On the regularity of the extremal solution of some Dirichlet problem

We consider the problem of regularity of the extremal positive solution of the Dirichlet problem for superlinear elliptic equations with a strong non-linearity and a positive parameter. We prove the smoothness of the extremal solution for some classes of nonlinearities. These classes are generalizations of the model examples of the exponential and of the power functions. We pay a special attention to cases where the regularity of the extremal solution holds for the same dimensions as for the model cases.