Precise asymptotic formulas for variational eigencurves of semilinear two-parameter elliptic eigenvalue problems |
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Authors: | Tetsutaro Shibata |
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Institution: | (1) The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan, e-mail: shibata@mis.hiroshima-u.ac.jp, JP |
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Abstract: | We consider the two-parameter nonlinear eigenvalue problem?−Δu = μu − λ(u + u
p
+ f(u)), u > 0 in Ω, u = 0 on ∂Ω,?where p>1 is a constant and μ,λ>0 are parameters. We establish the asymptotic formulas for the variational eigencurves λ=λ(μ,α) as
μ→∞, where α>0 is a normalizing parameter. We emphasize that the critical case from a viewpoint of the two-term asymptotics
of the eigencurve is p=3. Moreover, it is shown that p=5/3 is also a critical exponent from a view point of the three-term asymptotics when Ω is a ball or an annulus. This sort
of criticality for two-parameter problems seems to be new.
Received: February 9, 2002; in final form: April 3, 2002?Published online: April 14, 2003 |
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Keywords: | Mathematics Subject Classification (2000) 35P30 two-parameter variational eigencurves |
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