Existence and asymptotic behavior of normalized solutions for the modified Kirchhoff equations in ${\mathbb{R}}^3$

This paper is concerned with the following modified Kirchhoff type problem

$$\begin{array}{r}-\left(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\right)\mathrm{\Delta }u-u\mathrm{\Delta }(u^2)-\lambda u=|u{|}^{p-2}u,x\in {\mathbb{R}}^3,\end{array}$$

where $a,b>0$ are constants and $\lambda \in \mathbb{R}$. When $p=\frac{16}{3}$, we prove that the existence of normalized solution with a prescribed $L^2$-norm for the above equation by applying constrained minimization method. Moreover, when $p\in \left(\frac{10}{3},\frac{16}{3}\right)$, we prove the existence of mountain pass type normalized solution for the above modified Kirchhoff equation by using the perturbation method. And the asymptotic behavior of normalized solution as $b\to 0$ is analyzed. These conclusions extend some known ones in previous papers.