Uniqueness and nondegeneracy for a critical Choquard equation with Hardy potential

We study the critical Choquard equation \[ -Δu-μ|x|⁻²u =(|·|2-N*|u|^p)|u|ᵖ⁻²u R^N, \] where |·|2-N is the Newton kernel, p=N+2/N-2, and 0<μ<(N-2)²/4. For every N≥3, we prove that positive solutions in D1,2( RN) are radially symmetric about the origin, unique up to dilation, and nondegenerate. Our proof uses normalized logarithmic derivatives to obtain a cooperative system. The resulting monotonicity principle underlies both the uniqueness and nondegeneracy arguments, and remains effective in the subquadratic range $1