Advances in Differential Equations

Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities

Norimichi Hirano, Claudio Saccon, and Naoki Shioji

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Abstract

We study the existence of multiple positive solutions of $ -\Delta u = \lambda u^{-q} +u^p $ in $\Omega$ with homogeneous Dirichlet boundary condition, where $\Omega$ is a bounded domain in $\mathbb R^N$, $\lambda >0$, and $0 < q \leq 1 < p \leq (N+2)/(N-2)$. We show by a variational method that if $\lambda$ is less than some positive constant then the problem has at least two positive, weak solutions including the cases of $q=1$ or $p=(N+2)/(N-2)$. We also study the regularity of positive weak solutions.

Article information

Source
Adv. Differential Equations Volume 9, Number 1-2 (2004), 197-220.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867973

Mathematical Reviews number (MathSciNet)
MR2099611

Zentralblatt MATH identifier
05054519

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B65: Smoothness and regularity of solutions 35J20: Variational methods for second-order elliptic equations 47J30: Variational methods [See also 58Exx]

Citation

Hirano, Norimichi; Saccon, Claudio; Shioji, Naoki. Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Differential Equations 9 (2004), no. 1-2, 197--220. https://projecteuclid.org/euclid.ade/1355867973.


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