SINGULAR ELLIPTIC PROBLEMS WITH DIRICHLET OR MIXED DIRICHLET-NEUMANN NON-HOMOGENEOUS BOUNDARY CONDITIONS

Let Omega be a C-2 bounded domain in R-n such that partial derivative Omega = Gamma(1) boolean OR Gamma(2), where Gamma(1) and Gamma(2) are disjoint closed subsets of partial derivative Omega, and consider the problem -Delta u = g(. , u) in Omega, u = tau on Gamma(1), partial derivative u/partial derivative nu = eta on Gamma(2), where 0 <= t is an element of W-1/2,W-2(Gamma(1)), eta is an element of (H-0,Gamma 1(1), (Omega))', and g : Omega x(0,infinity) -> R is a nonnegative Caratheodory function. Under suitable assumptions on g and eta we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow g to be singular at s = 0 and also at x is an element of S for some suitable subsets S subset of (Omega) over bar. The Dirichlet problem -Delta u = g(., u) in Omega, u = sigma on partial derivative Omega is also studied in the case when 0 <= sigma is an element of W-1/2,W-2 (Omega).