Let (M, g) be a smooth compact Riemannian manifold of dimension $$n\ge 6$$ n≥6 , $$\xi _0\in M$$ ξ0∈M , and we are concerned with the following Hardy–Sobolev elliptic equations: 0.1$$\begin{aligned} -\Delta _gu+h(x)u=\frac{u^{2^{*}(s)-1-\epsilon }}{d_{g}(x,\xi _0)^s},\ \ \ \ u>0\ \ \mathrm{in} \ \ M, \end{aligned}$$ -Δgu+h(x)u=u2∗(s)-1-ϵdg(x,ξ0)s,u>0inM, where $$\Delta _g\,=\,\mathrm{div}_g(\nabla )$$ Δg=divg(∇) is the Laplace–Beltrami operator on M, h(x) is a $$C^1$$ C1 function on M, $$\epsilon $$ ϵ is a sufficiently small real parameter, $$2^{*}(s):=\frac{2(n-s)}{n-2}$$ 2∗(s):=2(n-s)n-2 is the critical Hardy–Sobolev exponent with $$s\in (0,2)$$ s∈(0,2) , and $$d_{g}$$ dg is the Riemannian distance on M. Performing the Lyapunov–Schmidt reduction procedure, we obtain the existence of blow-up families of positive solutions of problem (0.1).