Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases

Abstract

We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:{−ε2Δu+V(x)u+λ1|u|2u+λ2(K⁎|u|2)u=0 in R3,u>0, u∈H1(R3), where ε is a small positive parameter, λ1,λ2∈R, ⁎ denotes the convolution, K(x)=1−3cos2⁡θ|x|3 and θ=θ(x) is the angle between the dipole axis determined by (0,0,1) and the vector x. Under certain assumptions on (λ1,λ2)∈R2, we construct a family of positive solutions uε∈H1(R3) which concentrates around the local minima of V as ε→0. Our main results extend the results in J. Byeon and L. Jeanjean (2007) [6], which dealt with singularly perturbed Schrödinger equations with a local nonlinearity, to the nonlocal Gross–Pitaevskii type equation.