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

Let be a domain of ^N. We study the infimum ₁(h) of the functional |u|^p+h^–pV(x)|u|^p dx in W^1,p() for ||u||_LP()= 1 where h > 0 tends to zero and V is a measurable function on . When V is bounded, we describe the behaviour of ₁(h), in particular when V is radial and 'slowly' decaying to zero. We also study the limit of ₁(h) when h 0 for more general potentials and show a necessary and sufficient condition for ₁(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 ₁(h).