## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 14, Number 1 (2007), 15-24.

### On an Elliptic Equation Involving a Kirchhoff Term and a Singular Perturbation

#### Abstract

In this paper we consider the existence of positive solutions for the following class of singular elliptic nonlocal problems of Kirchhoff-type $$ \left\{\begin{array}{rclcc} -M(\|u\|^{2})\Delta u = \frac{h(x)}{u^{\gamma}}+k(x)u^{\alpha} \mbox{in} \Omega ,\\ u > 0 \mbox{in} \Omega ,\\ u = 0 \mbox{on} \partial\Omega ,\\ \end{array} \right. $$ where $\Omega \subset \mathbb R^{N}, N \geq 2,$ is a bounded smooth domain, $M:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function and $\|u\|^{2}=\int_{\Omega}|\nabla u|^{2}$ is the usual norm in $H^{1}_{0}(\Omega )$. The main tools used are the Galerkin method and a Hardy-Sobolev inequality.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 14, Number 1 (2007), 15-24.

**Dates**

First available in Project Euclid: 2 March 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1172852241

**Mathematical Reviews number (MathSciNet)**

MR2322319

**Zentralblatt MATH identifier**

1126.35031

**Subjects**

Primary: 34B15: Nonlinear boundary value problems 34B16, 35J65: Nonlinear boundary value problems for linear elliptic equations

**Keywords**

Kirchhoff equation Galerkin method Hardy-Sobolev inequality

#### Citation

Corrêa, Francisco Julio S.A. On an Elliptic Equation Involving a Kirchhoff Term and a Singular Perturbation. Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 15--24. https://projecteuclid.org/euclid.bbms/1172852241