Solutions of Schr\"odinger equations with symmetry in orientation preserving tetrahedral group

Abstract

We consider the nonlinear Schr\"odinger equation \begin{equation*} \Delta u = \big( 1 +\varepsilon V_1(|y|)\big)u - |u|^{p-1}u \quad \text{in} \quad \mathbb{R}^N, \quad N\ge 3, \quad p \in \left(1, \frac{N+2}{N-2}\right).\end{equation*} The phenomenon of pattern formation has been a central theme in the study of nonlinear Schr\"odinger equations. However, the following nonexistence of $O(N)$ symmetry breaking solution is well-known: if the potential function is radial and radially nondecreasing, any positive solution must be radial. Therefore, solutions of interesting patterns, such as those with symmetry in a discrete subgroup of $O(N)$, can only exist after violating the assumptions. For a potential function that is radial but asymptotically decreasing, a solution with symmetry merely in a discrete subgroup of $O(2)$ has been presented. These observations pose the question of whether patterns of higher dimensions can appear. In this study, the existence of nonradial solutions whose symmetry group is a discrete subgroup of $O(3)$, more precisely, the orientation-preserving regular tetrahedral group is shown.