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 $N\geq2$, $2<p<2+\frac{ 4}{ N}<4+\frac{ 4}{ N}<q<22^*$, $a>0$ 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 $2<p<4+\frac{ 4}{ N}\leq q<22^*$, 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 $2<p<2+\frac{ 4}{ N}<4+\frac{ 4}{ N}<q<22^*$, we obtain a local minimizer under the normalized constraint;\\ For $2<p<2+\frac{ 4}{ N}<4+\frac{ 4}{ N}<q\leq2^*$, 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.
Categories: math.AP