  The present paper is devoted to analysis of the lack of compactness of
bounded sequences in \emph{inhomogeneous} Sobolev spaces, where bounded
sequences might fail to be compact due to an isometric group action, that is,
\emph{translation}. It will be proved that every bounded sequence $(u_n)$ has
(possibly infinitely many) \emph{profiles}, and then the sequence is
asymptotically decomposed into a sum of translated profiles and a
double-suffixed 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 $(u_n)$ and $G$.
One also finds that a decomposition of the Sobolev norm into profiles is
bounded by the supremum of the norm of $u_n$. Moreover, the profile
decomposition leads to results of decomposition of integral 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.
