Vortex solutions for pseudo-relativistic Hartree equations

In this paper, we study k-vortex solutions of the form psi(t; x) = e(i(mu t+k theta(x1,x2)))u(x) of the pseudo-relativistic Hartree equation i psi(iota)(x,t) = (root-Delta+m2 - m) psi(xt) -(vertical bar x vertical bar-1 * vertical bar psi(x,t), (x,t) is an element of R-3 x R, (1) under the constraint integral(R3) vertical bar u vertical bar(2) dx = N: Such solutions are obtained as minimizers of the problem e(k)(N) = inf {E-k(u) : u is an element of H-s \ {0}; integral(R3) vertical bar u(x; 0)vertical bar(2) dx = N > 0} (2) with the associated functional E-k(u) of (1). We show that there is a threshold value N-c(k) > 0 such that problem (2) admits a nonnegative minimizer u(N) if 0 < N < Nc(k), and there exists no minimizer for e(k)(N) if N >= N-c(k). Moreover, the stability of the vortex solution is considered, and the limiting behavior of the minimizer u(N) as N -> N-c(k)(-) is described.