Normalized solutions for a Sobolev critical quasilinear Schr\"odinger equation

In this paper, we study the existence of normalized solutions for the following quasilinear Schr\"odinger equation with Sobolev critical exponent: \begin{eqnarray*} -\Delta u-u\Delta (u^2)+\lambda u=\tau|u|^{q-2}u+|u|^{2\cdot2^*-2}u,~~~~x\in\mathbb{R}^N, \end{eqnarray*} under the mass constraint $\int_{\mathbb{R}^N}|u|^2dx=c$ for some prescribed $c>0$. Here $\tau\in \mathbb{R}$ is a parameter, $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier, $N\ge3$, $2^*:=\frac{2N}{N-2}$ and $2<q<2\cdot2^*$. 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 $2+\frac{4}{N}\leq q<4+\frac{4}{N}$, we obtain the existence of normalized solutions of mountain pass type under different conditions on $c>0$; (c) For the case of $4+\frac{4}{N}\leq q<2\cdot2^*$, 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 $2<q<2\cdot2^*$ and all $c>0$. Our research provides a comprehensive analysis across the entire range $q\in(2, 2 \cdot 2^*)$ and for all $N\ge3$. The methods we have developed are flexible and can be extended to a broader class of nonlinearities.