论文标题
Schrödinger方程的解决方案具有对称性的定向，以保存四面体组
Solutions of Schrödinger equations with symmetry in orientation preserving tetrahedral group
论文作者
论文摘要
我们考虑非线性schrödinger方程\ begin {equation*}Δu= \ big（1 +\ varepsilon v_1（| y |）\ big） \ frac {n+2} {n-2} \ right）。\ end {equation*}模式形成的现象一直是非线性schrödinger方程的研究中的一个核心主题。但是，以下$ o（n）$对称性破坏解决方案的不存在是众所周知的：如果潜在函数是径向且径向不足的，则任何阳性解决方案都必须是径向。因此，仅在违反假设后才能存在有趣的模式的解决方案，例如$ O（n）$的离散子组中的对称性的解决方案。对于径向但渐近下降的潜在函数，已经提出了一种具有对称性的解决方案仅在$ O（2）$的离散亚组中。这些观察结果提出了一个问题，即是否会出现较高维度的模式。在这项研究中，其对称群的存在是$ O（3）$的离散亚组的存在，更确切地说是显示了定向定向的常规四面体组。
We consider the nonlinear Schrödinger equation \begin{equation*} Δu = \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 Schrödinger 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.
