Existence of positive solutions for a class of indefinite weight semilinear elliptic boundary value problems

The existence of positive classical solutions of the boundary value problems -?u = ?g(x) f(u) in O, (1-a)?u/?n+au = 0 on ?O, where ? and a are real parameters and O is an open bounded region of R^N, N=2 with smooth boundary ?O. It is assumed that a=1; thus a = 0 corresponds to the Neumann problem, a = 1 to the Dirichlet problem and 0<a<1 to the usual Robin problem. It is also assumed throughout that g: O¯?R is a smooth function which changes sign on O.