We show the nondegeneracy of positive bubble solutions for generalized
energy-critical Hartree equations (NLH) \begin{equation*}
-{\Delta u}\sts{x}
-{\bm\alpha}\sts{N,\lambda}
\int_{\R^N}
{ \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}{\lambda}} }\diff{y}\,
u^{p-1}\sts{x}
=0,\quad x\in \R^N \end{equation*} where u(x) is a real-valued function,
N≥3, 0<λ<N, p=N−22N−λ​ and
{\bm\alpha}\sts{N,\lambda} is a constant. It generalizes the results for the
whole range 0<λ<N in \cite{DY2019dcds, GWY2020na, LTX2021, MWX:Hartree}
and confirms an open nondegeneracy problem in \cite{GMYZ2022cvpde}.
Firstly, by the stereographic projection and sharp Hardy-Littlewood-Sobolev
inequality on the sphere §N in \cite{FL2012}, we give an alternative proof
of the existence of the extremizer of sharp Hardy-Littlewood-Sobolev inequality
in RN without use of the rearrangement inequalities in
\cite{lieb2001analysis}, which is related to
the existence of positive bubble solutions of (NLH). Secondly, by making use
of the Green function, we obtain an integral form in RN of the
corresponding linearized equation around positive bubble solutions under
suitable decay condition, and its equivalent integral form on the sphere §N
via the stereographic projection. Lastly, together with the key spherical
harmonic decomposition and the Funk-Hecke formula of the spherical harmonic
functions in \cite{AH2012, DX2013book, SteinW:Fourier anal}, we can obtain the
nondegeneracy of positive bubble solutions for generalized energy-critical
Hartree equation (NLH), which is inspired by Frank and Lieb in
\cite{FL2012am,FL2012}.Comment: 26 pages. Any comment is welcom