubin, Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry, Oper. Theory Adv. Appl., Vol. 110, Birkhäuser, Basel, 1999, pp. 185-226. MR 1747895 (2001c:58030)
  • 8. W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equat. 3 (1998), 441-472. MR 1751952 (2001g:58021)
  • 9. G. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math. 4 (2002), 763-776. MR 1938493 (2004b:35104)
  • 10. A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259-287. MR 2175045 (2006h:35087)
  • 11. A. Pankov, Lecture Notes on Schrödinger Equations, Nova Publ., 2007.
  • 12. P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270-291. MR 1162728 (93h:35194)
  • 13. M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978. MR 0493421 (58:12429c)
  • 14. B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447-526. MR 670130 (86b:81001a)
  • 15. C. A. Stuart, An introduction to elliptic equations on $ {\mathbb{R}}^n$, Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997, World Sci. Publ., River Edge, NJ, 1998, pp. 237-285. MR 1703533 (2000f:35025)
  • 16. M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. MR 1400007 (97h:58037)

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Additional Information

Alexander Pankov
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187–8795
Address at time of publication: Department of Mathematics, Morgan State University, Baltimore, Maryland 21251

Keywords: Nonlinear Schr\"odinger equation, exponential decay
Received by editor(s): September 18, 2006
Received by editor(s) in revised form: June 29, 2007
Published electronically: March 14, 2008
Communicated by: Michael Weinstein
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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