A new proof on quasilinear Schrödinger equations with prescribed mass and combined nonlinearities

In this work, we study the quasilinear Schrödinger equation $ \begin{equation*} \begin{aligned} -\Delta u-\Delta(u^2)u = |u|^{p-2}u+|u|^{q-2}u+\lambda u, \, \, x\in \mathbb{R}^N, \end{aligned} \end{equation*} $ under the mass constraint$ \begin{equation*} \displaystyle \int_{ \mathbb{R}^N}|u|^2\text{d}x = a, \end{equation*} $where $ N\geq2 $, $ 2 0 $ is a given mass, and $ \lambda $ is a Lagrange multiplier. As a continuation of our previous work (Chen et al., 2025, arXiv:2506.07346v1), we establish some results by means of a suitable change of variables as follows:$ {\bf{(i)}} $ Qualitative analysis of the constrained minimizationFor $ 2 0 $.$ {\bf{(ii)}} $ Existence of two distinct solutionsFor $ 2