Existence and multiplicity of normalized solutions for the quasi-linear Schr\"{o}dinger equations with mixed nonlinearities

In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -\Delta u-\Delta(|u|^2)u+\lambda u=|u|^{p-2}u+\tau|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4, \end{equation*} with prescribed mass $$\int_{\mathbb{R}^N}|u|^2dx=a ,$$ where $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier and the parameters $a,\tau$ are all positive constants. We are concerned about the mass-mixed case $2<q<2+\frac{4}{N}$ and $4+\frac{4}{N}<p<2\cdot2^*$, where $2^*:=\frac{2N}{N-2}$ for $N\geq3$, while $2^*:=\infty$ for $N=1,2$. We show the existence of normalized ground state solution and normalized solution of mountain pass type. Our results can be regarded as a supplement to Lu et al. ( Proc. Edinb. Math. Soc., 2024) and Jeanjean et al. ( arXiv:2501.03845).