Nondegeneracy of the positive solutions for critical nonlinear Hartree equation in $\R^6$

TL;DR

This paper proves the nondegeneracy of positive solutions to a critical nonlinear Hartree equation in six dimensions, employing spherical harmonics decomposition and Perron-Frobenius theory to analyze the linearized operator.

Contribution

It introduces a novel approach by decomposing the linear operator into one-dimensional components and analyzing their kernels to establish nondegeneracy.

Findings

The linear operator can be decomposed into a series of one-dimensional operators.

The kernel of each one-dimensional operator is finite.

The kernel of the full operator is the direct sum of the kernels of the one-dimensional operators.

Abstract

We prove that any positive solution for the critical nonlinear Hartree equation $$-\Laplacian\fct{u}{x} -\int_{\R^6} \frac{\abs{\fct{u}{y}}^2 }{ \abs{x-y}^4 }\odif{y} \,\fct{u}{x}=0,\qtq{} x\in\R^6.$$ 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.