In this work we consider the optimal control problem of a semilinear elliptic partial differential equation (PDE) with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated family of penalized problems, parametrized by epsilon > 0, whose solutions define a central path converging to the solution of the original problem. Our aim is to obtain an asymptotic expansion for the solutions of the penalized problems around the solution of the original problem. This approach allows us to recover some known error bounds for the logarithmic barrier approximation and to obtain some new ones for a rather general class of barrier functions. In this manner, we generalize the results of [F. Alvarez et al., Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems, Math. Program. Ser. A, to appear], which were obtained in the ordinary differential equation (ODE) framework.