A Global Compactness type result for Palais-Smale sequences in fractional Sobolev spaces
aa r X i v : . [ m a t h . A P ] D ec A GLOBAL COMPACTNESS TYPE RESULTFOR PALAIS-SMALE SEQUENCESIN FRACTIONAL SOBOLEV SPACES
GIAMPIERO PALATUCCI AND ADRIANO PISANTE
Abstract.
We extend the global compactness result by M. Struwe ([17]) to anyfractional Sobolev spaces ˙ H s (Ω), for 0 < s < N/ ⊂ R N a bounded domainwith smooth boundary. The proof is a simple direct consequence of the so-called profile decomposition of P. Gerard ([9]). To appear in
Nonlinear Anal. Introduction
Since the seminal paper [17], global compactness properties for Palais-Smale se-quences in the Sobolev space H have become very important tools in NonlinearAnalysis which have been crucial in many existence results, e. g. for ground states andblow-up solutions for nonlinear Schr¨odinger equations, for solutions of Yamabe-typeequations both in conformal and in CR geometry, for prescribing Q -curvature prob-lems, etc... Together with the aforementioned examples concerning single equationsin a scalar unknown function, more difficult systems of PDEs, often related to othergeometric problems, share similar compactness properties for their solutions; for in-stance, this is the case for parametric surfaces of constant mean curvature, harmonicmaps from Riemann surfaces into Riemannian manifolds, Yang-Mills connections overfour-manifolds, pseudo-holomorphic curves into symplectic manifolds, planar Todasystems, etc... The involved literature is really too wide to attempt any reasonableaccount here.In the present note, we aim to extend the global compactness result by M. Struwefor semilinear elliptic equation in H to the case of fractional Sobolev spaces ˙ H s of anyreal differentiability order 0 < s < N/ profile decomposition first obtained in [9] (see also [13] for an alternative slightly simpler and more abstractapproach).Let N ≥ < s < N/ H s ( R N ) the usual L -based homo-geneous fractional Sobolev spaces defined via Fourier transform as the completion of Mathematics Subject Classification.
Primary 35J60, 35C20, 35B33, 49J45.
Key words and phrases.
Profile decomposition, global compactness, fractional Sobolev, criticalSobolev exponent. For further details on the fractional Sobolev spaces, we refer to [4] and the references therein. C ∞ ( R N ) under the norm(1.1) k u k H s = Z R N | ξ | s | ˆ u ( ξ ) | d ξ . In view of the well known critical Sobolev embedding ˙ H s ֒ → L ∗ , where 2 ∗ = 2 N/ ( N − s ) is the critical Sobolev exponent, one has equivalently,˙ H s ( R N ) = n u ∈ L ∗ ( R N ) s. t. ( − ∆) s/ u ∈ L ( R N ) o , where, by definition, (( − ∆) s/ u ) b ( ξ ) := | ξ | s ˆ u ( ξ ).Let Ω be a bounded domain in R N with smooth boundary, N ≥
1, and define thehomogeneous Sobolev space ˙ H s (Ω) as the completion of C ∞ (Ω) under the norm (1.1),hence a closed subspace of ˙ H s ( R N ). Thus, we have a well defined fractional Laplacian( − ∆) s : ˙ H s (Ω) → ( ˙ H s (Ω)) ′ which is a bounded linear operator (isomorphism) givenby (( − ∆) s u ) b ( ξ ) := | ξ | s ˆ u ( ξ ), so that h v, ( − ∆) s u i H,H ′ = ( v, u ) H for any u, v ∈ H =˙ H s (Ω).Since in the entire space ˙ H s ֒ → L with compact embedding, we also have˙ H s (Ω) ֒ → L (Ω) ֒ → ( ˙ H s (Ω)) ′ , both with compact embedding. As a consequence,there is a well defined first eigenvalue λ = min (cid:8) k u k H s u ∈ ˙ H s (Ω) , k u k L = 1 (cid:9) ,with λ = λ (Ω) > s ≤
1; see, e. g., [6, Theorem 4.2]). Similarly, one has an increas-ing sequence of positive eigenvalues (repeated with multiplicities) going to infinity0 < λ ≤ λ ≤ . . . and corresponding eigenfunctions v , v , . . . giving an orthogonalbase both of L (Ω) and of ˙ H s (Ω), so that ( − ∆) s v k = λ k v k in ( ˙ H s (Ω)) ′ for any integer k ≥
1, i. e. ( v k , u ) H = λ k ( v k , u ) L for any u ∈ ˙ H s (Ω). Indeed it is enough to write( u, v ) L = ( Ku, v ) ˙ H s for some K ∈ L ( ˙ H s ) which is compact and self-adjoint and applythe spectral theorem.For any fixed λ ∈ R , consider the following nonlinear problem(P λ ) ( − ∆) s u − λu − | u | ∗ − u = 0 in ( ˙ H s (Ω)) ′ , i. e. the Euler-Lagrange equation d E ( u ) = 0 corresponding to the differentiable func-tional(1.2) E ( u ) = 12 Z R N | ( − ∆) s u | d x − λ Z Ω | u | d x − ∗ Z Ω | u | ∗ d x . It is worth noticing that when λ < λ , although the functional possess the MountainPass geometry (arguing as in [18], Chapter II, Section 6), the celebrated Mountain Passlemma does not apply because the Palais-Smale condition fails. More generally, when λ k < λ < λ k +1 the functional has a linking geometry (using the spectral decompositionabove and arguing again as in [18], Chapter II, Section 8) but the usual minimaxscheme still cannot be applied for the same reason. As it is well known when s = 1,this is due to the presence of a limiting nonlinearity in (1.2) and it is related to thelack of compactness for the associated critical Sobolev embedding ˙ H s ֒ → L ∗ , which isa consequence of the invariance of the ˙ H s - and L ∗ -norms with respect to the scaling(1.3) u ( · ) ˜ u x ,η ( · ) = η s − N u (cid:18) · − x η (cid:19) , for arbitrarily fixed η > x ∈ R N . LOBAL COMPACTNESS IN FRACTIONAL SOBOLEV SPACES 3
In the seminal paper [3] the authors circumvent this difficulty proving that, for s = 1, a local ( P S )-condition holds for λ < λ small enough. Soon after a decisivebreakthrough was obtained in [17], still in the local case s = 1, describing the precisemechanism responsible for the lack of the ( P S )-condition; i. e., in Author’s words,proving that compactness for Palais-Smale sequences holds “apart from jumps of thetopological type of admissible functions” , a sense we will make precise below. Thismajor advance paved the way to several extensions and to a huge number of applica-tions, e. g. in the case of problems involving biharmonic and polyharmonic operatorsbut also in other more complicated problems (see e. g. [18], [19] and the referencestherein).In order to state precisely our main result, consider the following limiting problem(P ) ( − ∆) s u − | u | ∗ − u = 0 in ( ˙ H s (Ω )) ′ , where Ω either the whole R N or a half-space; i. e. the Euler-Lagrange equation d E ∗ ( u ) = 0 corresponding to the energy functional E ∗ : ˙ H s (Ω ) → R ,(1.4) E ∗ ( u ) = 12 Z R N | ( − ∆) s u | d x − ∗ Z Ω | u | ∗ d x. We have the following extension of the result in [17], describing Palais-Smale sequencesfor (1.2) in the full range 0 < s < N/ Theorem 1.1.
Let E and E ∗ the functionals defined by (1.2) and (1.4) , respectively.Let { u n } be a sequence in ˙ H s (Ω) such that E ( u n ) ≤ c and (1.5) d E ( u n ) −→ n →∞ in ( ˙ H s (Ω)) ′ . Then, there exists a (possibly trivial) solution u (0) to (P λ ) such that, for a renumberedsubsequence of { u n } , we have u n ⇀ n →∞ u (0) in ˙ H s (Ω) . Moreover, either the convergence is strong, or there exist a finite set J = { , , ..., J } ,nontrivial solutions { u ( j ) } j ∈J to (P ) either in half spaces or in the entire space, u ( j ) ∈ ˙ H s (Ω ( j )0 ) , finitely many sequences of numbers { λ ( j ) n } j ∈J ⊂ (0 , ∞ ) converging tozero, and finitely many sequences of points { x ( j ) n } j ∈J ⊂ Ω such that, for a renumberedsubsequence of { u n } , we also have for any j ∈ J u ( j ) n ( · ) := λ ( j ) n N − s u n ( x ( j ) n + λ ( j ) n · ) ⇀ n →∞ u ( j ) ( · ) in ˙ H s ( R N ) In addition (1.6) u n ( · ) = u (0) ( · ) + J X j =1 λ ( j ) n s − N u ( j ) · − x ( j ) n λ ( j ) n ! + o (1) in ˙ H s ( R N ) , and (1.7) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log λ ( i ) n λ ( j ) n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ( i ) n − x ( j ) n ) λ ( i ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −→ n →∞ ∞ for i = j, i, j ∈ J (1.8) k u n k H s = J X j =0 k u ( j ) k H s + o (1) as n → ∞ , G. PALATUCCI AND A. PISANTE (1.9) E ( u n ) = E ( u (0) ) + J X j =1 E ∗ ( u ( j ) ) + o (1) as n → ∞ . In the local case s = 1, the original proof of Global Compactness in [17] consists ina subtle analysis concerning how the Palais-Smale condition fails for the functional E ,based on rescaling arguments, used in an iterated way to extract convergent subse-quences with nontrivial limit, together with some slicing and extension procedures onthe sequence of approximate solutions to (P λ ). That proof is difficult to extend evento the case s = 2 as was done in [8] and [10] (see, also, the forthcoming paper [7]for the analog on asymptotically hyperbolic Riemannian manifolds). The aforemen-tioned tools seem even more cumbersome to adapt to the more general frameworkpresented here and further difficulties appear due to the nonlocal structure of theinvolved fractional Sobolev spaces ˙ H s and the corresponding equations when s is notinteger.On the contrary, in the present note, we show how to deduce the results in Theo-rem 1.1 in a simple way from the so-called profile decomposition for bounded sequencesin ˙ H s spaces. The profile decomposition, as first proved by P. G´erard in [9], repre-sents a far-reaching functional analytic generalization of the aforementioned resultpreviously known only for Palais-Smale sequences and when s is an integer. Whileits essence is a somewhat general asymptotic orthogonal decomposition property, par-ticularly transparent e. g. in abstract Hilbert space setting (see [19, Chapter 3]), ithas been generalized to a wide range of Banach function spaces through wavelet de-composition (see [1] and the references therein). In view of its extreme flexibility, ithas become a common decisive tool in the study of properties of solutions of manyevolution equations and related issues.In the following, we state such result in the form as presented in [13, Theorem 4],whose proof has been derived by the authors combining a refinement of the criti-cal Sobolev inequality with remainder in Morrey spaces with the abstract Hilbertianapproach to profile decomposition in terms of dislocation spaces developed in [19]. Theorem 1.2.
Let { u n } be a bounded sequence in ˙ H s ( R N ) . Then, there exist a ( at most countable ) set I , a family of profiles { ψ j } ⊂ ˙ H s ( R N ) , a family of points { x ( j ) n } ∈ R N and a family of numbers { λ ( j ) n } ⊂ (0 , ∞ ) , such that, for a renumberedsubsequence of { u n } , we have (1.10) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log λ ( i ) n λ ( j ) n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ( i ) n − x ( j ) n ) λ ( i ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −→ n →∞ ∞ for i = j, i, j ∈ I , (1.11) u n ( x ) = X j ∈ I λ ( j ) n s − N ψ j x − x ( j ) n λ ( j ) n ! + r n ( x ) , where (1.12) lim n →∞ k r n k L ∗ = 0 , (1.13) and k u n k H s = X j ∈ I k ψ j k H s + k r n k H s + o (1) as n → ∞ . LOBAL COMPACTNESS IN FRACTIONAL SOBOLEV SPACES 5
Armed with such a strong characterization, the proof of Theorem 1.1 goes as follows.After having checked that the (
P S )-sequence { u n } is bounded in ˙ H s , we analyze thecorresponding profile decomposition in the sense of Theorem 1.2. Thus, we prove thatthe weak limit solves (P λ ) and all the other nontrivial profiles ψ ( j ) solve the limitingequation (P ) either in the entire space or in suitable half-spaces. As a consequence theprofiles are in finite number. Finally, we can conclude the proof by showing that thesequence of the remainders { r n } vanishes strongly in the ˙ H s -norm, which is a simpleconsequence of the orthogonality of scaling parameters together with an iterated useof the celebrated Brezis-Lieb lemma from [2].Finally, it is worth pointing out that when s = 1 the case Ω being a half spacecannot occur. This is a consequence of the standard higher regularity properties ofsolutions, the Pohozaev identity and the unique continuation property, which willassure that no trivial solutions of the limiting problem in such a case do exist (seealso the footnote on Page 7 below). In the general case when s is not integer, thoughhigher regularity could be presumably deduced (see, for istance, the papers [11, 12]for the case 0 < s <
1) and remarkable Pohozaev-type identities has been recentlyestablished (see the papers by Ros-Oton and Serra [15, 16]), a unique continuationproperty is still missing e. g. in the case when s ∈ (0 ,
1) and it is unknown to hold evenfor s = 2 (see however [14] for the case s = 2 in which unique continuation property isobtained under extra assumptions on the derivates of the solutions on the boundary).In our opinion, the contribution of this note is twofold: we extend Struwe’s GlobalCompactness result for Palais-Smale sequences to any fractional Sobolev spaces ˙ H s ,hence establishing a tool which could be useful for further investigations in this non-local framework; moreover, since we derive it from profile decomposition, we obtainan alternative and somewhat simpler proof of such compactness result even in theclassical case of integer s = 1 , Proof of the extended Global Compactness
Before starting with the proof of Theorem 1.1, we briefly fix some further notations.In the following, we denote for brevity by h u, v i and ( u, v ) the natural ( ˙ H s ) ′ − ˙ H s duality pairing and the scalar product in ˙ H s associated to the norm (1.1), respectively.Also, we recall that for any 0 < s < N/ S ∗ , depending only on N and s , namely(2.1) k u k ∗ L ∗ ≤ S ∗ k ( − ∆) s u k ∗ L ∀ u ∈ C ∞ (Ω) , where 2 ∗ = 2 N/ ( N − s ) is the critical Sobolev exponent; by a density argument, thesame inequality is valid on ˙ H s (Ω).Finally, in the rest of the paper we denote by ˜ u x ,η the rescaling of the function u with scaling parameters ( x , η ) as in (1.3); when clear from the context we do notspecify explicitly the choices of η and x there.2.1. Proof of Theorem 1.1.
For the reader’s convenience, we divide the proof in afew steps.
G. PALATUCCI AND A. PISANTE
Step 1. The sequence { u n } is bounded in ˙ H s (Ω) . This is a straightforward conse-quence of the fact that { u n } is a Palais-Smale sequence for E . Indeed (cid:18) − ∗ (cid:19) Z Ω | u | ∗ d x = E ( u n ) − h d E ( u n ) , u n i ≤ | c | + o (1) as n → ∞ , and in turn, since Ω is bounded,(2.2) Z Ω | u n | d x ≤ | Ω | sN ( | c | + o (1)) / ∗ as n → ∞ . Thus k u n k H s = 2 E ( u n ) + λ Z Ω | u n | d x + 22 ∗ Z Ω | u | ∗ d x ≤ | c | + | λ | · | Ω | sN ( | c | + o (1)) / ∗ + N − sN ( | c | + o (1)) as n → ∞ , which gives the claim. Step 2. The weak limit (up to subsequences) u (0) solves (P λ ) . From the previous step,passing to a subsequence we can assume that u n ⇀ u (0) weakly in ˙ H s (Ω) as n → ∞ ,thus also strongly in L (Ω) so that | u n | ∗ − u n → | u (0) | ∗ − u (0) strongly in L (Ω),hence weakly in L (2 ∗ ) ′ (Ω). Thus, by weak continuity of d E , the limit function u (0) isa (possibly trivial) solution to (P λ ). If the convergence is strong we are done. Step 3. If the convergence is not strong then { u n } contains further profiles . We assumethat u n ⇀ u (0) only weakly in ˙ H s (Ω) as n → ∞ , thus we can apply Theorem 1.2to the sequence { u n } and then we obtain a profile decomposition for a renumberedsubsequence. We set, from now on, u ( j ) = ψ j for a nonempty set of indices J ⊂ N corresponding to profiles different from u (0) , if any. Indeed, recall that, by definitionof profile, for fixed j ∈ J there exist sequences { x ( j ) n } ⊂ R N and λ ( j ) n ⊂ (0 , ∞ ) suchthat u n ( x ( j ) n + λ ( j ) n · ) ⇀ u ( j ) ( · ) in ˙ H s ( R N ) as n → ∞ . Thus, if u (0) = 0 then it can beregarded as profile corresponding to the trivial scaling x (0) n ≡ λ (0) n ≡ J is alwaysnot empty, as it always contains profiles associated to nontrivial scalings (postponingto Step 6 the proof that J is also a finite set).Arguing by contradiction, if { u n } contains no profiles (cid:0) except possibly u (0) (cid:1) then u n → u (0) strongly in L ∗ (see e. g. [13, Proposition 1]). On the other hand k u n − u (0) k H s = ( u n , u n − u (0) ) − ( u (0) , u n − u (0) )= h d E ( u n ) , u n − u (0) i − h d E ( u (0) ) , u n − u (0) i + λ k u n − u (0) k L + Z Ω (cid:16) | u n | ∗ − u n − | u (0) | ∗ − u (0) (cid:17) (cid:16) u n − u (0) (cid:17) d x = o (1)as n → ∞ in view of (1.5), Step 2 and the strong convergence u n → u (0) in L ∗ . Thusthe contradiction shows that J 6 = ∅ , i. e. { u n } contains further profiles. Step 4. The scaling parameters from Theorem 1.2 satisfy λ ( j ) n → and dist (cid:0) x ( j ) n , Ω (cid:1) = LOBAL COMPACTNESS IN FRACTIONAL SOBOLEV SPACES 7 O ( λ ( j ) n ) as n → ∞ for any j ≥ . We argue by contradiction and we first assumethat, up to a subsequence, λ ( j ) n → ∞ . Since the convergence defining the profiles isactually strong in L and u ( j ) = 0, we have0 < Z R N | u ( j ) | d x ≤ lim inf n Z R N λ ( j ) n N − s | u n ( x n + λ ( j ) n · ) | d x = lim inf n λ ( j ) n − s Z Ω | u n | d x which, in view of (2.2), yields a contradiction. Thus λ ( j ) n ≤ c ( j ) for each n ≥ λ ( j ) n ≥ c j > n ≥
1. Then, upto subsequences, | x n | → ∞ , since otherwise we would clearly have u (0) = 0, whichis however impossible because the two profiles u (0) and u ( j ) , j ≥
1, must obtainedalong distinct sequence of scaling parameters satisfying (1.10). We conclude that | x n | → ∞ , but then u n ( x ( j ) n + λ ( j ) n · ) ⇀ n ≥ λ ( j ) n − (Ω − x n ), so they are eventually zero on each compact set in R N since Ω is bounded and λ ( j ) n ≥ c j >
0. Since u ( j ) = 0 we have a contradiction andtherefore λ ( j ) n → n → ∞ . Finally, assuming dist (cid:0) x ( j ) n , Ω (cid:1) /λ ( j ) n → ∞ possiblyalong a subsequence, as n → ∞ would yield dist (cid:16) , λ ( j ) n − (cid:0) Ω − x n (cid:1)(cid:17) → ∞ and still u n ( x ( j ) n + λ ( j ) n · ) ⇀ Step 5. The profiles u ( j ) solve (P ) either in a half space or in the entire space andone may assume { x n } ⊂ Ω . This claim follows by the assumption in (1.5) together with the invariance ofthe ˙ H s - and L ∗ -norm with respect to the scaling in (1.3). According to Step 4 wejust need to distinguish two cases, depending whether dist (cid:0) x ( j ) n , ∂ Ω (cid:1) = O ( λ ( j ) n ) as n → ∞ (Case I) or, up to a subsequence, dist (cid:0) x ( j ) n , ∂ Ω (cid:1) /λ ( j ) n → ∞ (Case II). Notethat the latter happens only when { x ( j ) n } ⊂ Ω stay inside the domain or approachesthe boundary “slower than λ ( j ) n ”.Suppose that u ( j ) n ( · ) := u n ( x ( j ) n + λ ( j ) n · ) ⇀ u ( j ) ( · ) in ˙ H s ( R N ) as n → ∞ where,as in Step 4, u ( j ) n ∈ ˙ H s (cid:0) λ ( j ) n − (Ω − x n ) (cid:1) . Note that because of the condition onthe scaling parameters, arguing as in the appendix of [10], λ ( j ) n − (Ω − x n ) → Ω ( j )0 as n → ∞ , where Ω ( j )0 is either an open half space because of the smoothness of ∂ Ω(Case I) or the entire space (Case II), in the sense that for any compact set K ⊂ Ω ( j )0 we have K ⊂ λ ( j ) n − (Ω − x n ) for all n large enough. Since for the complements λ ( j ) n − ( C Ω − x n ) → C Ω ( j )0 as n → ∞ , we also have u ( j ) = 0 a. e. on C Ω ( j )0 , hence u ( j ) ∈ ˙ H s (Ω ( j )0 ). In the full range 0 < s < N/ can be either the whole R N or a half space.Indeed, as mentioned in the introduction, when s = 1 we can exclude the existence of nontrivialsolutions to the limiting problems in the half space by Pohozaev identity and unique continuation.Similarly, as shown in [5, Theorem 1.5], this is still the case when dealing with nonnegative solutions in the range s ∈ (0 , s = 2 this possibility cannot be apriori excluded. G. PALATUCCI AND A. PISANTE
Fix φ ∈ C ∞ (Ω ( j )0 ). By using the invariance in (1.3), with η = λ ( j ) n and x o = x ( j ) n there, together with the previous convergence we have ˜ φ ∈ C ∞ (Ω) and we can write(2.3) h d E ( u n ) , λ ( j ) n s − N φ · − x n λ ( j ) n ! i = h d E ∗ ( u ( j ) n ) , φ i − λ (cid:16) λ ( j ) n (cid:17) s Z R N u ( j ) n φ d x. In view of (1.5) the left hand-side in (2.3) goes to zero as n → ∞ , since k ˜ φ k ˙ H s = k φ k ˙ H s . On the other hand, since u ( j ) n → u ( j ) in L as n → ∞ and φ is smoothand compactly supported, the last term in (2.3) vanish as n → ∞ because λ ( j ) n → E ∗ we infer h d E ∗ ( u ( j ) ) , φ i = 0for each φ ∈ C ∞ (Ω ( j )0 ) and arguing by density the claim follows. Finally, we claimthat even in Case I we may always assume { x ( j ) n } ⊂ Ω. Indeed, if we fix ¯ x ( j ) ∈ Ω ( j )0 and we set ¯ x ( j ) n = x ( j ) n + λ ( j ) n ¯ x ( j ) , then { ¯ x ( j ) n } ⊂ Ω for n large enough and ¯ u ( j ) n ( · ) := u n (¯ x ( j ) n + λ ( j ) n · ) = u n ( x ( j ) n + λ ( j ) n ( · + ¯ x ( j ) )) ⇀ u ( j ) ( · + ¯ x ( j ) ) in ˙ H s ( R N ) as n → ∞ .Thus, the claim follows taking ¯ x ( j ) n as new traslation parameters and ¯ u ( j ) ( · ) := u ( j ) ( · + ¯ x ( j ) ) as new profiles. Step 6. The profiles u ( j ) are in finite number. This is a consequence of (1.13)in Theorem 1.2. It will just suffice to show that the ˙ H s -energy of the nontrivialprofiles u ( j ) , j ≥
1, is uniformly bounded from below. For this, we can use theprevious step by testing (P ) with φ = u ( j ) . From the Sobolev inequality (2.1) we get k u ( j ) k H s = k u ( j ) k ∗ L ∗ ≤ S ∗ (cid:16) k u ( j ) k H s (cid:17) ∗ . As a consequence k u ( j ) k ˙ H s ≥ S ∗ − ∗ and the sum in (1.13) must be finite. Step 7. The sequence of remainders { r n } converges strongly to 0 in ˙ H s ( R N ) . In viewof the asymptotic orthogonality of the scaled profiles corresponding to (1.10) and theinvariance of the ˙ H s -norm we have k r n k H s = (cid:13)(cid:13)(cid:13) u n − u (0) (cid:13)(cid:13)(cid:13) H s + J X j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) λ ( j ) n s − N u ( j ) · − x ( j ) n λ ( j ) n !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s − u n − u (0) , J X j =1 λ ( j ) n s − N u ( j ) · − x ( j ) n λ ( j ) n ! + o (1)= (cid:13)(cid:13)(cid:13) u n − u (0) (cid:13)(cid:13)(cid:13) H s + J X j =1 (cid:13)(cid:13)(cid:13) u ( j ) (cid:13)(cid:13)(cid:13) H s + o (1) − J X j =1 (cid:16) u ( j ) n , u ( j ) (cid:17) + 2 J X j =1 u (0) , λ ( j ) n s − N u ( j ) · − x ( j ) n λ ( j ) n !! = (cid:13)(cid:13)(cid:13) u n − u (0) (cid:13)(cid:13)(cid:13) H s − J X j =1 Z R N | ˜ u ( j ) | ∗ d x + o (1) . (2.4) LOBAL COMPACTNESS IN FRACTIONAL SOBOLEV SPACES 9 where in the last equality we also used the fact that the profiles solve (P ) becauseof Step 5, hence k u ( j ) k H s = R | u ( j ) | ∗ d x for each j ∈ J . On the other hand, since u n → u (0) strongly in L as n → ∞ , arguing as in Step 3 we have k u n − u (0) k H s = Z Ω (cid:16) | u n | ∗ − u n − | u (0) | ∗ − u (0) (cid:17) (cid:16) u n − u (0) (cid:17) d x + o (1) . As in Step 2, we have also u n ⇀ u (0) weakly in L ∗ (Ω) and | u n | ∗ − u n ⇀ | u (0) | ∗ − u (0) weakly in L (2 ∗ ) ′ (Ω) as n → ∞ , therefore the previous equality becomes k u n − u (0) k H s = Z Ω | u n | ∗ d x − Z Ω | u (0) | ∗ d x + o (1) , as n → ∞ , which combined with (2.4) finally gives(2.5) lim n →∞ k r n k H s = lim n →∞ Z Ω | u n | ∗ d x − Z Ω | u (0) | ∗ d x − J X j =1 Z R N | ˜ u ( j ) | ∗ d x . Finally, it remains to show that(2.6) Z Ω | u n | ∗ d x −→ n →∞ Z Ω | u (0) | ∗ d x + J X j =1 Z R N | u ( j ) | ∗ d x. By (1.11), we can write u n = u (0) + P j λ ( j ) n s − N/ u ( j ) ( · − x ( j ) n /λ ( j ) n ) + r n , and if weuse iteratively the Brezis-Lieb lemma and the invariance of the L ∗ -norm togetherwith (1.12) the conclusion follows (alternatively, see [9, Equation (1.11)], which givesthe same property for the general case of countably many profiles). Step 8. Completion of the proof.
Clearly (1.7) comes from (1.10) and (1.6), (1.8)follow from (1.13) in view of Step 7. It remains to show the validity of (1.9). Recallthat u n → u (0) in L (Ω) and u (0) solves ( P λ ) because of Step 2. Recall also that k u ( j ) k H s = R | u ( j ) | ∗ d x because of Step 5. Since(2.7) E ( u n ) = 12 Z R N | ( − ∆) s u n | d x − λ Z Ω | u n | ∗ d x − ∗ Z Ω | u n | ∗ d x , combining (1.8) and (2.6) the conclusion follows easily as n → ∞ . (cid:3) Acknowledgements.
The authors would like to thank Marco Squassina for havingpointed out to them the paper [5]. The first author has been supported by PRIN 2010-2011 “Calcolo delle Variazioni”. The first author is member of Gruppo Nazionale perl’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of IstitutoNazionale di Alta Matematica “F. Severi” (INdAM).
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Dipartimento di Matematica e Informatica, Universit`a degli Studidi Parma, Campus - Parco Area delle Scienze, 53/a, 43124 Parma, Italy, andSISSA, Via Bonomea, 256, 34136 Trieste, Italy
E-mail address : [email protected] (Adriano Pisante) Dipartimento di Matematica, Sapienza Universit`a di Roma, P. le AldoMoro, 5, 00185 Roma, Italia
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