### Abstract

We study singularly perturbed ID nonlinear Schrödinger equations (1.1). When V(x) has multiple critical points, (1.1) has a wide variety of positive solutions for small ε and the number of positive solutions increases to ∞ as ε → 0. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of V(x). Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

Original language | English |
---|---|

Pages (from-to) | 253-268 |

Number of pages | 16 |

Journal | Journal of the European Mathematical Society |

Volume | 8 |

Issue number | 2 |

Publication status | Published - 2006 |

### Fingerprint

### Keywords

- Adiabatic profiles
- Nonlinear Schrödinger equations
- Singular perturbations

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the European Mathematical Society*,

*8*(2), 253-268.

**On the number of positive solutions of singularly perturbed ID nonlinear Schrödinger equations.** / Felmer, Patricio; Martínez, Salomé; Tanaka, Kazunaga.

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 8, no. 2, pp. 253-268.

}

TY - JOUR

T1 - On the number of positive solutions of singularly perturbed ID nonlinear Schrödinger equations

AU - Felmer, Patricio

AU - Martínez, Salomé

AU - Tanaka, Kazunaga

PY - 2006

Y1 - 2006

N2 - We study singularly perturbed ID nonlinear Schrödinger equations (1.1). When V(x) has multiple critical points, (1.1) has a wide variety of positive solutions for small ε and the number of positive solutions increases to ∞ as ε → 0. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of V(x). Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

AB - We study singularly perturbed ID nonlinear Schrödinger equations (1.1). When V(x) has multiple critical points, (1.1) has a wide variety of positive solutions for small ε and the number of positive solutions increases to ∞ as ε → 0. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of V(x). Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

KW - Adiabatic profiles

KW - Nonlinear Schrödinger equations

KW - Singular perturbations

UR - http://www.scopus.com/inward/record.url?scp=33745670373&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745670373&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33745670373

VL - 8

SP - 253

EP - 268

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 2

ER -