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

In this work, we study the quasilinear Schrödinger equation equation* -Δu-Δ (u²) u=|u|^p-2u+|u|^q-2u+λu, \, \, xN, equation* under the mass constraint equation* ₍|u|²dx=a, equation* where N2, 20 is a given mass and λ 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: itemize (i) qualitative analysis of the constrained minimization\\ For 20; itemize itemize (ii) existence of two radial distinct normalized solutions\\ For 2<p<2+4N<4+4N<q<22^*, we obtain a local minimizer under the normalized constraint;\\ For 2<p<2+4N<4+4N<q2^*, we obtain a mountain pass type normalized solution distinct from the local minimizer. itemize 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 lean upon 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.