Consider the Kirchhoff equation with Hartree type nonlinearity \(\matrix{{ - \left( {a + b\int_{\mathbb{R}^{3}} {{{| {\nabla u} |}^2}} } \right)\Delta u - \lambda u = \mu {{| u |}^{q - 2}}u + ( {{I_\alpha } * {{| u |}^{3 + \alpha }}} ){{| u |}^{1 + \alpha }}u} & {{\rm{in}}\,{{\mathbb{R}}^3}}},\) where a, b > 0, λ, μ ∈ ℝ, 2 < q < 6, 0 < α < 3, and Iα is the Riesz potential integral operator of order α. Solutions with prescribed mass \({\|u\|_{{L^2}({{\mathbb{R}^3}})}} = c > 0\) , also known as normalized solutions, are of particular interest in the current paper. Under various assumptions on μ, c and q, we establish the existence, nonexistence and asymptotic behavior of normalized solutions for the above elliptic equation.
