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Solutions of Liouville equations with non-trivial profile in dimensions 2 and 4
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.jde.2020.10.011
Roberto Albesiano

We prove the existence of a family of non-trivial solutions of the Liouville equation in dimensions two and four with infinite volume. These solutions are perturbations of a finite-volume solution of the same equation in one dimension less. In particular, they are periodic in one variable and decay linearly to $-\infty$ in the other variables. In dimension two, we also prove that the periods are arbitrarily close to $\pi k, k \in \mathbb{N}$ (from the positive side). The main tool we employ is bifurcation theory in weighted H\"older spaces.

中文翻译:

在维度 2 和 4 中具有非平凡轮廓的 Liouville 方程的解

我们证明了具有无限体积的二维和四维 Liouville 方程的非平凡解族的存在。这些解是同一方程的有限体积解在一维上的微扰。特别是,它们在一个变量中是周期性的,而在其他变量中线性衰减到 $-\infty$。在维度二中,我们还证明了周期任意接近 $\pi k, k \in \mathbb{N}$(从积极的方面)。我们使用的主要工具是加权 H\" 旧空间中的分叉理论。
更新日期:2021-01-01
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