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

Abstract

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