  In this paper we find out some new blow-up estimates for the positive
explosive solutions of a paradigmatic class of elliptic boundary value problems
of superlinear indefinite type. These estimates are obtained by combining the
scaling technique of Guidas-Spruck together with a generalized De Giorgi-Moser
weak Harnack inequality found, very recently, by Sirakov. In a further step,
based on a comparison result of Amann and L\'opez-G\'omez, we will show how
these bounds provide us with some sharp a priori estimates for the classical
positive solutions of a wide variety of superlinear indefinite problems. It
turns out that this is the first general result where the decay rates of the
potential in front of the nonlinearity $a(x)$ do not play any role for getting
a priori bounds for the positive solutions when $N\geq 3$.
