Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations

Abstract

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, $N\geq 3$, $0<\lambda<N$, $p=\frac{2N-\lambda}{N-2}$ and {\bm\alpha}\sts{N,\lambda} is a constant. It generalizes the results for the whole range $0<\lambda<N$ 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 $\S^N$ in \cite{FL2012}, we give an alternative proof of the existence of the extremizer of sharp Hardy-Littlewood-Sobolev inequality in $\R^N$ 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 $\R^N$ of the corresponding linearized equation around positive bubble solutions under suitable decay condition, and its equivalent integral form on the sphere $\S^N$ 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