On positive solutions for a class of infinite semipositone problems

Document Type : Research Paper

Authors

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

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