Normalized solutions of quasilinear Schrödinger equations in the general $L^2$-supercritical case

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schr\"odinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=h(u) \quad\mathrm{in}\ \mathbb{R}^{3}, \end{aligned} \end{equation*} where $\lambda$ appears as a Lagrange multiplier, $h$ is a $L^2$-supercritical and Sobolev subcritical nonlinearity. The solutions correspond to critical points of the energy functional subject to the $L^2$-norm constraint $\int_{\mathbb{R}^3}|u|^2dx=a^2>0$. Taking into account the Pohozaev manifold and perturbation method, we obtain the existence of ground state normalized solutions and infinitely many normalized solutions. Moreover, our results cover several relevant existing results in \cite{LZ2023}. And in the end, we get the asymptotic properties of energy as $a$ tends to $+\infty$ and $a$ tends to $0^+$.