Studies of normalized solutions to Schrödinger equations with Sobolev critical exponent and combined nonlinearities

Abstract

We consider the Sobolev critical Schrödinger equation with combined nonlinearities \begin{equation*} \begin{cases} -Δu=λu+|u|^{2^*-2}u+μ|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 $N\ge 3$, $\mu >0$, $\lambda \in \mathbb{R}$, $a>0$ and $q\in (2,2^{\ast })$. We prove in this paper (1) Multiplicity and stability of solutions for $q\in (2,2+\frac{4}{N})$ and $\mu a^{\frac{q(1-{\gamma }_q)}{2}}\le (2K{)}^{\frac{q{\gamma }_q-2^{\ast }}{2^{\ast }-2}}$ 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 $\mu a^{\frac{q(1-{\gamma }_q)}{2}}<(2K{)}^{\frac{q{\gamma }_q-2^{\ast }}{2^{\ast }-2}}$ to the case $\mu a^{\frac{q(1-{\gamma }_q)}{2}}\le (2K{)}^{\frac{q{\gamma }_q-2^{\ast }}{2^{\ast }-2}}$. (2) Nonexistence of ground states for $q=2+\frac{4}{N}$ and $\mu a^{\frac{q(1-{\gamma }_q)}{2}}\ge {\bar{a}}_N$ 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}.

arXiv admin note: text overlap with arXiv:2104.09317