Existence and Limiting Profiles of Boosted Ground States for the Pseudo-Relativistic Schrodinger Equation with Focusing Power Type Nonlinearity

We study the following constrained minimization problem d(a,q)(1) := inf(phi is an element of S1) Ea,q(phi), (0.1) where the boosted energy functional E-a,E-q(phi) is given by E-a,E-q(phi) := 1/2 integral(3)(R) (phi) over bar(root-Delta + m(2) - m)phi dx + i/2 integral(R3)(phi) over bar (v center dot del)phi dx - a/q+2 integral(R3)vertical bar phi vertical bar(q+2)dx, and phi is a complex function, the parameters m, a > 0, v is an element of R-3 with vertical bar v vertical bar < 1, and the constraint S-1 is defined as S-1 := {phi is an element of H-1/2(R-3) : integral(R3)vertical bar phi vertical bar(2)dx = 1}. We prove that the problem Eq. 0.1 has at least one minimizer if (a,q) is an element of D :=(0, +infinity) x (0, 2/3]\[a*, +infinity) x 2/3 for some a* > 0, and there is no minimizer if (a,q) is an element of (0, +infinity) x (0, +infinity)\D. Moreover, we analyse the asymptotic behavior of minimizers as (a, q) is an element of D -> (a(0), q(0)) is an element of D, and find that the minimizer of d(a,q)(1) converges to some minimizer of d(a0,q0)(1) in H-1/2(R-3). In addition, when (a, q) is an element of D -> ((sic), 2/3) with (sic) is an element of [a*, +infinity), we show that all minimizers must blow up and present the detailed asymptotic behavior of minimizers.