Existence of Electrically Charged Structures with Regular Center in Nonlinear Electrodynamics Minimally Coupled to Gravity

We address the question of correct description of Lagrange dynamics for regular electrically charged structures in nonlinear electrodynamics coupled to gravity. Regular spherically symmetric configuration satisfying the weak energy condition has obligatory de Sitter center in which the electric field vanishes while the energy density of electromagnetic vacuum achieves its maximal value. The Maxwell weak field limit L(F) --> F as r --> infinity requires vanishing electric field at infinity. A field invariant. evolves between two minus zero in the center and at infinity which makes a Lagrangian L(F) with nonequal asymptotic limits inevitably branching. We formulate the appropriate nonuniform variational problem including the proper boundary conditions and present the example of the spherically symmetric Lagrangian describing electrically charged structure with the regular center.