  In this paper, we consider the following problem involving fractional
Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u=
|u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0
\,\, {\rm on}\, \, \partial\Omega, \end{equation} where $\Omega$ is a smooth
bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_\alpha-2)$,
$0<\alpha<1,\, 2^*_\alpha = \frac {2N}{N-2\alpha}$. We show that for any
sequence of solutions $u_n$ of \eqref{eq:0.1} corresponding to
$\varepsilon_n\in [0, 2^*_\alpha-2)$, satisfying $\|u_n\|_{H}\le C$ in the
Sobolev space $H$ defined in \eqref{eq:1.1a}, $u_n$ converges strongly in $H$
provided that $N>6\alpha$ and $\lambda>0$. An application of this compactness
result is that problem \eqref{eq:0.1} possesses infinitely many solutions under
the same assumptions.
