Existence and multiplicity of semiclassical states for Gross–Pitaevskii equation in dipolar quantum gases

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 115; no. 2
• Main Authors: Zhang, Hui, Xu, Junxiang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Convolution; Dipoles; Ground state; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Original Paper; Theoretical; Variational methods; Gross–Pitaevskii equation; 35J20; Ground states; 35A15; 35J60; Multiplicity of solutions; Nehari manifold; Concentration
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-021-01012-8

In this paper, we study the singularly perturbed Gross–Pitaevskii equation - ϵ 2 Δ u + V ( x ) u + λ 1 | u | 2 u + λ 2 ( K ∗ | u | 2 ) u = 0 , u ∈ H 1 ( R 3 ) , where ϵ > 0 is a parameter, the potential V is a positive function which possesses global minimum points, λ 1 , λ 2 ∈ R , ∗ denotes the convolution, K ( x ) = 1 - 3 cos 2 θ | x | 3 and θ = θ ( x ) is the angle between the dipole axis determined by (0, 0, 1) and the vector x . Using variational methods, we show the existence of ground states for ϵ small, and describe the concentration phenomena of ground states as ϵ → 0 . We also investigate the relationship between the number of positive solutions and the profile of the potential V .