Electron. J. Diff. Equ., Vol. 2013 (2013), No. 155, pp. 1-6.

Infinite semipositone problems with indefinite weight and asymptotically linear growth forcing-terms

Ghasem A. Afrouzi, Saleh Shakeri

Abstract:
In this work, we study the existence of positive solutions to the singular problem
$$\displaylines{ -\Delta_{p}u = \lambda m(x)f(u)-u^{-\alpha} \quad \hbox{in }\Omega,\cr u = 0 \quad \hbox{on }\partial \Omega, }$$
where $\lambda$ is positive parameter, $\Omega $ is a bounded domain with smooth boundary, $ 0 <\alpha<1 $, and $ f:[0,\infty] \to\mathbb{R}$ is a continuous function which is asymptotically p-linear at $\infty $. The weight function is continuous satisfies $m(x)>m_0>0$, $\|m\|_{\infty}<\infty$. We prove the existence of a positive solution for a certain range of $\lambda$ using the method of sub-supersolutions.

Submitted September 17, 2012. Published July 8, 2013.
Math Subject Classifications: 35J55, 35J25.
Key Words: Infinite semipositone problem; indefinite weight; forcing term; asymptotically linear growth; sub-supersolution method.

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Ghasem Alizadeh Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences
University of Mazandaran, Babolsar, Iran
email: afrouzi@umz.ac.ir
Saleh Shakeri
Department of Mathematics
Faculty of Mathematical Sciences
University of Mazandaran, Babolsar, Iran
email: s.shakeri@umz.ac.ir

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