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.
Categories: math.AP