Local uniqueness of ground states for the generalized Choquard equation

We consider the generalized Choquard equation of the type \(\begin{aligned} -\Delta Q + Q = I(|Q|^p)|Q|^{p-2} Q, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }Q+Q={I(|Q|}^p{)|Q|}^{p-2}Q,\end{array}$$ for \(3\le n\le 5\) $3\le n\le 5$ , with \(Q \in H^1_{rad}(\mathbb {R}^n),\) $Q\in H_{\mathit{rad}}^1({\mathbb{R}}^n),$ where the operator I is the classical Riesz potential defined by \(I(f)(x) =(-\Delta )^{-1}f(x)\) $I(f)(x)={(-\mathrm{\Delta })}^{-1}f(x)$ and the exponent \(p \in (2,1+4/(n-2))\) $p\in (2,1+4/(n-2))$ is energy subcritical. We consider Weinstein-type functional restricted to rays passing through the ground state. The corresponding real valued function of the path parameter has an appropriate analytic extension. We use the properties of this analytic extension in order to show local uniqueness of ground state solutions. The uniqueness of the ground state solutions for the case \(p=2\) $p=2$ , i.e. for the case of Hartree–Choquard, is well known. The main difficulty for the case \(p > 2\) $p>2$ is connected with a possible lack of control on the \(L^p\) $L^p$ norm of the ground states as well on the lack of Sturm’s comparison argument.