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arXiv · 2002.09920

Solutions of Liouville equations with non-trivial profile in dimensions 2 and 4

Roberto Albesiano
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Abstract

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.

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Roberto Albesiano. 2020-02-23. Solutions of Liouville equations with non-trivial profile in dimensions 2 and 4. https://doi.org/10.1016/j.jde.2020.10.011

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