Existence and multiplicity of normalized solutions for the quasi-linear Schrödinger 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*} -Δu-Δ(|u|^2)u+λu=|u|^{p-2}u+τ|u|^{q-2}u, $\text{ in }$$\mathbb{R}$^N,~ 1\leq N\leq4, \end{equation*} with prescribed mass $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=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 $4+\frac{4}{N}<p<2\cdot 2^{\ast }$, where $2^{\ast }:=\frac{2N}{N-2}$ for $N\ge 3$, while $2^{\ast }:=\mathrm{\infty }$ 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).