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