Due the migration of mobile molecules and ions, a thin diffusive layer of distributed charge - the electric double layer - forms at the interface between a polyelectrolyte gel and a liquid ionic solution.  When two polyelectrolyte gels are brought closely together, the electric double layers overlap and interact with each other, resulting in an effective repulsion.  The multiphysics coupling nature of soft gels makes their surface interactions significantly different from the interactions between rigid solids.  Using the recently formulated nonlinear theory, this paper develops a continuum model to study the surface interactions between two like-charged polyelectrolyte gels, accounting for the coupled electric, concentration, and deformation fields in both the gels and the liquid.  Numerical solutions of the surface interactions are obtained and compared with a qualitative scaling law derived via linearization.  The results suggest that the structure of double layers, as well as their interactions, depends not only on the concentration of liquid solutions, but more on the bulk properties of the gels such as stiffness and fixed-charge density.  This model also provides insights to the mechanism of the low-friction phenomena on the surface of a polyelectrolyte gel.