On Dirichlet problems with singular nonlinearity of indefinite sign

Let \(\Omega\) be a smooth bounded domain in \(\mathbb{R}^{N}\), \(N\geq1\), let \(K\), \(M\) be two nonnegative functions and let \(\alpha,\gamma>0\). We study existence and nonexistence of positive solutions for singular problems of the form \(-\Delta u=K\left( x\right) u^{-\alpha}-\lambda M\le...

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Bibliographic Details
Published in:arXiv.org 2015-03
Main Authors: Godoy, Tomás, Kaufmann, Uriel
Format: Article
Language:English
Subjects:
Dirichlet problem
Online Access:Get full text
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Summary:Let \(\Omega\) be a smooth bounded domain in \(\mathbb{R}^{N}\), \(N\geq1\), let \(K\), \(M\) be two nonnegative functions and let \(\alpha,\gamma>0\). We study existence and nonexistence of positive solutions for singular problems of the form \(-\Delta u=K\left( x\right) u^{-\alpha}-\lambda M\left( x\right) u^{-\gamma}\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\lambda>0\) is a real parameter. We mention that as a particular case our results apply to problems of the form \(-\Delta u=m\left( x\right) u^{-\gamma}\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(m\) is allowed to change sign in \(\Omega\).
ISSN:2331-8422