Uniqueness of positive radial ground states for the massive pseudo-relativistic nonlinear Schrödinger equation

We study the semilinear pseudo-relativistic equation \[ \sqrt{-Δ+m^{2}}\,Q+ωQ=Q^{p-1}\qquad\text{in }\R^{d}, \] with $m>0$, $ω>0$, $d\ge1$ and $2<p<2^{*}:=\frac{2d}{d-1}$ (with $2^{*}=\infty$ when $d=1$). We prove that the positive radial variational ground state is unique. The proof rests on a mass