  For $\Omega\subset \mathbb{R}^2$ a smooth and bounded domain, we derive a
sharp universal energy estimate for non-negative solutions of free boundary
problems on $\Omega$ arising in plasma physics. As a consequence, we are able
to deduce new universal estimates for this class of problems. We first come up
with a sharp positivity threshold which guarantees that there is no free
boundary inside $\Omega$ or either, equivalently, with a sharp necessary
condition for the existence of a free boundary in the interior of $\Omega$.
Then we derive an explicit bound for the $L^{\infty}$-norm of non-negative
solutions and also obtain explicit estimates for the thresholds relative to
other neat density boundary values. At least to our knowledge, these are the
first explicit estimates of this sort in the superlinear case.
