Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solutions of Schr\"odinger equations\\ with symmetry in orientation
  preserving tetrahedral group   We consider the nonlinear Schr\"odinger equation \begin{equation*} \Delta u =
\big( 1 +\ve 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.

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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: chao-dyn