A new proof on quasilinear Schr\"{o}dinger equations with prescribed mass and combined nonlinearities

In this work, we study the quasilinear Schr"{o}dinger equation \begin{equation*} \aligned -\Delta u-\Delta(u^2)u=|u|^{p-2}u+|u|^{q-2}u+\lambda u,,, x\in\R^N, \endaligned \end{equation*} under the mass constraint \begin{equation*} \int_{\R^N}|u|^2$\text{d}$x=a, \end{equation*} where $N\ge 2$, $2<p<2+\frac{4}{N}<4+\frac{4}{N}<q<{22}^{\ast }$, $a>0$ 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: \begin{itemize} \item[{\bf(i) }] {\bf qualitative analysis of the constrained minimization}\ For $2<p<4+\frac{4}{N}\le q<{22}^{\ast }$, we provide a detailed study of the minimization problem under some appropriate conditions on $a>0$; \end{itemize} \begin{itemize} \item[{\bf(ii)}]{\bf existence of two radial distinct normalized solutions}\ For $2<p<2+\frac{4}{N}<4+\frac{4}{N}<q<{22}^{\ast }$, we obtain a local minimizer under the normalized constraint;\ For $2<p<2+\frac{4}{N}<4+\frac{4}{N}<q\le 2^{\ast }$, we obtain a mountain pass type normalized solution distinct from the local minimizer. \end{itemize} Notably, the second result {\bf (ii)} resolves the open problem {\bf(OP1)} posed by (Chen et al., 2025, arXiv:2506.07346v1). Unlike previous approaches that rely on constructing Palais-Smale-Pohozaev sequences by [Jeanjean, 1997, Nonlinear Anal. {\bf 28}, 1633-1659], we obtain the mountain pass solution employing a new method, which lean upon the monotonicity trick developed by (Chang et al., 2024, Ann. Inst. H. Poincar'{e} C Anal. Non Lin'{e}aire, {\bf 41}, 933-959). We emphasize that the methods developed in this work can be extended to investigate the existence of mountain pass-type normalized solutions for other classes of quasilinear Schr"{o}dinger equations.