Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential

In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in},, \R^n, \end{align*} where $n\ge 1$, $0<s<1$, $\omega >-{\lambda }_{1,s}$, $2<p<\frac{2n}{(n-2s{)}^{+}}$, and ${\lambda }_{1,s}>0$ is the lowest eigenvalue of the operator $(-\mathrm{\Delta }{)}^s+\mathrm{\mid }x{\mathrm{\mid }}^2$. This solves an open question raised in \cite{SS} concerning the uniqueness of solutions to the equation.