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

Abstract In this paper, we study the validity of the sub-supersolution method for the equation \begin{equation*}\begin{cases}-\mbox{div}(K(x)\nabla u)=K(x)|x|^{\alpha-2}f(x,u) \,\mbox{in } {\mathbb{R}}^{N},\\u \gt 0 \,\mbox{in } {\mathbb{R}}^{N},\end{cases}\end{equation*} where $N \geq 3$ , $K(x)=exp(|x|^{\alpha}/4)$ , $\alpha\geq 2$ 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.