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Existence and uniqueness of a positive solution to a rapidly growing problem via sub-supersolution method
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 13 November 2025, pp. 1-15
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In this paper, we study the validity of the sub-supersolution method for the equation
where
\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*}
$N \geq 3$, 
$K(x)=exp(|x|^{\alpha}/4)$, 
$\alpha\geq 2$ 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.
