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

Byeon, 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 -