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Existence and regularity of minimizers in elliptic free boundary problemshave been extensively studied in the literature. The corresponding study ofhigher critical points was recently initiated in Jerison and Perera [30, 31].In particular, the existence and nondegeneracy of a mountain pass point in asuperlinear and subcritical free boundary problem related to plasma confinementwas proved in [30]. In this paper we study ground states of a critical freeboundary problem related to the Brezis-Nirenberg problem [5]. We extend theresults of [30] to this problem by combining the method introduced there withthe concentration compactness principle to overcome the difficulties arisingfrom lack of compactness.

Existence and nondegeneracy of ground states in critical free boundary problems