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July/August 2018 Positive solutions of indefinite semipositone problems via sub-super solutions
Uriel Kaufmann, Humberto Ramos Quoirin
Differential Integral Equations 31(7/8): 497-506 (July/August 2018).

Abstract

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem \[ \left\{ \begin{array} [c]{lll} -\Delta u=\lambda m(x)(f(u)-k) & \mathrm{in} & \Omega,\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where $\lambda,k>0$ and $f$ is either sublinear at infinity with $f(0)=0$, or $f$ has a singularity at $0$. We prove the existence of a positive solution for certain ranges of $\lambda$ provided that the negative part of $m$ is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.

Citation

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Uriel Kaufmann. Humberto Ramos Quoirin. "Positive solutions of indefinite semipositone problems via sub-super solutions." Differential Integral Equations 31 (7/8) 497 - 506, July/August 2018.

Information

Published: July/August 2018
First available in Project Euclid: 11 May 2018

zbMATH: 06890401
MathSciNet: MR3801821

Subjects:
Primary: 35B09, 35J25, 35J60

Rights: Copyright © 2018 Khayyam Publishing, Inc.

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Vol.31 • No. 7/8 • July/August 2018
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