Geodesic
Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation
Journal of the European Mathematical Society, Tome 14 (2012) no. 6, pp. 1923-1953.

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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 RN, u∈H1(RN), 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 Dk​×O(N−2) where Dk​⊂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).
DOI : 10.4171/jems/351
Classification : 35-XX, 00-XX
Keywords: Nonradial bound states, nonlinear Schrödinger equations, balancing condition, Lyapunov–Schmidt reduction method 
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     author = {Monica Musso and Frank Pacard and Juncheng Wei},
     title = {Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear {Schr\"odinger} equation},
     journal = {Journal of the European Mathematical Society},
     pages = {1923--1953},
     publisher = {mathdoc},
     volume = {14},
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     year = {2012},
     doi = {10.4171/jems/351},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/351/}
}
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Monica Musso; Frank Pacard; Juncheng Wei. Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation. Journal of the European Mathematical Society, Tome 14 (2012) no. 6, pp. 1923-1953. doi : 10.4171/jems/351. http://geodesic.mathdoc.fr/articles/10.4171/jems/351/

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