  Existence and regularity of minimizers in elliptic free boundary problems
have been extensively studied in the literature. The corresponding study of
higher critical points was recently initiated in Jerison and Perera [30, 31].
In particular, the existence and nondegeneracy of a mountain pass point in a
superlinear and subcritical free boundary problem related to plasma confinement
was proved in [30]. In this paper we study ground states of a critical free
boundary problem related to the Brezis-Nirenberg problem [5]. We extend the
results of [30] to this problem by combining the method introduced there with
the concentration compactness principle to overcome the difficulties arising
from lack of compactness.
