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<p<2+\frac{4}{N}<4+\frac{4}{N}\leq q<2\cdot2^* $, $ 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:

$ {\bf{(i)}} $ Qualitative analysis of the constrained minimization

For $ 2<p<4+\frac{4}{N}\leq q<2\cdot2^* $, we provide a detailed study of the minimization problem under some appropriate conditions on $ a>0 $.

$ {\bf{(ii)}} $ Existence of two distinct solutions

For $ 2<p<2+\frac{4}{N}<4+\frac{4}{N}<q\leq2^* $, we obtain a radial local minimizer under the normalized constraint.

For $ 2<p<2+\frac{4}{N}<4+\frac{4}{N}<q\leq2^* $, we obtain a radial mountain-pass-type normalized solution distinct from the local minimizer.

Notably, the second result (ⅱ) resolves the open problem (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. 28, 1633-1659], we obtain the mountain pass solution employing a new method, which relies on the monotonicity trick developed by (Chang et al., 2024, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 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 Schrödinger equations via the dual method.