Document Type : Research Paper

Authors

• M. B. Ghaemi ¹
• M. Choubin ²

¹ Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

² Department of Mathematics, Faculty of Basic Sciences, Payame Noor University, Tehran, Iran

Abstract

We discuss the existence of a positive solution to the innite semipositone problem
$$\mathrm{\Delta }u=au-b{u}^{\gamma }-f(u)-\frac{c}{u^{\alpha }},x\in \mathrm{\Omega },u=0,x\in \partial \mathrm{\Omega },$$
where $\mathrm{\Delta }$ is the Laplacian operator,  $\gamma >1,\alpha \in (0,1),a,b$ and $c$ are positive constants, $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with smooth boundary $\partial \mathrm{\Omega }$, and $f:[0;1)\to \mathbb{R}$ is a continuous function such that $f(u)\to \mathrm{\infty }$ as $u\to \mathrm{\infty }$. Also we assume that there exist $A>0$ and $\beta >1$ such that $f(s)\le A{s}^{\beta }$, for all $s\ge 0$. We obtain our result via the method of sub- and supersolutions.

Keywords

• positive solution
• Innite semipositone
• Sub- and supersolutions