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Existence and regularity of minimizers in elliptic free boundary problemshave been extensively studied in the literature. We initiate the correspondingstudy of higher critical points by considering a superlinear free boundaryproblem related to plasma confinement. The associated energy functional isnondifferentiable, and therefore standard variational methods cannot be useddirectly to prove the existence of critical points. Here we obtain a nontrivialgeneralized solution $u$ of mountain pass type as the limit of mountain passpoints of a suitable sequence of $C^1$-functionals approximating the energy. Weshow that $u$ minimizes the energy on the associated Nehari manifold and usethis fact to prove that it is nondegenerate. We use the nondegeneracy of $u$ toshow that it satisfies the free boundary condition in the viscosity sense.Moreover, near any free boundary point that has a measure-theoretic normal, thefree boundary is a smooth surface, and hence the free boundary condition holdsin the classical sense.

Existence and regularity of higher critical points for elliptic free boundary problems