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(u^2)u=|u|ᵖ⁻²u+|u|q-2u+λu,\,\, x∈^N, {equation*} under the mass constraint {equation*} ∫^N|u|^2dx=a, {equation*} where N≥2, 2<p<2+4/N<4+4/N<q<22^*, $a>0$ 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 2<p<4+4/N≤ q<22^*, we provide a detailed study of the minimization problem under some appropriate conditions on $a>0$; {itemize} {itemize} [{(ii)}]{ existence of two radial distinct normalized solutions}\\ For 2<p<2+4/N<4+4/N<q<22^*, we obtain a local minimizer under the normalized constraint;\\ For 2<p<2+4/N<4+4/N<q≤2^*, 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.