论文标题
liouville方程的解决方案在维度2和4中具有非平凡性的解决方案
Solutions of Liouville equations with non-trivial profile in dimensions 2 and 4
论文作者
论文摘要
我们证明了liouville方程的非平凡解决方案的存在，其尺寸为二和四。这些解决方案是相同方程的有限体积解决方案的扰动。特别是，它们在一个变量中定期为周期性，并在其他变量中线性衰减至$ - \ infty $。在第二个维度中，我们还证明了该周期接近$πk，k \ in \ mathbb {n} $（从正面）。我们采用的主要工具是加权Hölder空间中的分叉理论。
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 $πk, k \in \mathbb{N}$ (from the positive side). The main tool we employ is bifurcation theory in weighted Hölder spaces.
