This paper deals with the following fractional elliptic equation with critical exponent
\[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\]
where $\lambda$, $\bar {\nu }\in {{\mathfrak R}}$, $s\in (0,1)$, $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$, $(-\Delta )^{s}$ is the fractional Laplace operator, $\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and $\varphi _{1}$ is the first positive eigenfunction of the fr...
