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