Vortex solutions for pseudo-relativistic Hartree equations

Abstract

In this paper, we study k-vortex solutions of the form of the pseudo-relativistic Hartree equation 1 iψt(x,t)=(−Δ+m2−m)ψ(x,t)−(|x|−1∗|ψ|2)ψ(x,t), (x,t)∈R3×R, under the constraint Such solutions are obtained as minimizers of the problem 2 ek(N)=inf{Ek(u):u∈Hs∖{0},∫ℝ3|u(x,0)|2dx=N>0} with the associated functional of (1). We show that there is a threshold value such that problem (2) admits a nonnegative minimizer u N if , and there exists no minimizer for if . Moreover, the stability of the vortex solution is considered, and the limiting behavior of the minimizer u N as is described.