Normalized solutions to the quasilinear Schrödinger equations with combined nonlinearities

We consider the radially symmetric positive solutions to quasilinear problem \begin{equation*}-\triangle u-u\triangle u^{2}+\lambda u=f(u),\quad{\rm in} \ \mathbb{R}^{N},\end{equation*} having prescribed mass $\int_{\mathbb{R}^{N}}|u|^2 =a^2,$ where a > 0 is a constant, λ appears as a Lagrange multiplier. We focus on the pure L2-supercritical case and combination case of L2-subcritical and L2-supercritical nonlinearities \begin{equation*}f(u)=\tau |u|^{q-2}u+|u|^{p-2}u,\quad \tau \gt 0,\qquad{\rm where}\ \ 2 \lt q \lt 2+\frac{4}{N} \ {\rm and} \quad \ p \gt \bar{p},\end{equation*} where $\bar{p}:=4+\frac{4}{N}$ is the L2-critical exponent. Our work extends and develops some recent results in the literature.