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

In this work, we study the quasilinear Schrdinger equationwhere, a > 0 is a given mass and 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: (i) qualitative analysis of the constrained minimization For 2 < p < 4 + 4 N q < 2 2 * , we provide a detailed study of the minimization problem under some appropriate conditions on a > 0;(ii) existence of two distinct radial normalized solutions For 2 < p < 2 + 4 N < 4 + 4 N < q 2 * , we obtain a radial local minimizer under the normalized constraint; For 2 < p < 2 + 4 N < 4 + 4 N < q 2 * , 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,