Ground state and multiple normalized solutions of quasilinear Schr\"odinger equations in the $L^2$-supercritical case and the Sobolev critical 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=|u|^{p-2}u \quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N=3,4$, $\lambda$ appears as a Lagrange multiplier and $p \in (4+\frac{4}{N},2\cdot2^*]$. The solutions correspond to critical points of the energy functional subject to the $L^2$-norm constraint $\int_{\mathbb{R}^N}|u|^2dx=a^2>0$. In the Sobolev critical case $p=2\cdot 2^*$, the energy functional has no critical point. As for $L^2$-supercritical case $p \in (4+\frac{4}{N},2\cdot2^*)$: on the one hand, taking into account Pohozaev manifold and perturbation method, we obtain the existence of ground state normalized solutions for the non-radial case; on the other hand, we get the existence of infinitely many normalized solutions in $H^1_r(\mathbb{R}^N)$. Moreover, our results cover several relevant existing results. And in the end, we get the asymptotic properties of energy as $a$ tends to $+\infty$ and $a$ tends to $0^+$.