Profile decomposition in Sobolev spaces and decomposition of integral functionals II: homogeneous case

Abstract

The present paper is devoted to a theory of profile decomposition for bounded sequences in \emph{homogeneous} Sobolev spaces, and it enables us to analyze the lack of compactness of bounded sequences. For every bounded sequence in homogeneous Sobolev spaces, the sequence is asymptotically decomposed into the sum of profiles with dilations and translations and a double suffixed residual term. One gets an energy decomposition in the homogeneous Sobolev norm. The residual term becomes arbitrarily small in the critical Lebesgue or Sobolev spaces of lower order, and then, the results of decomposition of integral functionals are obtained, which are important strict decompositions in the critical Lebesgue or Sobolev spaces where the residual term is vanishing.