Abstract: | Let be a domain of N. We study the infimum 1(h) of the functional ![int](/content/tp4gu642g887p422/xxlarge8747.gif) | u|p+h–pV(x)|u|p dx in W1,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). |