论文标题
$ \ mathbb {r} $的非本地liouville方程的非唯一性和应用程序
Non-uniqueness for the nonlocal Liouville equation in $\mathbb{R}$ and applications
论文作者
论文摘要
我们构建了多个解决方案，用于非局部liouville方程\ begin {qore} \ label {eqk} \ tag {l}（-Δ）^{\ frac {\ frac {1} {2} {2}} u = k（x） \ end {equation}更准确地说，对于$ k（x）= 1+ \varepsilonκ（x）$的$ k $，带有$ \ varepsilon \ in（0,1）$ small small和$κ\ in C^{1，α}（1，α}（1，α}） $α> 0 $，我们证明存在多种解决方案，以从气泡中分叉\ eqref {eqk}。这些解决方案在半平面中提供了平面平面指标的示例，其边界上有规定的地球曲率$ k（x）$。此外，它们暗示存在Calogero-Moser衍生物NLS的多个基态孤子溶液。
We construct multiple solutions to the nonlocal Liouville equation \begin{equation} \label{eqk} \tag{L} (-Δ)^{\frac{1}{2}} u = K(x) e^u \quad \mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form $K(x) = 1+\varepsilon κ(x)$ with $\varepsilon \in (0,1)$ small and $κ\in C^{1,α}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ for some $α> 0$, we prove existence of multiple solutions to \eqref{eqk} bifurcating from the bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $K(x)$ on its boundary. Furthermore, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS.
