In this work, we study a class of critical problems driven by the fractional Laplacian operator with a Hardy potential in the form \(\begin{aligned} {\left\{ \begin{array}{ll} \left( -\Delta \right) ^{s}u - \mu \frac{ u }{\left| x \right| ^{2s}} = \lambda u+ \left| u\right| ^{2^{*}_{s} -2} u & \text { in } \Omega , \\ u=0 & \text { in } {\mathbb {R}}^{n} \backslash \Omega , \end{array}\right. } \end{aligned}\) where \(\left( -\Delta \right) ^{s}\) is the fractional Laplace operator, \(s \in (0,1)\) , \(\Omega \) is a smooth bounded domain in \({\mathbb {R}}^{n}\) containing the origin with \(n > 2s\) , \(0< \mu < {\overline{\mu }}_{H}\) , \(\lambda >0\) and \(2^{*}_{s}=\frac{2n}{n-2s}\) is the critical fractional Sobolev exponent. Using asymptotic analysis, we prove that the existence and multiplicity of solutions depend on the value of \(\mu \) and the position of \(\lambda \) relative to the spectrum of the operator \((\left( -\Delta \right) ^{s} - {\mu }{\left| x \right| ^{-2s}})\) with Dirichlet boundary conditions.
