Ground state and multiple normalized solutions of quasilinear Schrödinger equations in the $L^2$-supercritical case and the Sobolev critical case

Abstract

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schrödinger equation \begin{equation*} \begin{aligned} -Δu-uΔu^2 +λu=|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 $p\in (4+\frac{4}{N},2\cdot 2^{\ast }]$. The solutions correspond to critical points of the energy functional subject to the $L^2$-norm constraint ${\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=a^2>0$. In the Sobolev critical case $p=2\cdot 2^{\ast }$, the energy functional has no critical point. As for $L^2$-supercritical case $p\in (4+\frac{4}{N},2\cdot 2^{\ast })$: 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 $H_r^1({\mathbb{R}}^N)$. 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 $0^{+}$.