Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities (2102.04030v1)

Abstract: In this paper, we consider the following nonlinear Schr\"{o}dinger equations with mixed nonlinearities: \begin{eqnarray*} \left{\aligned &-\Delta u=\lambda u+\mu |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 $N\geq3$, $\mu>0$, $\lambda\in\mathbb{R}$ 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 $\mu\>0$ large. \item[$(3)$]\quad Precisely asymptotic behaviors of ground states and mountain-pass solutions as $\mu\to0$ and $\mu$ 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}.