  In this note we construct self-dual cosmic strings from a gauge field theory
with a generalized linear formation of potential energy density. By integrating
the Einstein equation, we obtain a nonlinear elliptic equation which is equal
with the sources. We prove the existence of a solution in the broken symmetry
category on the full plane and the multiple string solutions are valid under a
sufficient condition imposed only on the total string number N. The technique
of upper-lower solutions and the method of regularization are employed to show
the existence of a solution when there are at least two distant string centers.
When all the string centers are identical, fixed point theorem are used to
study the properties of the nonlinear elliptic equation. Finally, We give the
sharp asymptotic estimate for the solution at infinity.
