Differential and Integral Equations

Positive solutions of indefinite semipositone problems via sub-super solutions

Uriel Kaufmann and Humberto Ramos Quoirin

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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.

Article information

Source
Differential Integral Equations, Volume 31, Number 7/8 (2018), 497-506.

Dates
First available in Project Euclid: 11 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.die/1526004027

Mathematical Reviews number (MathSciNet)
MR3801821

Zentralblatt MATH identifier
06890401

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations 35J60: Nonlinear elliptic equations 35B09: Positive solutions

Citation

Kaufmann, Uriel; Quoirin, Humberto Ramos. Positive solutions of indefinite semipositone problems via sub-super solutions. Differential Integral Equations 31 (2018), no. 7/8, 497--506. https://projecteuclid.org/euclid.die/1526004027


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