The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem

Abstract

In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p−laplacian in the whole R n .

Multiple solutions for a class of noncooperative critical nonlocal system with variable exponents

“…For such problems, the concentration-compactness principles introduced by Lions 19,20 and its variant at infinity [21][22][23] have played a decisive role in showing a minimizing sequence or a Palais-Smale sequence is precompact. By using these concentration-compactness principles or extending them to the Sobolev spaces with fractional order or variable exponents, many authors have been successful to deal with critical problems involving p-Laplacian or p(•)-Laplacian or fractional p-Laplacian, see, for example, other studies 15,[24][25][26][27][28][29][30][31][32][33][34][35][36][37] and references therein. Recently, Ho and Kim 38 proved the concentration-compactness principles for fractional Sobolev spaces with variable exponents and obtained the existence of many solutions for a class of critical nonlocal problems with variable exponents.…”

Existence of Small-Energy Solutions to Nonlocal Schrödinger-Type Equations for Integrodifferential Operators in ℝN

“…Lemma 6. Let {Z n } ∞ n=1 be a sequence of positive numbers satisfying recursion inequality [52]). Next, we show the following assertion, which is a regularity-type result via the De Giorgi technique and the localization method.…”

Three solutions for a nonlocal problem with critical growth

“…For the local case it can be found in [12,28]. For the non local case it follows similarly, see [10] for the details. Now, we can prove the Palais-Smale condition for the restricted functional.…”