论文标题
$ \ mathbb {r} $中非本地liouville方程的唯一性
Uniqueness for the Nonlocal Liouville Equation in $\mathbb{R}$
论文作者
论文摘要
我们证明了非局部liouville方程$$（-Δ）^{1/2} w = k e^w \ quad \ mbox {in $ \ mathbb {r} $} $}的独特性。在这里，规定的$ q $ - curvaturature函数$ k = k（| x |）> 0 $被认为是一个正面的，对称降低的函数，满足适当的规律性和衰减界限。特别是，我们在高斯案例中获得了$ k（x）= \ exp（-x^2）$的独特性。我们的独特性证明利用了非局部liouville方程与Calogero-Moser衍生物NLS的基态孤子的联系，这是P.Gérard和第二作者最近研究的完全可集成的PDE。
We prove uniqueness of solutions for the nonlocal Liouville equation $$ (-Δ)^{1/2} w = K e^w \quad \mbox{in $\mathbb{R}$} $$ with finite total $Q$-curvature $\int_{\mathbb{R}} K e^w \, dx< +\infty$. Here the prescribed $Q$-curvature function $K=K(|x|) > 0$ is assumed to be a positive, symmetric-decreasing function satisfying suitable regularity and decay bounds. In particular, we obtain uniqueness of solutions in the Gaussian case with $K(x) = \exp(-x^2)$. Our uniqueness proof exploits a connection of the nonlocal Liouville equation to ground state solitons for Calogero--Moser derivative NLS, which is a completely integrable PDE recently studied by P. Gérard and the second author.
