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In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential,\[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \]where $n \geq 1$
, $0< s<1$
, $\omega >-\lambda _{1,s}$
, $2< p< {2n}/{(n-2s)^+}$
, $\lambda _{1,s}>0$
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 $\xi \in \mathbb {R}^n$
, where $\mathcal {F}$
denotes the Fourier transform. This solves an open question in [M. Stanislavova and A. G. Stefanov. J. Evol. Equ. 21 (2021), 671–697.] concerning the uniqueness of ground states.
In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations
where 
\begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*}
$0<\alpha <1$, 
$\Omega = \mathbb {R}^{N}$ or 
$\Omega$ is a smooth bounded domain, 
$\Gamma$ is a singular subset of 
$\Omega$ with fractional capacity zero, 
$f(t)$ is locally bounded and positive for 
$t\in [0,\,\infty )$, and 
$f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in 
$t$ for large 
$t$, rather than for every 
$t>0$. Our main result is that the solutions satisfy the estimate
This estimate is new even for 
\begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*}
$\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.
