Normalized ground states for nonlinear Schrödinger equations with general Sobolev critical nonlinearities

In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger equation \begin{equation*} \left{ \begin{aligned} &-Δu=f(u)+ λu\quad \mbox{in}\ $\mathbb{R}$^{N},\ &u\in H^1($\mathbb{R}$^N), ~~~\int_{$\mathbb{R}$^N}|u|^2dx=c, \end{aligned} \right. \end{equation*} where $N\ge 3$, $c>0$, $\lambda \in \mathbb{R}$ and $f$ has a Sobolev critical growth at infinity but does not satisfies the Ambrosetti-Rabinowitz condition. By analysing the monotonicity of the ground state energy with respect to $c$, we develop a constrained minimization approach to establish the existence of normalized ground state solutions for all $c>0$.