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New asymptotic formula for eigenvalues of nonlinear Sturm-Liouville problems

Authors:

Institution:

(1) Department of Applied Mathematics Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

Abstract:

We study the nonlinear Sturm-Liouville problem

$- u'(t) + f(u(t)) = \lambda u(t), u(t) > 0, t \in {\rm I}: = (0,1), u(0) = u(1) = 0,$

where λ > 0 is an eigenvalue parameter and f(u) is a rapidly increasing function. For better understanding of the global behavior of the bifurcation branch in R+ × L ²(I), we establish precise asymptotic formulas up to the third term for the eigenvalue λ(α) associated with the eigenfunction u _α with ‖u _α‖2 = α, as α → ∞. We show that there exists a new type of asymptotic formula for λ (α) as α → ∞.

Keywords:

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