Existence and asymptotic behavior of normalized solutions for Kirchhoff equation with Hardy nonlinearities

In this paper, we consider the qualitative analysis of solutions for the following Kirchhoff equation with Hardy nonlinearities: $$ \left \{\begin{aligned} &-(a+b\int _{\mathbb{R}^{N}}|\nabla u|^{2} dx)\Delta u=\lambda u+\frac{|u|^{p - 2}u}{|x|^{\beta}}+\mu \frac{|u|^{q - 2}u}{|x|^{\beta}}, &&\text{ in } \mathbb{R}^{N},\ &\int _{\mathbb{R}^{N}} u^{2} dx = c^{2}, \end{aligned}\right . $$ where $N\geq 3,~0<\beta <2,~2<q<p<2^{*}_{\beta}:=\frac{2(N-\beta )}{N-2}$, $a,b,c>0,~\mu >0$ and $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. By developing a perturbed Pohozaev constraint approach, we show the existence of normalized solutions under the mixed $L^{2}$-critical case where $2< q<\frac{2(N+4-\beta )}{N}<p<2^{*}_{\beta}$ and ground state solutions in the $L^{2}$-supercritical case where $\frac{2(N+4-\beta )}{N}< q< p<2^{*}_{\beta}$, respectively. Moreover, the asymptotic behaviors of the normalized solutions are also obtained.