### 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

Mathematical Reviews number (MathSciNet)
MR2122698

Zentralblatt MATH identifier
1284.35151

#### 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.