### Abstract

We study the nonlinear Schrödinger equations: (P_{λ}) -Δu+(λ^{2}a(x)+1)u = |u|^{p-1}u, u ε H ^{1}(R^{N}), where p > 1 is a subcritical exponent, a(x) is a continuous function satisfying a(x) ≥ 0, 0 < lim inf _{|x|-∞} a(x) ≤ lim sup_{|x|-∞} a(x) < ∞ and a-^{1}(0) consists of 2 connected bounded smooth components Ω1 and Ω2. We study the existence of solutions (uλ) of (Pλ) which converge to 0 in R^{N} \ (Ω_{1}Ω _{2}) and to a prescribed pair (v_{1}(x), v_{2}(x)) of solutions of the limit problem: -Δv_{i} + v_{i} = |v _{i}|^{p-1}v_{i} in Ω_{i} (i = 1, 2) as λ → ∞.

Original language | English |
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Pages (from-to) | 6205-6253 |

Number of pages | 49 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2009 Dec |

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### Keywords

- Critical frequency
- Nonlinear schrödinger equations
- Sign-changing solutions
- Singular perturbation

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Sign-Changing multi-bump solutions for nonlinear schrödinger equations with steep potential wells.** / Sato, Yohei; Tanaka, Kazunaga.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 361, no. 12, pp. 6205-6253. https://doi.org/10.1090/S0002-9947-09-04565-6

}

TY - JOUR

T1 - Sign-Changing multi-bump solutions for nonlinear schrödinger equations with steep potential wells

AU - Sato, Yohei

AU - Tanaka, Kazunaga

PY - 2009/12

Y1 - 2009/12

N2 - We study the nonlinear Schrödinger equations: (Pλ) -Δu+(λ2a(x)+1)u = |u|p-1u, u ε H 1(RN), where p > 1 is a subcritical exponent, a(x) is a continuous function satisfying a(x) ≥ 0, 0 < lim inf |x|-∞ a(x) ≤ lim sup|x|-∞ a(x) < ∞ and a-1(0) consists of 2 connected bounded smooth components Ω1 and Ω2. We study the existence of solutions (uλ) of (Pλ) which converge to 0 in RN \ (Ω1Ω 2) and to a prescribed pair (v1(x), v2(x)) of solutions of the limit problem: -Δvi + vi = |v i|p-1vi in Ωi (i = 1, 2) as λ → ∞.

AB - We study the nonlinear Schrödinger equations: (Pλ) -Δu+(λ2a(x)+1)u = |u|p-1u, u ε H 1(RN), where p > 1 is a subcritical exponent, a(x) is a continuous function satisfying a(x) ≥ 0, 0 < lim inf |x|-∞ a(x) ≤ lim sup|x|-∞ a(x) < ∞ and a-1(0) consists of 2 connected bounded smooth components Ω1 and Ω2. We study the existence of solutions (uλ) of (Pλ) which converge to 0 in RN \ (Ω1Ω 2) and to a prescribed pair (v1(x), v2(x)) of solutions of the limit problem: -Δvi + vi = |v i|p-1vi in Ωi (i = 1, 2) as λ → ∞.

KW - Critical frequency

KW - Nonlinear schrödinger equations

KW - Sign-changing solutions

KW - Singular perturbation

UR - http://www.scopus.com/inward/record.url?scp=77950659711&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950659711&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-09-04565-6

DO - 10.1090/S0002-9947-09-04565-6

M3 - Article

VL - 361

SP - 6205

EP - 6253

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 12

ER -