On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory

We study a system of elliptic equations from the Abelian Born-Infeld system coupled with the Einstein equations under the boundary condition of symmetric vacuum (nontopological type). When the total string number satisfies 1 ? N< 1/4? G, where Gis the gravitational constant, we construct a family of solutions to the system. The qualitative properties of the solutions are quite different from the solutions with the boundary condition of broken vacuum symmetry.