We consider the following fractional Hénon-type equation with critical growth: \(\begin{aligned} \left\{ \begin{array}{lll} (-\Delta )^{s} u = K(|y|)u^{\frac{N+2s}{N-2s}},u>0, & y\in B_{1}(0), \\ \displaystyle u =0,& y\in B^c_{1}(0), \end{array}\right. \end{aligned}\) where \(B_{1}(0)\) is the unit ball in \(\mathbb {R}^{N}\) , \(K:[0,1] \rightarrow \mathbb {R}\) is a bounded and non-negative function. We first prove a non-degeneracy result of the bubble solutions to the above problem. Then we apply the non-degeneracy result to obtain the new existence results for the problem. This paper contains many technique estimates which we believe that they are useful in solving other nonlocal problems involving nondegeneracy and construction of solutions.
