Ground States of Nonlinear Schrödinger Equations with potentials vanishing at infinity

Antonio Ambrosetti; Veronica Felli; Andrea Malchiodi

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 2, page 81-86
  • ISSN: 1120-6330

Abstract

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In this preliminary Note we outline the results of the forthcoming paper [2] dealing with a class on nonlinear Schrödinger equations with potentials vanishing at infinity. Working in weighted Sobolev spaces, the existence of a ground state is proved. Furthermore, the behaviour of such a solution, as the Planck constant tends to zero (semiclassical limit), is studied proving that it concentrates at a point.

How to cite

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Ambrosetti, Antonio, Felli, Veronica, and Malchiodi, Andrea. "Ground States of Nonlinear Schrödinger Equations with potentials vanishing at infinity." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.2 (2004): 81-86. <http://eudml.org/doc/252395>.

@article{Ambrosetti2004,
abstract = {In this preliminary Note we outline the results of the forthcoming paper [2] dealing with a class on nonlinear Schrödinger equations with potentials vanishing at infinity. Working in weighted Sobolev spaces, the existence of a ground state is proved. Furthermore, the behaviour of such a solution, as the Planck constant tends to zero (semiclassical limit), is studied proving that it concentrates at a point.},
author = {Ambrosetti, Antonio, Felli, Veronica, Malchiodi, Andrea},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Nonlinear Schrödinger equations; Weighted Sobolev spaces; Critical point theory; nonlinear Schrödinger equations; weighted Sobolev spaces; critical point theory},
language = {eng},
month = {6},
number = {2},
pages = {81-86},
publisher = {Accademia Nazionale dei Lincei},
title = {Ground States of Nonlinear Schrödinger Equations with potentials vanishing at infinity},
url = {http://eudml.org/doc/252395},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Ambrosetti, Antonio
AU - Felli, Veronica
AU - Malchiodi, Andrea
TI - Ground States of Nonlinear Schrödinger Equations with potentials vanishing at infinity
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/6//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 2
SP - 81
EP - 86
AB - In this preliminary Note we outline the results of the forthcoming paper [2] dealing with a class on nonlinear Schrödinger equations with potentials vanishing at infinity. Working in weighted Sobolev spaces, the existence of a ground state is proved. Furthermore, the behaviour of such a solution, as the Planck constant tends to zero (semiclassical limit), is studied proving that it concentrates at a point.
LA - eng
KW - Nonlinear Schrödinger equations; Weighted Sobolev spaces; Critical point theory; nonlinear Schrödinger equations; weighted Sobolev spaces; critical point theory
UR - http://eudml.org/doc/252395
ER -

References

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  1. AMBROSETTI, A. - BADIALE, M., Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann. Inst. H. Poincaré Analyse Non Linéaire, 15, 1998, 233-252. Zbl1004.37043MR1614571DOI10.1016/S0294-1449(97)89300-6
  2. AMBROSETTI, A. - FELLI, V. - MALCHIODI, A., Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. Preprint S.I.S.S.A., n. 16/2004/M, Febr. 2004; to appear on Journal Europ. Math. Soc. Zbl1064.35175MR2148536
  3. AMBROSETTI, A. - MALCHIODI, A. - SECCHI, S., Multiplicity results for some nonlinear singularly perturbed elliptic problems on R n . Arch. Rat. Mech. Anal., 159, 2001, 253-271. Zbl1040.35107MR1857674DOI10.1007/s002050100152
  4. LI, Y.Y., On a singularly perturbed elliptic equation. Adv. Diff. Eqns., 2, 1997, 955-980. Zbl1023.35500MR1606351
  5. NOUSSAIR, E.S. - SWANSON, C.A., Decaying solutions of semilinear elliptic equations in R N . SIAM J. Math. Anal., 20, 6, 1989, 1336-1343. Zbl0696.35051MR1019304DOI10.1137/0520088
  6. OPIC, B. - KUFNER, A., Hardy-type inequalities. Pitman Res. Notes in Math. Series219, Longman Scientific & Technical, Harlow1990. Zbl0698.26007MR1069756
  7. SCHNEIDER, M., Entire solutions of semilinear elliptic problems with indefinite nonlinearities. PhD Thesis, Universität Mainz, 2001. Zbl1134.35347

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