The aim of this paper is to prove the nondegeneracy of the unique positive solutions for the following critical Hartree type equations when $\mu >0$ is close to 0, $-\Delta u=\left(I_{\mu }\ast u^{2_{\mu }^{\ast }}\right)u^{2_{\mu }^{\ast }-1},x\in {\mathbb{R}}^N,$ where $I_{\mu }\left(x\right)=\frac{\Gamma \left(\frac{\mu }{2}\right)}{\Gamma \left(\frac{N-\mu }{2}\right){\pi }^{\frac{N}{2}}{2}^{N-\mu }{\left|x\right|}^{\mu }}$ is the Riesz potential and $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}$ is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality.

中文翻译：

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临界Hartree方程解的非退化性

本文的目的是证明以下临界Hartree型方程唯一正解的非退化性 $\mu >0$ 接近0， $-\Delta ü=\left({\mathrm{一世}}_{\mu }\ast {ü}^{2_{\mu }^{\ast }}\right){ü}^{2_{\mu }^{\ast }-\mathrm{1个}}，X\in {\mathbb{[R}}^{ñ}，$ 哪里 ${\mathrm{一世}}_{\mu }（X）=\frac{\Gamma （\frac{\mu }{2}）}{\Gamma （\frac{ñ-\mu }{2}）{\pi }^{\frac{ñ}{2}}{2}^{ñ-\mu }{\left|X\right|}^{\mu }}$ 是里兹的潜力， $2_{\mu }^{\ast }=\frac{2ñ-\mu }{ñ-2}$ 是由Hardy–Littlewood–Sobolev不等式引起的最高临界指数。