Advances in Differential Equations

Multiple solutions for the Brezis-Nirenberg problem

Mónica Clapp and Tobias Weth

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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

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

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations 47J30: Variational methods [See also 58Exx]


Clapp, Mónica; Weth, Tobias. Multiple solutions for the Brezis-Nirenberg problem. Adv. Differential Equations 10 (2005), no. 4, 463--480.

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