Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

In this paper, we consider the following nonlinear Schrödinger equations with mixed nonlinearities: \begin{eqnarray*} \left\{\aligned &-Δu=λu+μ|u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ &u\in H^1(\bbr^N),\quad\int_{\bbr^N}u^2=a^2, \endaligned\right. \end{eqnarray*} where $N\geq3$, $μ>0$, $λ\in\mathbb{R}$ and $2<q<2^*$. We prove in this paper \begin{enumerate} \item[$(1)$]\quad Existence of solutions of mountain-pass type for $N=3$ and $2<q<2+\frac{4}{N}$. \item[$(2)$]\quad Existence and nonexistence of ground states for $2+\frac{4}{N}\leq q<2^*$ with $μ>0$ large. \item[$(3)$]\quad Precisely asymptotic behaviors of ground states and mountain-pass solutions as $μ\to0$ and $μ$ goes to its upper bound. \end{enumerate} Our studies answer some questions proposed by Soave in \cite[Remarks~1.1, 1.2 and 8.1]{S20}.

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