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 L\'{e}vy flights. We prove that there exists a positive constant $N^*$, such that if $0<N<N^*$ and the L\'{e}vy 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 $N>N^*$ and $\alpha$ closed to $2$. We also study the asymptotic behavior of $u_\alpha$ and $v_\alpha$ as $\alpha\to 2_-$.