  Self-consistent solutions of the Ginzburg-Landau system of equations, which
describe the order parameter and the magnetic field distribution in a long
superconducting cylinder of finite radius R, in external magnetic field H, when
vortex line, carrying m flux quanta, is situated on the cylinder axis (a giant
m-vortex state), are studied numerically. If the field H exceeds some critical
value H_s, the giant m-vortex solution becomes unstable and passes to a new
stable edge-suppressed form. The quantum number m in this state does not
change, but the order parameter diminishes by a jump (almost to zero) near the
cylinder surface; however, superconductivity remains in the deep, at some
distance from a cylinder axis. This edge-suppressed state exist in the fields
H_s<H<H_c, where H_c is the field, in which the second order phase transition
into the normal state occures. (In the case of large radii R and finite m the
field H_c coincides with critical field H_{c2}, in which superconductivity
vanishes in the bulk.) If the vorticity m is large, the edge-suppressed state
degenerates into the usual state of surface superconductivity, and may survive
up to the field H_{c3}=1.69H_{c2}. The magnetic moment, total magnetic flux,
the Gibbs free energies are found, as functions of the field H, for different
radii R, vorticities m and parameters \kappa of GL-theory. The intervals of R,
\kappa, m are found, where the edge-suppressed solutions exist. The giant
m-vortex states exist in both type-I and type-II superconductors, but the
edge-suppressed solutions are possible only in type-II superconductors (with
\kappa > 1/sqrt{2}). The paramagnetic effect in mesoscopic samples and also the
possible connection of the theory and experiment are shortly discussed.
