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

Jaeyoung Byeon; Kazunaga Tanaka

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 5, page 1859-1899
- ISSN: 1435-9855

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topByeon, Jaeyoung, and Tanaka, Kazunaga. "Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential." Journal of the European Mathematical Society 015.5 (2013): 1859-1899. <http://eudml.org/doc/277634>.

@article{Byeon2013,

abstract = {We consider a singularly perturbed elliptic equation $\epsilon ^2 \Delta u-V(x)u+f(u)=0, u(x)>0$ on $\mathbb \{R\}^N$, $\texttt \{lim\}_\{\left|x\right|\rightarrow \infty \} u(x)=0$, where $V(x)>0$ for any $x\in \mathbb \{R\}^N$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f(U)=0, U(x)>0$ on $\mathbb \{R\}^N$, $\texttt \{lim\}_\{\left|x\right|\rightarrow \infty \} U(x)=0, c>0$. Berestycki-Lions found almost necessary and sufficient conditions on 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 potential $V$ under possibly general conditions on $f$. In this paper, we prove that under the optimal conditions of Berestycki-Lions on $f\in C^1$, there exists a solution concentrating around topologically stable positive critical points of $V$, whose critical values are characterized by minimax methods.},

author = {Byeon, Jaeyoung, Tanaka, Kazunaga},

journal = {Journal of the European Mathematical Society},

keywords = {variational method; critical points; nonlInear Schrödinger equations; Berestycki–Lions conditions; gradient flows; linking; variational method; critical points; nonlinear Schrödinger equations; Berestycki-Lions conditions; gradient flows; linking},

language = {eng},

number = {5},

pages = {1859-1899},

publisher = {European Mathematical Society Publishing House},

title = {Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential},

url = {http://eudml.org/doc/277634},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Byeon, Jaeyoung

AU - Tanaka, Kazunaga

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

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 5

SP - 1859

EP - 1899

AB - We consider a singularly perturbed elliptic equation $\epsilon ^2 \Delta u-V(x)u+f(u)=0, u(x)>0$ on $\mathbb {R}^N$, $\texttt {lim}_{\left|x\right|\rightarrow \infty } u(x)=0$, where $V(x)>0$ for any $x\in \mathbb {R}^N$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f(U)=0, U(x)>0$ on $\mathbb {R}^N$, $\texttt {lim}_{\left|x\right|\rightarrow \infty } U(x)=0, c>0$. Berestycki-Lions found almost necessary and sufficient conditions on 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 potential $V$ under possibly general conditions on $f$. In this paper, we prove that under the optimal conditions of Berestycki-Lions on $f\in C^1$, there exists a solution concentrating around topologically stable positive critical points of $V$, whose critical values are characterized by minimax methods.

LA - eng

KW - variational method; critical points; nonlInear Schrödinger equations; Berestycki–Lions conditions; gradient flows; linking; variational method; critical points; nonlinear Schrödinger equations; Berestycki-Lions conditions; gradient flows; linking

UR - http://eudml.org/doc/277634

ER -

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