In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \(\begin{aligned} -\Delta u-\Delta (|u|^2)u+\lambda u=|u|^{p-2}u+\tau |u|^{q-2}u, \text { in }\mathbb {R}^N,~ 1\le N\le 4, \end{aligned}\) with prescribed mass \(\begin{aligned} \int \limits _{\mathbb {R}^N}|u|^2{\textrm{d}}x=a , \end{aligned}\) where \(\lambda \in \mathbb {R}\) appears as a Lagrange multiplier and the parameters \(a,\tau \) are all positive constants. We are concerned about the mass-mixed case \(2<q<2+\frac{4}{N}\) and \(4+\frac{4}{N}<p<2\cdot 2^*\) , where \(2^*:=\frac{2N}{N-2}\) for \(N\ge 3\) , while \(2^*:=\infty \) for \(N=1,2\) . We show the existence of normalized ground state solution and normalized solution of mountain pass type. Here we establish the existence of two normalized solution by studying two minimization problems constrained on the closed ball, different from previous ones: First, we use the weak closedness to prove that the constrained minimization problems on the closed ball are attained. Next, we show that the attainable minimizers are the solutions to the above equation, though it may not satisfy the \(L^2\) constraint condition. Then, we prove that the Lagrange multiplier \(\lambda \) is non-zero. Finally, using the fact that \(\lambda \ne 0\) , we deduce that the attainable minimizers must lie on the sphere. Our results can be seen as a supplement to Mao et al. (Proc Edinb Math Soc 67(2):349–387, 2024) and Jeanjean et al. (Existence and limiting profile of energy ground states for a quasi-linear Schrödinger equations: mass super-critical case, arXiv:2501.03845 ).
