The principle of concentration compactness in 𝒟1,p

In this paper, we consider the concentration-compactness principle in [Formula: see text], which is applied to treat the problems associated with the determination of extremal functions in functional inequalities. A more precise equality of [Formula: see text] is obtained under the conditions, that, [Formula: see text] (as stated in Theorem 3.1). It is an improvement of the limit inequality “[Formula: see text]” originally proposed by [P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoam. 1(1) (1985) 145–201] and [D. Smets, A concentration-compactness lemma with applications to singular eigenvalue problems, J. Funct. Anal. 167(2) (1999) 463–480]. Moreover, we establish that the additional term [Formula: see text] may only possess the “oscillation”, excluding the “concentration” and “translation to infinity”. Consequently, due to Brézis–Lieb Lemma, we reach [Formula: see text] if [Formula: see text] belongs to some regular space (e.g. [Formula: see text]). This follows the solvability of the minimization problem (2.2).