Existence and non-existence of normalized solutions for a nonlinear fractional Schrödinger system
Chungen Liu, Zhigao Zhang, Jiabin Zuo
TL;DR
The paper investigates the existence and nonexistence of $L^2$-normalized solutions to a one-dimensional two-component fractional Schrödinger system with nonlocal operator $(-\Delta)^{1/2}$ and trapping potentials. It employs a variational approach on a mass-constrained manifold, introducing the constrained energy $E_{a_1,a_2,\beta}$ and auxiliary functionals $\hat{e}$ and $\Gamma$ to establish sharp existence/nonexistence thresholds tied to $a^*=$$\|Q\|_2^2$, $\beta_*$, and $\beta^*$, with $Q$ solving $\sqrt{-\Delta}Q+Q=Q^3$. The analysis combines compactness results under confinement, concentration-compactness arguments, and precise energy estimates, yielding a dichotomy: minimizers exist for subcritical intra-species strengths and small interspecies coupling, while minimizers fail to exist beyond the identified thresholds or in strong coupling/negative beta regimes. In symmetric/critical regimes, minimizing sequences concentrate to the scalar ground state, providing a detailed description of limiting profiles and grounding the threshold phenomenon in the underlying fractional Gagliardo–Nirenberg structure.
Abstract
This paper is concerned with a nonlinear fractional Schördinger system in $\mathbb{R}$ with intraspecies interactions $a_{i}>0 \ (i=1,2)$ and interspecies interactions $β\in\mathbb{R}$. We study this system by solving an associated constrained minimization problem (i.e., $L^2-$norm constrains). Under certain assumptions on the trapping potentials $V_i(x) \ (i=1,2),$ we derive some delicate estimates for the related energy functional and establish a criterion for the existence and non-existence of solutions, in which way several existence results are obtained.