In this paper, we study a fractional Hardy–Sobolev problem with a concave nonlinearity \(\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\alpha u= \vert u\vert ^{2_\alpha ^{*}-2} u +\frac{\vert u\vert ^{2_\alpha ^{*}(s)-2}u}{\vert x\vert ^{s}}+ \lambda \vert u\vert ^{q-2}u & \text{ in } \Omega , \\ u=0 & \text{ on } \partial \Omega , \end{array}\right. \end{aligned}\) where \(N \ge 4\alpha \) , \(0<\alpha <1\) , \(0<s<2\alpha \) , \(1<q\le 2\) , \(\lambda >0\) , and \(\Omega \subset \mathbb {R}^N\) is a bounded domain satisfying suitable geometric conditions. We introduce subcritical perturbations of this problem and prove the strong convergence of any bounded sequence of solutions in the associated fractional Sobolev space, based on estimates in fractional safe regions combined with a localized Pohozaev-type identity. As a consequence, we obtain multiplicity results. In the case \(1<q<2\) , the problem admits infinitely many solutions with negative energy, while for \(1<q\le 2\) , there exist infinitely many solutions with positive energy tending to \(+\infty \) .
