Quasilinear Schr\"odinger equations: ground state and infinitely many normalized solutions

In the present paper, we study the normalized solutions for the following quasilinear Schr\"odinger equations: $$-\Delta u-u\Delta u^2+\lambda u=|u|^{p-2}u \quad \text{in}~\mathbb R^N,$$ with prescribed mass $$\int_{\mathbb R^N} u^2=a^2.$$ We first consider the mass-supercritical case $p>4+\frac{4}{N}$, which has not been studied before. By using a perturbation method, we succeed to prove the existence of ground state normalized solutions, and by applying the index theory, we obtain the existence of infinitely many normalized solutions. Then we turn to study the mass-critical case, i.e., $p=4+\frac{4}{N}$, and obtain some new existence results. Moreover, we also observe a concentration behavior of the ground state solutions.