In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity

$$\begin{array}{r}\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u=\lambda u+|u{|}^{p-2}u+|u{|}^{2_s^{\ast }-2}u,& x\in {\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}{u}^2\mathrm{d}x=a^2,u\in H^s({\mathbb{R}}^N),\end{array}\end{array}$$

where $N\ge 2$, $s\in (0,1)$, $a>0$, $2<p<2_s^{\ast }\triangleq \frac{2N}{N-2s}$ and $(-\mathrm{\Delta }{)}^s$ is the fractional Laplace operator. In the purely $L^2$-subcritical perturbation case $2<p<2+\frac{4s}{N}$, we prove the existence of a second normalized solution under some conditions on $a$, $p$, $s$, and $N$. This is a continuation of our previous work (Z. Angew. Math. Phys., 73 (2022) 149) where only one solution is obtained.

In this paper, we investigate the existence of solutions to a generalized quasilinear Schrödinger equation with concave-convex nonlinearities and potentials vanishing at infinity. Using the mountain pass theorem, we get the existence of a positive solution.