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Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

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Abstract

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu λ as λ→∞.

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References

  • [BH1] P. A. Binding and Y. X. Huang,The principal eigencurves for the p-Laplacian, Differential Integral Equations8 (1995), 405–414.

    MATH  MathSciNet  Google Scholar 

  • [BH2] P. A. Binding and Y. X. Huang,Bifurcation from eigencurves of the p-Laplacian, Differential Integral Equations8 (1995), 415–428.

    MATH  MathSciNet  Google Scholar 

  • [BK] A. I. Bobenko and S. B. Kuksin,The nonlinear Klein-Gordon equation on an interval as a perturbed Sine-Gordon equation, Comment. Math. Helv.70 (1995), 63–112.

    Article  MATH  MathSciNet  Google Scholar 

  • [B] H. Brezis,Analyse Fonctionnelle, Masson, Paris, 1983.

    MATH  Google Scholar 

  • [GNN] B. Gidas, W. M. Ni and L. Nirenberg,Symmetry and related properties via the maximum principle, Comm. Math. Phys.68 (1979), 209–243.

    Article  MATH  MathSciNet  Google Scholar 

  • [O] R. E. O'Malley, Jr.Phase-plane solutions to some singular perturbation problems, J. Math. Anal. Appl.54 (1976), 449–466.

    Article  MATH  MathSciNet  Google Scholar 

  • [RR] M. Renardy and R. C. Rogers,An Introduction to Partial Differential Equations, Springer, New York, 1993.

    MATH  Google Scholar 

  • [R1] B. P. Rynne,Bifurcation from eigenvalues in nonlinear multiparameter problems, Nonlinear Anal.15 (1990), 185–198.

    Article  MATH  MathSciNet  Google Scholar 

  • [R2] B. P. Rynne,Uniform convergence of multiparameter eigenfunction expansions, J. Math. Anal. Appl.147 (1990), 340–350.

    Article  MATH  MathSciNet  Google Scholar 

  • [S1] T. Shibata,Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm-Liouville problems, J. Analyse Math.66 (1995), 277–294.

    Article  MATH  MathSciNet  Google Scholar 

  • [S2] T. Shibata,Spectral asymptotics and bifurcation for nonlinear multiparameter elliptic eigenvalue problems, Bull. Belg. Math. Soc.3 (1996), 501–515.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Division of Mathematical and Information Sciences Faculty of Integrated Arts and Sciences, Hiroshima University, 739-8521, Higashi-Hiroshima, Japan

    Tetsutaro Shibata

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  1. Tetsutaro Shibata
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Correspondence to Tetsutaro Shibata.

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This research has been supported by the Japan Society for the Promotion of Science.

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Shibata, T. Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval. J. Anal. Math. 83, 109–120 (2001). https://doi.org/10.1007/BF02790258

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  • DOI: https://doi.org/10.1007/BF02790258

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