New approaches for Schrödinger equations with prescribed mass: The Sobolev subcritical case and The Sobolev critical case with mixed dispersion

In this paper, we prove the existence of normalized solutions for the following Schrödinger equation \begin{equation*} \left{ \begin{array}{ll} -Δu-λu=f(u), & x\in \R^N, \int_{\R^N}u^2$\mathrm{d}$x=c \end{array} \right. \end{equation*} with $N\ge 3$, $c>0$, $\lambda \in \mathbb{R}$ and $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ in the Sobolev subcritical case with weaker $L^2$-supercritical conditions and in the Sobolev critical case when $f(u)=\mu \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u$ with $\mu >0$ and $2<q<2^{\ast }=\text{f}2NN-2$ allowing to be $L^2$-subcritical, critical or supercritical. Our approach is based on several new critical point theorems on a manifold, which not only help to weaken the previous $L^2$-supercritical conditions in the Sobolev subcritical case, but present an alternative scheme to construct bounded (PS) sequences on a manifold when $f(u)=\mu \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u$ technically simpler than the Ghoussoub minimax principle involving topological arguments, as well as working for all $2<q<2^{\ast }$. In particular, we propose new strategies to control the energy level in the Sobolev critical case which allow to treat, in a unified way, the dimensions $N=3$ and $N\ge 4$, and fulfill what were expected by Soave and by Jeanjean-Le . We believe that our approaches and strategies may be adapted and modified to attack more variational problems in the constraint contexts.