A Global Compactness type result for Palais–Smale sequences in fractional Sobolev spaces

Abstract

We extend the global compactness result by Struwe (1984) to any fractional Sobolev spaces H ̇ s(⌦), for 0 < s < N/2 and ⌦ ⇢ RN a bounded domain with smooth boundary. The proof is a simple direct consequence of the so-called profile decomposition of Gérard (1998)

Bifurcation properties for a class of fractional Laplacian equations in

“…Motivated by the fact that the fractional Laplacian appears in a lot of application, several authors have dedicated a special attention for problems involving this important operator. The reader can find some recent results in Alves and Miyagaki [1,2], Brändler, Colorado and Sánchez [10], Cabré and Sire [14], Caffarelli and Silvestre [15], Chang and Wang [16], Cotsiolis and Tavoularis [18], Dávila, del Pino and Wei [19], Dipierro, Palatucci and Valdinoci [21], Fall, Mahmoudi and Valdinoci [25], Felmer, Quaas and Tan [26], Palatucci and Pisante [30], Secchi [32], Silvestre [33] and their references. In the most part of the above references the variational method was used to show the existence of solution.…”

Classification of finite energy solutions to the fractional Lane–Emden–Fowler equations with slightly subcritical exponents

“…First of all, as mentioned before, the Struwe's concentration-compactness principle-type result (Step 1 in Introduction) can be obtained as in [2,31,48]. Besides the moving plane argument in Sect.…”

“…Then, it is not hard to draw analogous results to Lemma 2.2 (cf. [48]) and (2.18 The aim of this section is to obtain a sharp pointwise upper bound of solutions U to (2.4).…”

“…This renowned principle is found by Struwe [59] for Eq. (1.9), and recently extended to problem (1.1) by Almaraz [2] for s = 1 2 , and by Fang and González [31], and Palatucci and Pisante [48] for all 0 < s < 1 (in slightly different setting). It makes possible to decompose solutions {u n } n∈N of (1.1) as…”

Further Results on the Fractional Yamabe Problem: The Umbilic Case

“…On the other hand, compactness and asymptotic behavior results for Palais-Smale sequences for fractional Laplacian equations with critical nonlinearities such as (1.7) were considered in [17,18,7].…”