Nondegeneracy of the positive solutions for critical nonlinear Hartree equation in ℝ⁶

Abstract

We prove that any positive solution for the critical nonlinear Hartree equation − △ ⁡ u ( x ) − ∫ R 6 | u ( y ) | 2 | x − y | 4 d y u ( x ) = 0 , x ∈ R 6 , \begin{equation*} -\operatorname {\bigtriangleup }{u}\left (x\right ) -\int _{\mathbb {R}^6} \frac {\left |{u}\left (y\right )\right |^2 }{ \left |x-y\right |^4 }\mathrm {d}y \,{u}\left (x\right )=0,\quad \quad x\in \mathbb {R}^6, \end{equation*} is nondegenerate. Firstly, in terms of spherical harmonics, we show that the corresponding linear operator can be decomposed into a series of one dimensional linear operators. Secondly, by making use of the Perron-Frobenius property, we show that the kernel of each one dimensional linear operator is finite. Finally, we show that the kernel of the corresponding linear operator is the direct sum of the kernel of all one dimensional linear operators.