Normalized ground states for the critical fractional NLS equation with a perturbation

Abstract

In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-Δ)^{s}u=λu +μ|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N}, \int_{\mathbb{R}^{N}}u^{2}dx=a^{2},\\ \end{cases} \end{equation*} where $(-\Delta {)}^s$ is the fractional Laplacian, $0<s<1$, $N>2s$, $2<q<2_s^{\ast }=2N/(N-2s)$ is a fractional critical Sobolev exponent, $a>0$, $\mu \in \mathbb{R}$. By using Jeanjean's trick in \cite{Jeanjean}, and the standard method which can be found in \cite{Brezis} to overcome the lack of compactness, we first prove several existence and nonexistence results for a $L^2$-subcritical (or $L^2$-critical or $L^2$-supercritical) perturbation $\mu |u{|}^{q-2}u$, then we give some results about the behavior of the ground state obtained above as $\mu \to 0^{+}$. Our results extend and improve the existing ones in several directions.

37 pages