Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

In this paper we study the existence, nonexistence and multiplicity of positive solutions for the family of problems (Delta) u = f_lambda (x,u), u in H^1_0 (Omega), where Omega is a bounded domain in mathbb{R}^N, N\geq 3 and lambda 0 is a parameter. The results include the well-known nonlinearities of the Ambrosetti-Brezis-Cerami type in a more general form, namely(lambda) a (x)u^q + b(x) u^p, where 0 leq q<1