Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential

Abstract: In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, $$ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, $$ where $n \geq 1$, $0<s\<1$, $\omega>-\lambda_{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, $\lambda_{1,s}\>0$ is the lowest eigenvalue of $(-\Delta)^s + |x|^2$. The fractional Laplacian $(-\Delta)^s$ is characterized as $\mathcal{F}((-\Delta)^{s}u)(\xi)=|\xi|^{2s} \mathcal{F}(u)(\xi)$ for $\xi \in \R^n$, where $\mathcal{F}$ denotes the Fourier transform. This solves an open question in \cite{SS} concerning the uniqueness of ground states.