Asymptotic Estimates for a Variational Problem Involving a Quasilinear Operator in the Semi-Classical Limit - Archive ouverte HAL Access content directly
Journal Articles Annals of Global Analysis and Geometry Year : 2004

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

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Yves Belaud
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Abstract

Let Ω be a domain of ℝN. We study the infimum λ1(h) of the functional ∫Ω|∇u|p+h −p V(x)|u|p dx in W 1,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).
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Dates and versions

hal-03922178 , version 1 (04-01-2023)

Identifiers

  • HAL Id : hal-03922178 , version 1

Cite

Yves Belaud. Asymptotic Estimates for a Variational Problem Involving a Quasilinear Operator in the Semi-Classical Limit. Annals of Global Analysis and Geometry, 2004, 26, pp.271-313. ⟨hal-03922178⟩
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