Studies of normalized solutions to Schr\"{o}dinger equations with Sobolev critical exponent and combined nonlinearities

We consider the Sobolev critical Schr\"{o}dinger equation with combined nonlinearities \begin{equation*} \begin{cases} -\Delta u=\lambda u+|u|^{2^*-2}u+\mu|u|^{q-2}u,\ \ x\in\mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N),\ \int_{\mathbb{R}^N}|u|^2dx=a, \end{cases} \end{equation*} where $N\geq 3$, $\mu>0$, $\lambda\in \mathbb{R}$, $a>0$ and $q\in (2,2^*)$. We prove in this paper (1) Multiplicity and stability of solutions for $q\in (2,2+\frac{4}{N})$ and $\mu a^{\frac{q(1-\gamma_q)}{2}}\leq (2K)^{\frac{q\gamma_q-2^*}{2^*-2}}$ with $\gamma_q:=\frac{N}{2}-\frac{N}{q}$ and $K$ being some positive constant. This result extends the results obtained in Jeanjean et al. \cite{JEANJEAN-JENDREJ} and Jeanjean and Le \cite{Jeanjean-Le} for the case $\mu a^{\frac{q(1-\gamma_q)}{2}}<(2K)^{\frac{q\gamma_q-2^*}{2^*-2}}$ to the case $\mu a^{\frac{q(1-\gamma_q)}{2}}\leq (2K)^{\frac{q\gamma_q-2^*}{2^*-2}}$. (2) Nonexistence of ground states for $q=2+\frac{4}{N}$ and $\mu a^{\frac{q(1-\gamma_q)}{2}}\geq\bar{a}_N$ with $\bar{a}_N$ being some positive constant. We give a new proof to this result different with Wei and Wu \cite{Wei-Wu 2021}.