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

Abstract

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 $N^*$, such that if $0<N<N^*$ and the Lévy index $α$ closed to $2$, the fractional Gross-Pitaevskii equation admits a local minimal normalized solution $u_α$ and a mountain pass solution $v_α$, but there does not exist positive local minimal solution if $N>N^*$ and $α$ closed to $2$. We also study the asymptotic behavior of $u_α$ and $v_α$ as $α\to 2_-$.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3