Existence and asymptotic behavior of normalized solutions for Choquard equations with Kirchhoff perturbation

In this paper, we consider the following Choquard equation involving the Kirchhoff type perturbation: \(\begin{aligned} -(a+b\int _{\mathbb {R}^{N}}|\nabla u|^{2}\textrm{d}x)\Delta u = \lambda u + (I_{\alpha }*|u|^{p})|u|^{p-2}u+\mu (I_{\alpha }*|u|^{q})|u|^{q-2}u \text{ in } \mathbb {R}^{N} \end{aligned}\) $$\begin{array}{c}\hfill -(a+b{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^2\text{d}x)\mathrm{\Delta }u=\lambda u+(I_{\alpha }{\ast |u|}^p{)|u|}^{p-2}u+\mu (I_{\alpha }{\ast |u|}^q{)|u|}^{q-2}u\text{in}{\mathbb{R}}^N\end{array}$$ under the constraint \(\begin{aligned} \int _{\mathbb {R}^{N}}u^{2}\textrm{d}x=c^{2}, \end{aligned}\) $$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{u}^2\text{d}x=c^2,\end{array}$$ where \(N\ge 3\) $N\ge 3$ , \(a,b,c>0\) $a,b,c>0$ , \(\mu >0\) $\mu >0$ , \((N+\alpha )/N<q<(N+4+\alpha )/N<p<2^{*}_{\alpha }:=(N+\alpha )/(N-2)\) $(N+\alpha )/N<q<(N+4+\alpha )/N<p<2_{\alpha }^{\ast }:=(N+\alpha )/(N-2)$ or \((N+4+\alpha )/N<q<p<2^{*}_{\alpha }\) $(N+4+\alpha )/N<q<p<2_{\alpha }^{\ast }$ , \(\mu >0\) $\mu >0$ and \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. By developing a new perturbed Pohozaev constraint approach, we establish the existence of two normalized solutions for the above problem in the mixed critical case where \((N+\alpha )/N<q<(N+4+\alpha )/N<p<2^{*}_{\alpha }\) $(N+\alpha )/N<q<(N+4+\alpha )/N<p<2_{\alpha }^{\ast }$ , and the existence of ground state solutions in the \(L^{2}\) $L^2$ –supercritical case where \((N+4+\alpha )/N<q<p<2^{*}_{\alpha }\) $(N+4+\alpha )/N<q<p<2_{\alpha }^{\ast }$ . Moreover, the asymptotic behavior for the normalized solutions was also explored.