Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms

We establish results concerning non-existence and existence of entire positive solutions for the nonlinear elliptic problem {−Δpu=a(x)um+λb(x)unin RN,u>0in RN,u(x)⟶|x|→∞0, where −∞0λ>0 appropriately.