JournalsjemsVol. 14, No. 6pp. 1923–1953

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

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δuu+f(u)=0\Delta u - u + f(u) =0 in RN\mathbb R^N, uH1(RN)u \in H^1 (\mathbb R^N), where N2N\geq 2. Under natural conditions on the nonlinearity ff, we prove the existence of infinitely many nonradial solutions in any dimension N2N \geq 2. Our result complements earlier works of Bartsch and Willem (N=4N=4 or N6N \geq 6) and Lorca-Ubilla (N=5N=5) where solutions invariant under the action of O(2)×O(N2)O(2) \times O(N-2) are constructed. In contrast, the solutions we construct are invariant under the action of Dk×O(N2)D_k \times O(N-2) where DkO(2)D_k \subset O(2) denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with kk sides invariant, for some integer k7k\geq 7, but they are not invariant under the action of O(2)×O(N2)O(2) \times O(N-2).

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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