## Advances in Differential Equations

- Adv. Differential Equations
- Volume 10, Number 4 (2005), 463-480.

### Multiple solutions for the Brezis-Nirenberg problem

#### Abstract

We establish the existence of multiple solutions to the Dirichlet problem for the equation \[ -\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u \] on a bounded domain $\Omega$ of $\mathbb{R}^{N},$ $N\geq4.$ We show that, if $\lambda>0$ is not a Dirichlet eigenvalue of $-\Delta$ on $\Omega,$ this problem has at least $\frac{N+1}{2}$ pairs of nontrivial solutions. If $\lambda$ is an eigenvalue of multiplicity $m$ then it has at least $\frac{N+1-m}{2}$ pairs of nontrivial solutions.

#### Article information

**Source**

Adv. Differential Equations, Volume 10, Number 4 (2005), 463-480.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867873

**Mathematical Reviews number (MathSciNet)**

MR2122698

**Zentralblatt MATH identifier**

1284.35151

**Subjects**

Primary: 35J20: Variational methods for second-order elliptic equations

Secondary: 35J60: Nonlinear elliptic equations 47J30: Variational methods [See also 58Exx]

#### Citation

Clapp, Mónica; Weth, Tobias. Multiple solutions for the Brezis-Nirenberg problem. Adv. Differential Equations 10 (2005), no. 4, 463--480. https://projecteuclid.org/euclid.ade/1355867873