### Abstract

We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε^{2}u″ + V(x)u = |u|^{p-1}u in R, p > 1, when s is a small parameter. Given any finite set of points x_{1} < X_{2} < ⋯ < x_{m} constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n _{i}, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each x_{i} as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

Original language | English |
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Pages (from-to) | 1653-1671 |

Number of pages | 19 |

Journal | Nonlinearity |

Volume | 15 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2002 Sep 1 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics

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## Cite this

*Nonlinearity*,

*15*(5), 1653-1671. https://doi.org/10.1088/0951-7715/15/5/315