### Abstract

We consider a singularly perturbed elliptic equation ε ^{2}Δu - V (x)u + f(u) = 0, u(x) > 0 on ℝ^{N}, lim _{|x|→∞} u(x) = 0, where V (x) > 0 for any x ε ℝ^{N}: The singularly perturbed problem has corresponding limiting problems ΔU - cU + f(U) = 0, U(x) > 0 on ℝ^{N}, lim _{|x|→∞}U(x) = 0, c > 0: Berestycki-Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f ε C^{1} In this paper, we prove that under the optimal conditions of Berestycki-Lions on f 2 C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods.

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

Pages (from-to) | 1859-1899 |

Number of pages | 41 |

Journal | Journal of the European Mathematical Society |

Volume | 15 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential.** / Byeon, Jaeyoung; Tanaka, Kazunaga.

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 15, no. 5, pp. 1859-1899. https://doi.org/10.4171/JEMS/407

}

TY - JOUR

T1 - Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

AU - Byeon, Jaeyoung

AU - Tanaka, Kazunaga

PY - 2013

Y1 - 2013

N2 - We consider a singularly perturbed elliptic equation ε 2Δu - V (x)u + f(u) = 0, u(x) > 0 on ℝN, lim |x|→∞ u(x) = 0, where V (x) > 0 for any x ε ℝN: The singularly perturbed problem has corresponding limiting problems ΔU - cU + f(U) = 0, U(x) > 0 on ℝN, lim |x|→∞U(x) = 0, c > 0: Berestycki-Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f ε C1 In this paper, we prove that under the optimal conditions of Berestycki-Lions on f 2 C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods.

AB - We consider a singularly perturbed elliptic equation ε 2Δu - V (x)u + f(u) = 0, u(x) > 0 on ℝN, lim |x|→∞ u(x) = 0, where V (x) > 0 for any x ε ℝN: The singularly perturbed problem has corresponding limiting problems ΔU - cU + f(U) = 0, U(x) > 0 on ℝN, lim |x|→∞U(x) = 0, c > 0: Berestycki-Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f ε C1 In this paper, we prove that under the optimal conditions of Berestycki-Lions on f 2 C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods.

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U2 - 10.4171/JEMS/407

DO - 10.4171/JEMS/407

M3 - Article

VL - 15

SP - 1859

EP - 1899

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 5

ER -