Existence and blow‐up behavior of standing waves for the Gross–Pitaevskii functional with a higher order interaction

We study the constraint minimization problem related to the Gross–Pitaevskii functional with a higher order interactionI a δ : = infE a δ ( ϕ ) : ϕ ∈ H ( R 2 ) , ‖ ϕ ‖L 22 = 1 , where δ >0, a >0,E a δ ( ϕ ) : = ∫R 2| ∇ ϕ | 2 d x + ∫R 2V | ϕ | 2 d x + δ 2∫R 2| ∇ ( | ϕ | 2 ) | 2 d x − a 2∫R 2| ϕ | 4 d x and V is a continuous periodic potential. Thanks to the concentration‐compactness principle, we show the existence of minimizers forI a δwith a ≥ a * : = ‖ Q ‖L 22and δ sufficiently small, where Q is the unique positive radial solution to − Δ Q + Q − Q 3 = 0 . The blow‐up behaviors of minimizers forI a δas δ ↘0 are described in details with an additional assumption on the external potential in the case a = a * .