Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Musso, Monica, Pacard, Frank, and Wei, Juncheng. "Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation." Journal of the European Mathematical Society 014.6 (2012): 1923-1953. <http://eudml.org/doc/277632>.

@article{Musso2012,
abstract = {We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f(u)=0$ in $\mathbb \{R\}^N$, $u\in H^1(\mathbb \{R\}^N)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\textit \{infinitely many nonradial solutions\}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4 \left.\texttt \{or\} \right.N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O(2)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k\times O(N-2)$ where $D_k\subset O(2)$ denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with $k$ sides invariant, for some integer $k\ge 7$, but they are not invariant under the action of $O(2)\times O(N-2)$.},
author = {Musso, Monica, Pacard, Frank, Wei, Juncheng},
journal = {Journal of the European Mathematical Society},
keywords = {nonradial bound states; nonlinear Schrödinger equations; balancing condition; Lyapunov-Schmidt reduction method; nonradial bound states; nonlinear Schrödinger equation; balancing condition; Lyapunov-Schmidt reduction method},
language = {eng},
number = {6},
pages = {1923-1953},
publisher = {European Mathematical Society Publishing House},
title = {Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation},
url = {http://eudml.org/doc/277632},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Musso, Monica
AU - Pacard, Frank
AU - Wei, Juncheng
TI - Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 6
SP - 1923
EP - 1953
AB - We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f(u)=0$ in $\mathbb {R}^N$, $u\in H^1(\mathbb {R}^N)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\textit {infinitely many nonradial solutions}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4 \left.\texttt {or} \right.N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O(2)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k\times O(N-2)$ where $D_k\subset O(2)$ denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with $k$ sides invariant, for some integer $k\ge 7$, but they are not invariant under the action of $O(2)\times O(N-2)$.
LA - eng
KW - nonradial bound states; nonlinear Schrödinger equations; balancing condition; Lyapunov-Schmidt reduction method; nonradial bound states; nonlinear Schrödinger equation; balancing condition; Lyapunov-Schmidt reduction method
UR - http://eudml.org/doc/277632
ER -