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 Sobolev critical exponent: \begin{eqnarray*} -Δu-uΔ(u^2)+λu=τ|u|^{q-2}u+|u|^{2\cdot2^-2}u,~~~~x\in$\mathbb{R}$^N, \end{eqnarray} under the mass constraint ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=c$ for some prescribed $c>0$. Here $\tau \in \mathbb{R}$ is a parameter, $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier, $N\ge 3$, $2^{\ast }:=\frac{2N}{N-2}$ and $2<q<2\cdot 2^{\ast }$. By deriving precise energy level estimates and establishing new convergence theorems, we apply the perturbation method to establish several existence results for $\tau >0$ in the Sobolev critical regime: (a) For the case of $2<q<2+\frac{4}{N}$, we obtain the existence of two solutions, one of which is a local minimizer, and the other is a mountain pass type solution, under explicit conditions on $c>0$; (b) For the case of $2+\frac{4}{N}\le q<4+\frac{4}{N}$, we obtain the existence of normalized solutions of mountain pass type under different conditions on $c>0$; © For the case of $4+\frac{4}{N}\le q<2\cdot 2^{\ast }$, we obtain the existence of a ground state normalized solution under different conditions on $c>0$. Moreover, when $\tau \le 0$, we derive the non-existence result for $2<q<2\cdot 2^{\ast }$ and all $c>0$. Our research provides a comprehensive analysis across the entire range $q\in (2,2\cdot 2^{\ast })$ and for all $N\ge 3$. The methods we have developed are flexible and can be extended to a broader class of nonlinearities.