A PRIORI ESTIMATES FOR ELLIPTIC PROBLEMS WITH A STRONGLY SINGULAR GRADIENT TERM AND A GENERAL DATUM

In this paper we show approximation procedures for studying singular elliptic problems whose model is {-Delta u = b(u) vertical bar(Sic)u vertical bar(2) + f(x), in Omega; u = 0, on partial derivative Omega; where b(u) is singular in the u-variable at u = 0, and f epsilon L-m(Omega), with m > N/2, is a function that does not have a constant sign. We will give an overview of the landscape that occurs when different problems (classified according to the sign of b(s)) are considered. So, in each case and using different methods, we will obtain a priori estimates, prove the convergence of the approximate solutions, and show some regularity properties of the limit.