吴元泽: Normalized solutions for Schr\"odinger equations with critical Sobolev exponent and mixed nonlinearities-火狐棋牌官网

## 吴元泽: Normalized solutions for Schr\"odinger equations with critical Sobolev exponent and mixed nonlinearities

：吴元泽 博士

In this talk, I will introduce our recent results, joint work with professor Juncheng Wei, on the following nonlinear Schr\"{o}dinger equations with mixed nonlinearities:

\begin{eqnarray*}

\left\{\aligned

\endaligned\right.

\end{eqnarray*}

where $N\geq3$, $\mu>0$, $\lambda\in\bbr$ and $2<q<2^*=\frac{2N}{N-2}$.  We prove in this paper

\begin{enumerate}

\item[$(1)$]\quad Existence of solutions of mountain-pass type for $N\geq 3$ 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 open questions proposed by Soave in \cite{S20}.