Nondegeneracy of solutions for a critical Hartree equation

Abstract

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 $I_{\mu }(x)=\frac{\Gamma (\frac{\mu }{2})}{\Gamma (\frac{N-\mu }{2}){\pi }^{\frac{N}{2}}{2}^{N-\mu }|x{|}^{\mu }}$ 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.

arXiv admin note: text overlap with arXiv:1810.11186