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

TL;DR

This paper proves the existence of nonradial solutions to a nonlinear Schr"odinger equation with tetrahedral symmetry, expanding understanding of pattern formation beyond radial solutions.

Contribution

It demonstrates for the first time the existence of solutions with tetrahedral symmetry in a nonlinear Schr"odinger equation, under specific potential conditions.

Findings

Existence of nonradial solutions with tetrahedral symmetry.

Solutions are found for potentials that are radial but asymptotically decreasing.

Expands the class of known symmetric solutions in nonlinear Schr"odinger equations.

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.