Existence of normalized solutions for fractional coupled Hartree-Fock type system

TL;DR

This paper proves the existence of normalized solutions for a fractional coupled Hartree-Fock system with specific potential and nonlinearity conditions, expanding understanding of such systems in mathematical physics.

Contribution

It establishes the existence and positivity of normalized solutions for the fractional coupled Hartree-Fock system under certain parameter restrictions, a novel result in this context.

Findings

Existence of solutions under specified conditions.

Positivity of solutions for certain nonlinearities.

Extension of results to fractional Laplacian systems.

Abstract

In this paper, we consider the existence of solutions for the following fractional coupled Hartree-Fock type system \begin{align*} \left\{\begin{aligned} &(-\Delta)^s u+V_1(x)u+\lambda_1u=\mu_1(I_{\alpha}\star |u|^p)|u|^{p-2}u+\beta(I_{\alpha}\star |v|^r)|u|^{r-2}u\\ &(-\Delta)^s v+V_2(x)v+\lambda_2v=\mu_2(I_{\alpha}\star |v|^q)|v|^{q-2}v+\beta(I_{\alpha}\star |u|^r)|v|^{r-2}v \end{aligned} \right.~\quad x\in\mathbb{R}^N, \end{align*} under the constraint \begin{align*} \int_{\mathbb{R}^N}|u|^2=a^2,~\int_{\mathbb{R}^N}|v|^2=b^2. \end{align*} where $s\in(0,1),~N\ge3,~\mu_1>0,~\mu_2>0,~\beta>0,~\alpha\in(0,N),~1+\frac{\alpha}{N}<p,~q,~r<\frac{N+\alpha}{N-2s}$ and $I_{\alpha}(x)=|x|^{\alpha-N}$. Under some restrictions of $N,\alpha,p,q$ and $r$, we give the positivity of normalized solutions for $p,q,r\le 1+\frac{\alpha+2s}{N}$.