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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 hasbeen a central theme in the study of nonlinear Schr\"odinger equations.However, the following nonexistence of $O(N)$ symmetry breaking solution iswell-known: if the potential function is radial and radially nondecreasing, anypositive solution must be radial. Therefore, solutions of interesting patterns,such as those with symmetry in a discrete subgroup of $O(N)$, can only existafter violating the assumptions. For a potential function that is radial butasymptotically decreasing, a solution with symmetry merely in a discretesubgroup of $O(2)$ has been presented. These observations pose the question ofwhether patterns of higher dimensions can appear. In this study, the existenceof nonradial solutions whose symmetry group is a discrete subgroup of $O(3)$,more precisely, the orientation-preserving regular tetrahedral group is shown.

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