Fractional Gross-Pitaevskii equations in non-Gaussian attractive Bose-Einstein condensates

In this paper, we investigate normalized solutions of a fractional Gross-Pitaevskii equation, which arises in an attractive Bose-Einstein condensation consisting of $N$ bosons moving by Lévy flights. We prove that there exists a positive constant $N^{\ast }$, such that if $0<N<N^{\ast }$ and the Lévy index $\alpha$ closed to $2$, the fractional Gross-Pitaevskii equation admits a local minimal normalized solution $u_{\alpha }$ and a mountain pass solution $v_{\alpha }$, but there does not exist positive local minimal solution if $N>N^{\ast }$ and $\alpha$ closed to $2$. We also study the asymptotic behavior of $u_{\alpha }$ and $v_{\alpha }$ as $\alpha \to 2_{-}$.