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Existence and multiplicity of normalized solutions for the quasi-linear Schrödinger equations with mixed nonlinearities
In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -Δu-Δ(|u|^2)u+λu=|u|^{p-2}u+τ|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4, \end{equation*} with prescribed mass $$\int_{\mathbb{R}^N}|u|^2dx=a ,$$ where $λ\in\mathbb{R}$ appears as a Lagrange multiplier and the parameters $a,τ$ are all positive constants. We are concerned about the mass-mixed case $2

Normalized solutions for a Sobolev critical quasilinear Schrödinger equation
In this paper, we study the existence of normalized solutions for the following quasilinear Schrödinger equation with critical exponent: \begin{equation*} -Δu-uΔ(u^2)+λu=τ|u|^{q-2}u+|u|^{2\cdot2^*-2}u,~~~~x\in\R^N, \end{equation*} under the mass constraint $\int_{\R^N}|u|^2dx=c$ for some prescribed $c>0$. Here $τ\in \mathbb{R}$ is a parameter, $λ\in\R$ appears as a Lagrange multiplier, $N\ge3$, $2^*:=\frac{2N}{N-2}$ and $2