## Differential and Integral Equations

### Existence of multiple positive solutions for a nonlinear elliptic problem with the critical exponent and a Hardy term

#### Abstract

In this paper, we show that if $\mu>0$ is small enough, the problem \begin{equation*} \left\{ \begin{aligned} -\Delta u -\mu\frac{u}{|x|^2} & =|u|^{2^\ast-2}u & & \text{in $\Omega$,}\\ u & =0 & & \text{on $\partial\Omega$} \end{aligned} \right. \end{equation*} has at least cat $\Omega -1$ positive solutions, where $\Omega$ is a noncontractible, bounded domain in $\mathbb R^N (N\geq 4)$ such that its boundary is smooth and $0 \in \Omega$.

#### Article information

Source
Differential Integral Equations, Volume 21, Number 9-10 (2008), 971-980.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356038595

Mathematical Reviews number (MathSciNet)
MR2483344

Zentralblatt MATH identifier
1224.35134

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B33: Critical exponents 47J30: Variational methods [See also 58Exx]

#### Citation

Hirano, Norimichi; Shioji, Naoki. Existence of multiple positive solutions for a nonlinear elliptic problem with the critical exponent and a Hardy term. Differential Integral Equations 21 (2008), no. 9-10, 971--980. https://projecteuclid.org/euclid.die/1356038595