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

### Juncheng Wei

University of British Columbia, Vancouver, Canada### Monica Musso

Ponificia Universidad Catolica de Chile, Santiago, Chile### Frank Pacard

École Polytechnique, Palaiseau, France

## Abstract

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $Δu−u+f(u)=0$ in $R_{N}$, $u∈H_{1}(R_{N})$, where $N≥2$. Under natural conditions on the nonlinearity $f$, we prove the existence of *infinitely many nonradial solutions* in any dimension $N≥2$. Our result complements earlier works of Bartsch and Willem ($N=4$ or $N≥6$) and Lorca-Ubilla ($N=5$) where solutions invariant under the action of $O(2)×O(N−2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_{k}×O(N−2)$ where $D_{k}⊂O(2)$ denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with $k$ sides invariant, for some integer $k≥7$, but they are not invariant under the action of $O(2)×O(N−2)$.

## Cite this article

Juncheng Wei, Monica Musso, Frank Pacard, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation. J. Eur. Math. Soc. 14 (2012), no. 6, pp. 1923–1953

DOI 10.4171/JEMS/351