Normalized solutions for a Schrödinger equation with critical growth in ${\mathbb{R}}^N$

Abstract

In this paper we study the existence of normalized solutions to the following nonlinear Schrödinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-Δu=λu+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a>0$, $\lambda \in \mathbb{R}$ and $f$ has an exponential critical growth when $N=2$, and $f(t)=\mu |u{|}^{q-2}u+|u{|}^{2^{\ast }-2}u$ with $q\in (2+\frac{4}{N},2^{\ast })$, $\mu >0$ and $2^{\ast }=\frac{2N}{N-2}$ when $N\ge 3$. Our main results complement some recent results for $N\ge 3$ and it is totally new for $N=2$.