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

Abstract

In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{Δu}\sts{x} -{\bmα}\sts{N,λ} \int_{\R^N} { \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}λ} }\diff{y}\, u^{p-1}\sts{x} =0,\quad x\in \R^N \end{equation*} where $N\ge 3$, $0<\lambda <N$, $p=\frac{2N-\lambda }{N-2}$ and ${\bmα}\sts{N,λ}$ is a normalized constant such that $u(x)={\left(1+|x{|}^2\right)}^{-\frac{N-2}{2}}$ is a bubble solution of the equation \eqref{NLH}. It solves an open nondegeneracy problem in \cite{MWX:Hartree, GMYZ2022cvpde} and generalizes the partial nondegeneracy results in \cite{DY2019dcds, GWY2020na, LTX2021} to the full range $0<\lambda <N$. The key observation is that by use of the stereographic projection $\mathcal{S}$, the weighted pushforward map ${\mathcal{S}}_{\ast }$ is one-to-one map between the null space of the linearized operator and the spherical harmonic function subspace ${\mathcal{H}}_1^{N+1}$ of degree one.

17 pages. All comments are welcome