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

Monica Musso; Frank Pacard; Juncheng Wei

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 6, page 1923-1953
- ISSN: 1435-9855

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topMusso, 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 -

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