Title: Nondegeneracy of bubble solutions to the Choquard equation in two dimension Show Chinese title

Title: 双维Choquard方程的泡解的非退化性

Abstract: In this paper, we study the following Choquard equation with exponential nonlinearity \begin{equation*} -\Delta u=\left(\int_{\R^{2}}\frac{e^{u(y)}}{|x-y|^{\alpha}}dy\right)e^{u(x)},\quad \text{~in~}\R^{2}, \end{equation*} where $\alpha\in (0,2)$. Although the classification of solutions to this equation has been established recently, the nondegeneracy of its solutions remains open. Here, we prove the nondegeneracy by combining the integral representation of solutions with the spherical harmonic decomposition. The main result of this paper can be viewed as an extension of the nondegeneracy of solutions for both the planar Liouville equation and the higher-dimensional upper critical Choquard equation. Show Chinese abstract

Abstract: 在本文中，我们研究以下具有指数非线性的Choquard方程\begin{equation*} -\Delta u=\left(\int_{\R^{2}}\frac{e^{u(y)}}{|x-y|^{\alpha}}dy\right)e^{u(x)},\quad \text{~in~}\R^{2}, \end{equation*}，其中$\alpha\in (0,2)$。尽管该方程解的分类最近已被确立，但其解的非退化性仍然未被解决。在这里，我们通过结合解的积分表示与球谐分解来证明非退化性。本文的主要结果可以视为对平面Liouville方程和高维上临界Choquard方程解的非退化性的扩展。