Existence and nondegeneracy of ground states in critical free boundary problems

Abstract

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 (2018, 0000). 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 Jerison and Perera (2018). In this paper we study ground states of a critical free boundary problem related to the Brézis–Nirenberg problem (1983). We extend the results of Jerison and Perera (2018) to this problem by combining the method introduced there with the concentration compactness principle to overcome the difficulties arising from lack of compactness.