  In this paper, we study the existence and uniqueness of positive solutions
for the following nonlinear fractional elliptic equation: \begin{eqnarray*}
(-\Delta)^\alpha u=\lambda a(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*}
where $ \alpha\in(0,1) $, $ N\ge 2 $, $\lambda >0$, $a$ and $b$ are positive
smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm
and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \]
Our proof is based on a comparison principle and existence, uniqueness and
asymptotic behaviors of various boundary blow-up solutions for a class of
elliptic equations involving the fractional Laplacian.
