A Local Linking Theorem for Relativistic Action Functionals

We establish an analogue of the Brezis-Nirenberg local linking theorem for a class of Szulkin-type functionals arising from relativistic action principles.

In this framework, compactness of Palais-Smale sequences is formulated with respect to a topology induced by the effective domain of the functional, replacing the classical strong Palais-Smale condition.

The proof combines the original construction of the min-max geometry, based on a negative gradient flow, with the Ekeland-Lasry regularization.

The main difficulty is that the regularized functional is naturally associated with the strong topology of the underlying functional space, whereas compactness for the original functional is formulated in the topology induced by the effective domain.

We overcome this obstacle through a new perturbative construction that recovers the required min-max structure.

We apply our abstract multiplicity result to two representative relativistic models: the Lorentz force equation, describing the dynamics of a charged particle in an electromagnetic field, and the Dirichlet problem for the prescribed mean curvature operator in Minkowski space.

As a consequence, under natural assumptions, each problem admits at least two non-constant solutions.