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Asymptotic formulas and critical exponents for two-parameter nonlinear eigenvalue problems

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  • Tetsutaro Shibata

Abstract

We study the nonlinear two-parameter problem − u ″ ( x ) + λ u ( x ) q = μ u ( x ) p , u ( x ) > 0 , x ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . Here, 1 < q < p are constants and λ , μ > 0 are parameters. We establish precise asymptotic formulas with exact second term for variational eigencurve μ ( λ ) as λ → ∞ . We emphasize that the critical case concerning the decaying rate of the second term is p = ( 3 q − 1 ) / 2 and this kind of criticality is new for two-parameter problems.

Suggested Citation

  • Tetsutaro Shibata, 2003. "Asymptotic formulas and critical exponents for two-parameter nonlinear eigenvalue problems," Abstract and Applied Analysis, Hindawi, vol. 2003, pages 1-14, January.
    • Handle: RePEc:hin:jnlaaa:490369
    DOI: 10.1155/S1085337503212045
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