Normalized solutions of Kirchhoff equations with Hartree-type nonlinearity

In the present paper, we prove the existence of the solutions (λ, u) ∈ ℝ × H¹(ℝ³) to the following Kirchhoff equations with the Hartree-type nonlinearity under the general mass supercritical settings, { -(a+b∫ℝ3| ∇u |2dx)Δu-λu=[ Iα*(K(x)F(u)) ]K(x)f(u),u∈H1(ℝ3), \left\{ {\matrix{{ - \left( {a + b\int\limits_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}dx} } \right)\Delta u - \lambda u = \left[ {{I_\alpha }*\left( {K\left( x \right)F\left( u \right)} \right)} \right]K\left( x \right)f\left( u \right),} \hfill \cr {u \in {H^1}\left( {{\mathbb{R}^3}} \right),} \hfill \cr } } \right. where a, b > 0 are prescribed, I_α = |x|^α−3 is the riesz potential where α ∈ (0, 3), K ∈ 𝒞¹(ℝ³, ℝ⁺) satisfies an explicit assumption and f ∈𝒞 (ℝ, ℝ) satisfies some weak conditions, we develop some new tricks for dealing with the Hartree-type term to overcome the difficulties produced by the appearance of non-constant potential K(x). This paper extends and promotes the previous results on prescribed L²-norm solutions of the Kirchhoff-type equation.