  We are concerned with the global bifurcation analysis of positive solutions
to free boundary problems arising in plasma physics. We show that in general,
in the sense of domain variations, the following alternative holds: either the
shape of the branch of solutions resembles the monotone one of the model case
of the two-dimensional disk, or it is a continuous simple curve without
bifurcation points which ends up at a point where the boundary density
vanishes. On the other hand, we deduce a general criterion ensuring the
existence of a free boundary in the interior of the domain. Application to a
classic nonlinear eigenvalue problem is also discussed.
