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

TL;DR

This paper investigates the existence and properties of normalized ground states for a critical fractional nonlinear Schrödinger equation with a perturbation, extending previous results and analyzing the behavior as the perturbation parameter approaches zero.

Contribution

It introduces new existence and nonexistence results for ground states in the fractional NLS with perturbations and studies their asymptotic behavior as the perturbation vanishes.

Findings

Established existence of ground states for various perturbation regimes.

Demonstrated nonexistence results under certain conditions.

Analyzed the limiting behavior of solutions as the perturbation parameter tends to zero.

Abstract

In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-\Delta)^{s}u=\lambda u +\mu|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 $\mu\rightarrow 0^{+}$. Our results extend and improve the existing ones in several directions.