Differential and Integral Equations

Existence of positive solutions for singular Dirichlet problems

Norimichi Hirano and Naoki Shioji

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Abstract

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.

Article information

Source
Differential Integral Equations, Volume 14, Number 12 (2001), 1531-1540.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356123009

Mathematical Reviews number (MathSciNet)
MR1859920

Zentralblatt MATH identifier
1021.35031

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations

Citation

Hirano, Norimichi; Shioji, Naoki. Existence of positive solutions for singular Dirichlet problems. Differential Integral Equations 14 (2001), no. 12, 1531--1540. https://projecteuclid.org/euclid.die/1356123009


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