Existence and nondegeneracy of ground states in critical free boundary problems

TL;DR

This paper investigates the existence and nondegeneracy of ground states in a critical free boundary problem, extending previous results to a more complex setting using advanced mathematical techniques.

Contribution

It extends prior work on free boundary problems by establishing ground states for a critical case using concentration compactness methods.

Findings

Proved existence of ground states in the critical free boundary problem.

Established nondegeneracy of these ground states.

Extended previous results to a more challenging critical setting.

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 [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.