Existence and uniqueness of a positive solution to a rapidly growing problem via sub-supersolution method

In this paper, we study the validity of the sub-supersolution method for the equation {equation*}{cases}-div(K(x)∇ u)=K(x)|x|α-2f(x,u) \,in {R}N,\ > 0 \,in {R}N,{cases}{equation*} where N ≥ 3 , K(x)=exp(|x|α/4) , α≥ 2 and f is a continuous function, with hypotheses that will be given later. We apply the method to cases where f is singular, where f behaves like a logistic function, showing in both cases the existence and uniqueness of a positive solution.