GLOBAL EXISTENCE AND SCATTERING OF EQUIVARIANT DEFOCUSING CHERN-SIMONS-SCHRODINGER SYSTEM

In this paper, we consider the following equivariant defocusing Chern-Simons-Schrodinger system, i partial derivative(t)phi + Delta phi = 2m/r(2) A(theta)phi + A(0)phi + 1/r(2) A(theta)(2)phi - lambda vertical bar phi vertical bar(p-2)phi, partial derivative(r)A(0) = 1/r (m + A(theta)) vertical bar phi vertical bar(2), partial derivative(t)A(theta) = rIm((phi) over bar partial derivative(r)phi), partial derivative(r)A(theta) = --1/2 vertical bar phi vertical bar(2)r, A(r) = 0. where phi(t, x(1), x(2)) : R1+2 -> R is a complex scalar field, A mu(t , x(1), x(2)) : R1+2 -> R is the gauge field for mu = 0, 1, 2, A(r) = x(1)/vertical bar x vertical bar A(1) + x(2)/vertical bar x vertical bar A(2), A(theta) = - x(2)A(1) + x(1)A(2), lambda < 0 and p > 4. When p > 4, the system is in the mass supercritical and energy subcrtical range. By using the conservation law of the system and the concentration compactness method introduced in [17], we show that the solution of the system exists globally and scatters.