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2001 Existence of positive solutions for singular Dirichlet problems
Norimichi Hirano, Naoki Shioji
Differential Integral Equations 14(12): 1531-1540 (2001). DOI: 10.57262/die/1356123009


We study the existence of positive solutions for singular Dirichlet problems. By the variational method, we show that if $\lambda_1$ is contained in some interval, then a singular Dirichlet problem has a radially symmetric, positive solution on an annulus $\Omega$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with homogeneous Dirichlet boundary condition. Since we consider a problem with singularity, we have difficulties that either the functional $I$ associated with the problem is not differentiable even in the sense of Gâteaux or the strong maximum principle is not applicable. But we show that a function which attains some minimax value for $I$ is a solution. We also treat the case that $\Omega$ is a ball.


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Norimichi Hirano. Naoki Shioji. "Existence of positive solutions for singular Dirichlet problems." Differential Integral Equations 14 (12) 1531 - 1540, 2001.


Published: 2001
First available in Project Euclid: 21 December 2012

zbMATH: 1021.35031
MathSciNet: MR1859920
Digital Object Identifier: 10.57262/die/1356123009

Primary: 35J65
Secondary: 35B05 , 35J20 , 35J60

Rights: Copyright © 2001 Khayyam Publishing, Inc.

Vol.14 • No. 12 • 2001
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