  We consider a Schr\"odinger operator H with a non-vanishing radial magnetic
field B=dA and Dirichlet boundary conditions on the unit disk. We assume growth
conditions on B near the boundary which guarantee in particular the compactness
of the resolvent of this operator. Under some assumptions on an additional
radial potential V the operator H + V has a discrete negative spectrum and we
obtain an upper bound on the number of negative eigenvalues. As a consequence
we get an upperbound of the number of eigenvalues of H smaller than any
positive value, which involves the minimum of B and the square of the L^2 -norm
of A(r)/r, where A(r) is the specific magnetic potential defined as the flux of
the magnetic field through the disk of radius r centerde in the origin.
