Profile decomposition in Sobolev spaces and decomposition of integral functionals I: Inhomogeneous case

摘要

The present paper is devoted to analyzing the lack of com-pactness of bounded sequences in inhomogeneous Sobolev spaces, where bounded sequences might fail to be compact due to an isometric group action, that is, translation. It will be proved that every bounded sequence (un) has (possibly infinitely many) profiles, and then the sequence is asymp-totically decomposed into a sum of translated profiles and a double-suffix residual term, where the residual term becomes arbitrarily small in appropriate Lebesgue or Sobolev spaces of lower order. To this end, functional analytic frameworks are established in an abstract way by making use of a group action G, in order to characterize profiles by (un) with G. One also finds that a decomposition of the Sobolev norm into profiles is bounded by the supremum of the norm of un. Moreover, the profile decomposition leads to results of decomposition of inte-gral functionals of subcritical order. It is noteworthy that the space where the decomposition of integral functionals holds is the same as that where the residual term is vanishing.(c) 2022 Elsevier Inc. All rights reserved.