Uniqueness and nondegeneracy for a critical Choquard equation with Hardy potential

Abstract

We study the critical Choquard equation \[ -Δu-μ|x|^{-2}u =\bigl(|\cdot|^{2-N}*|u|^p\bigr)|u|^{p-2}u \qquad\text{in }\mathbb R^N, \] where $|\cdot|^{2-N}$ is the Newton kernel, $p=\frac{N+2}{N-2}$, and $0<μ<\frac{(N-2)^2}{4}$. For every $N\ge3$, we prove that positive solutions in $D^{1,2}(\mathbb R^N)$ 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 $1<p<2$, corresponding to $N\ge7$. Consequently, the linearized quadratic form has Morse index one and the sharp inequality admits a global quadratic remainder estimate. The best constant is strictly concave in the Hardy parameter on $\bigl[0,\frac{(N-2)^2}{4}\bigr)$, is of class $C^2$ with strictly negative second derivative in the interior, and satisfies two-sided bounds that agree to first order at $μ=0$.