Uniqueness and nondegeneracy of positive solutions for a critical Choquard-Kirchhoff equation

In this paper, we investigate positive solutions of the critical Choquard–Kirchhoff equation :[Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] is the Hardy–Littlewood–Sobolev critical exponent. We first classify positive solutions by parameter dependence: (i) unique solution for [Formula: see text] and all [Formula: see text]; (ii) unique solution for [Formula: see text] if and only if [Formula: see text] (with [Formula: see text] the ground state of the non-Kirchhoff critical Choquard equation); (iii) two distinct solutions for [Formula: see text] (single solution at critical [Formula: see text]). We then analyze asymptotic behavior of solutions as [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text], we prove nondegeneracy of positive solutions via spherical harmonics expansion on [Formula: see text], decomposing the linearized operator into radial differential operators. Combining Perron–Frobenius property and spherical harmonic orthogonality, we show the linearized operator’s kernel is spanned by scaling ([Formula: see text]) and translation ([Formula: see text]) perturbations of positive solutions.