The effect of the variational framework on the spectral asymptotics for two–parameter nonlinear eigenvalue problems

✍ Scribed by Tetsutaro Shibata

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 206 KB
• Volume: 248-249
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310012

✦ Synopsis

Abstract

We consider the nonlinear two–parameter problem

u″(x) + μu(x)^p^ = λu(x)^p^,

u(x) > 0, x ∈ I = (0, 1),

u(0) = u(1) = 0,

where 1 < q < p < 2__q__ + 3 and λ, μ > 0 are parameters. We establish the three–term spectral asymptotics for the eigencurve λ = λ(μ) as μ → ∞ by using a variational method on the general level set due to Zeidler. The first and second terms of λ(μ) do not depend on the relationship between p and q. However, the third term depends on the relationship between p and q, and the critical case is p = (3__q__ – 1)/2.