This paper presents a rigorous treatment on a two-dimensional problem of an elliptic hole or a crack in an infinite electrostrictive solid subjected to remote electric fields based on the complex variable method. Firstly, the general solutions are obtained for the electric fields inside the elliptical hole and the complex potentials in the solid, respectively. Secondly, numerical results of stresses around the hole are given in order to discuss the effect of the electric fields inside the hole on the fracture behaviors of the solid. Finally, explicit and closed-form solutions are obtained for an electrically permeable/impermeable crack when the hole degenerates to a crack. It is found that: (1) in general, for a permeable crack, the total stress may have a traditional r Γ1=2 type singularity at the crack tip under pure electric loads, and the applied electric loads may enhance or retard crack propagation, which is dependent on the Maxwell stresses on the crack faces and at infinity; (2) when the interior of the permeable crack is filled with the same medium as that at infinity, the magnitude of the Maxwell stress on the crack faces is equal but opposite to that on the solid surface at infinity, and as a result the applied electric field has no effects on crack growth; and (3) for an impermeable crack, the total stresses still have a r Γ1=2 type singularity at the crack tips under pure electric loads, and the applied electric field perpendicular to the crack surface may enhance crack propagation.