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Annals of Global Analysis and Geometry

, Volume 26, Issue 3, pp 271–313 | Cite as

Asymptotic Estimates for a Variational Problem Involving a Quasilinear Operator in the Semi-Classical Limit

  • Y. Belaud
Article

Abstract

Let Ω be a domain of ℝ N . We study the infimum λ1(h) of the functional ∫Ω|∇u| p +hpV(x)|u| p dx in W1,p(Ω) for ||u||LP(Ω)= 1 where h > 0 tends to zero and V is a measurable function on Ω. When V is bounded, we describe the behaviour of λ1(h), in particular when V is radial and 'slowly' decaying to zero. We also study the limit of λ1(h) when h→ 0 for more general potentials and show a necessary and sufficient condition for λ1(h) to be bounded. A link with the tunelling effect is established. We end with a theorem of existence for a first eigenfunction related to λ1(h).

nonlinear equation p-Laplacian semi-classical limits first eigenvalue tunelling effect 

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© Kluwer Academic Publishers 2004

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  • Y. Belaud

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