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

In this paper, we study the singularly perturbed Gross-Pitaevskii equation -epsilon(2)Delta u + V(x)u + lambda(1)vertical bar u vertical bar(2)u + lambda(2)(K*vertical bar u vertical bar(2))u = 0, u is an element of H-1(R-3), where epsilon > 0 is a parameter, the potential V is a positive function which possesses global minimum points, lambda(1), lambda(2) is an element of R, * denotes the convolution, K(x) = 1-3cos(2)theta/vertical bar x vertical bar(3) and theta = theta(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 epsilon small, and describe the concentration phenomena of ground states as epsilon -> 0. We also investigate the relationship between the number of positive solutions and the profile of the potential V.

Automated summary

This paper investigates the existence of ground states for the singularly perturbed Gross-Pitaevskii equation with small ε, and describes the concentration phenomena of ground states as ε approaches 0. The relationship between the number of positive solutions and the profile of the potential V is also explored.