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

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 [1] (N = 4 or N ≥ 6) and Lorca and Ubilla [13] (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 D _k ⊂ O(2) denotes the dihedral group of rotations and reflections 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).