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

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