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

Yves Belaud (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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).

Domaines: Equations aux dérivées partielles [math.AP]

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⟩