Precise Spectral Asymptotics for the Dirichlet Problem − u″(t) + g(u(t)) = λsinu(t) |
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Authors: | Tetsutaro Shibata |
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Affiliation: | Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, Japanf1shibata@mis.hiroshima-u.ac.jpf1 |
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Abstract: | We consider the nonlinear eigenvalue problem on an interval−u″(t)+g(u(t))=λsinu(t),u(t)>0,t∈I:=(−T,T),u(±T)=0,where λ > 0 is a parameter and T > 0 is a constant. It is known that if λ ? 1, then the corresponding solution has boundary layers. In this paper, we characterize λ by the boundary layers of the solution when λ ? 1 from a variational point of view. To this end, we parameterize a solution pair (λ, u) by a new parameter 0 < ?< T, which characterizes the boundary layers of the solution, and establish precise asymptotic formulas for λ(?) with exact second term as ? → 0. It turns out that the second term is a constant which is explicitly determined by the nonlinearity g. |
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