## Differential and Integral Equations

### Existence and boundary behaviour of solutions for a nonlinear Dirichlet problem in the annulus

#### Abstract

In this paper, we mainly study the following semilinear Dirichlet problem $-\Delta u=q(x)f(u),\; u > 0,\;x\in \Omega ,$ $u_{|\partial \Omega }=0,$ where $\Omega$ is an annulus in $\mathbb{R}^{n},\;\big( n\geq 2\big) .$ The function $f$ is nonnegative in $\mathcal{C}^{1}(0,\infty )$ and $q\in \mathcal{C}_{loc}^{\gamma }(\Omega ),\;(0 < \gamma < 1),$ is positive and satisfies some required hypotheses related to Karamata regular variation theory. We establish the existence of a positive classical solution to this problem. We also give a global boundary behavior of such solution.

#### Article information

Source
Differential Integral Equations, Volume 30, Number 11/12 (2017), 929-946.

Dates
First available in Project Euclid: 1 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.die/1504231280

Mathematical Reviews number (MathSciNet)
MR3693992

Zentralblatt MATH identifier
06819585

#### Citation

Ben Makhlouf, Sonia; Zribi, Malek. Existence and boundary behaviour of solutions for a nonlinear Dirichlet problem in the annulus. Differential Integral Equations 30 (2017), no. 11/12, 929--946. https://projecteuclid.org/euclid.die/1504231280