On regularity of the extremal solution of the Dirichlet problem for some semilinear elliptic equations of the second order

We consider the problem on the regularity of the extremal positive solution of the Dirichlet problem for superlinear elliptic equations with a positive parameter. We prove the smoothness of the extremal solution for a certain class of nonlinearities. In this class we distinguish a subclass of functions for which the extremal solution is classical in the space of any dimension