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

TL;DR

This paper proves the nondegeneracy of positive bubble solutions for a class of energy-critical Hartree equations, resolving an open problem and extending previous partial results to the full parameter range.

Contribution

It establishes the nondegeneracy of solutions for generalized energy-critical Hartree equations across all relevant parameters, using stereographic projection techniques.

Findings

Confirmed nondegeneracy of bubble solutions for all 0<λ<N

Connected null space of linearized operator to spherical harmonics

Resolved open problem in the mathematical analysis of Hartree equations

Abstract

In this paper, 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 $N\geq 3$, $0<\lambda<N$, $p=\frac{2N-\lambda}{N-2}$ and ${\bm\alpha}\sts{N,\lambda}$ 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 $\mathcal{S}_*$ 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.