Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent

Abstract In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents $$\begin{aligned} -\Delta u=(|x|^{-(n-2)}*u^{p-\varepsilon })u^{p-1-\varepsilon }\quad \text{ in }~~\Omega ,~~ u=0\quad \text{ on }~~\partial \Omega , \end{aligned}$$ - Δ u = ( | x | - ( n - 2 ) ∗ u p - ε ) u p - 1 - ε in Ω , u = 0 on ∂ Ω , where $$\Omega $$ Ω is a smooth bounded domain in $$\mathbb {R}^n$$ R n for $$n=3,4,5$$ n = 3 , 4 , 5 , $$*$$ ∗ denotes the standard convolution, $$\varepsilon >0$$ ε > 0 is a small parameter and $$p=\frac{n+2}{n-2}$$ p = n + 2 n - 2 is $$\mathcal {D}^{1,2}$$ D 1 , 2 energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first $$(n+2)$$