CHARACTERIZATION OF THE LACK OF COMPACTNESS OF H2 (R4) INTO THE ORLICZ SPACE rad INES BEN AYED AND MOHAMED KHALIL ZGHAL Abstract. This paper is devoted to the description of the lack of compactness of the Sobolev space H2 (R4) in the Orlicz space (R4). The approach that we rad L adopt to establish this characterization is in the spirit of the one adopted in the case of H1 (R2) into the Orlicz space (R2) in [5]. rad L 3 1 0 Contents 2 n 1. Introduction 1 a 1.1. Development in critical Sobolev embedding 1 J 1.2. Critical 4D Sobolev embedding 2 8 1 1.3. Lack of compactness in 4D critical Sobolev embedding in Orlicz space 3 1.4. Statement of the results 7 ] P 1.5. Structure of the paper 11 A 2. Proof of the main theorem 11 . h 2.1. Scheme of the proof 11 t 2.2. Preliminaries 12 a m 2.3. Extraction of the first scale and the first profile 13 [ 2.4. Conclusion 18 1 3. Appendix 20 v References 22 5 7 4 4 . 1. Introduction 1 0 1.1. Development in critical Sobolev embedding. Due to the scaling invari- 3 1 ance, the critical Sobolev embedding : v (1.1) H˙ s(Rd) ֒ Lp(Rd), i → X when 0 s < d and 1 = 1 s, is not compact. r ≤ 2 p 2 − d a After the pioneering works of P. Lions [13] and [14], P. G´erard described in [8] the lack of compactness of (1.1) by means of profiles in the following terms: a sequence (u ) bounded in H˙ s(Rd) can be decomposed, up to a subsequence extraction, on n n a finite sum of orthogonal profiles such that the remainder converges to zero in Lp(Rd) as the number of the sum and n tend to infinity. This question was later I. Ben Ayed & M.- K. Zghal are grateful to the Laboratory of PDE and Applications at the Faculty of Sciences of Tunis. 1 2 INES BEN AYEDAND MOHAMED KHALILZGHAL investigated by S. Jaffard in the more general case of Hs,q(Rd) ֒ Lp(Rd), 0 < s < d → p and 1 = 1 s by the use of nonlinear wavelet and recently in an abstract frame p q − d X ֒ Y including Sobolev, Besov, Triebel-Lizorkin, Lorentz, H¨older and BMO → spaces. (One can consult [7] and the references therein for an introduction to these spaces). In addition, in [3], [4] and [5] H. Bahouri, M. Majdoub and N. Masmoudi characterized the lack of compactness of H1(R2) in the Orlicz space (see Definition 1.1) H1(R2) ֒ (R2), → L in terms of orthogonal profiles generalizing the example by Moser: α log x n g (x) := ψ − | | , n 2π α r n (cid:16) (cid:17) where α := (α ), called the scale, is a sequence of positive real numbers going to n infinity and ψ, called the profile, belongs to the set ψ L2(R,e 2sds); ψ L2(R), ψ = 0 . − ′ ] ,0] ∈ ∈ |−∞ n o The study of the lack of compactness of critical Sobolev embedding was at the origin of several works concerning the understanding of features of solutions of nonlinear partial differential equations. Among others, one can mention [2], [10], [11], [12] and [18]. 1.2. Critical 4D Sobolev embedding. TheSobolevspaceH2(R4)iscontinuously embedded in all Lebesgue spaces Lp(R4) for all 2 p < . On the other hand, it is also known that H2(R4) embed inBMO(R4) L2(≤R4), w∞here BMO(Rd) denotes the ∩ space of bounded mean oscillations which is the space of locally integrable functions f such that 1 1 f = sup f f dx < with f = f dx. BMO B B k k B | − | ∞ B B | | ZB | | ZB The above supremum being taken over the set of Euclidean balls B, denoting | · | the Lebesgue measure. In this paper, our goal is to investigate the lack of compactness of the Sobolev space H2 (R4) in the Orlicz space (R4) defined as follows: rad L Definition 1.1. Let φ : R+ R+ be a convex increasing function such that → φ(0) = 0 = lim φ(s), lim φ(s) = . s 0+ s ∞ → →∞ We say that a measurable function u : Rd C belongs to Lφ if there exists λ > 0 → such that u(x) φ | | dx < . λ ∞ ZRd (cid:18) (cid:19) We denote then u(x) u = inf λ > 0, φ | | dx 1 . k kLφ λ ≤ (cid:26) ZRd (cid:18) (cid:19) (cid:27) CHARACTERIZATION OF THE LACK OF... 3 In what follows we shall fix d = 4, φ(s) = es2 1 and denote the Orlicz space Lφ by − endowed with the norm where the number 1 is replaced by the constant κ Linvolved in (1.3). It is easykt·oksLee that ֒ Lp for every 2 p < . L → ≤ ∞ The 4D Sobolev embedding in Orlicz space states as follows: L 1 (1.2) u u . (R4) H2(R4) k kL ≤ √32π2k k Inequality (1.2) derives immediately from the following proposition due to Ruf and Sani in [17]: Proposition 1.2. There exists a finite constant κ > 0 such that (1.3) sup e32π2u(x)2 1 dx := κ. | | − u∈H2(R4),kukH2(R4)≤1 ZR4 (cid:16) (cid:17) Let us notice that if we only require that ∆u 1 then the following result L2(R4) k k ≤ established in [15] holds. Proposition 1.3. Let β [0,32π2[, then there exists C > 0 such that β ∈ (1.4) eβu(x)2 1 dx C u 2 u H2(R4) with ∆u 1, | | − ≤ βk kL2(R4) ∀ ∈ k kL2 ≤ R4 Z (cid:16) (cid:17) and this inequality is false for β 32π2. ≥ Remarks 1.4. The well-known following properties can be found in [15] and [17]. a) The inequality (1.3) is sharp. b) There exists a positive constant C such that for any domain Ω R4 ⊆ sup e32π2 u(x)2 1 dx C. | | − ≤ u∈H2(Ω),k(−∆+I)ukL2(Ω)≤1ZΩ(cid:16) (cid:17) c) In dimension 2, the inequality (1.4) is replaced by the following Trudinger-Moser type inequality (see [1] and [16]): Let α [0,4π[. A constant C exists such that α ∈ (1.5) eαu(x)2 1 dx C u 2 u H1(R2) with u 1. | | − ≤ αk kL2(R2) ∀ ∈ k∇ kL2(R2) ≤ R2 Z (cid:16) (cid:17) Moreover, if α 4π then (1.5) is false. ≥ 1.3. Lackof compactness in4D criticalSobolev embedding in Orliczspace. The embedding of H2(R4) into the Orlicz space is non compact. Firstly, we have a lack of compactness at infinity as shown by the following example: u (x) = ϕ(x+x ), ϕ (R4) 0 and x . k k k ∈ D \{ } | | k−→ ∞ →∞ 4 INES BEN AYEDAND MOHAMED KHALILZGHAL Secondly, we have a lack of compactness generated by a concentration phenomenon as illustrated by the following example (see [17] for instance): 8απ2 + 1√−3|x2|π22e2αα if |x| ≤ e−α  p (1.6) fα(x) =  −√l8oπg2|αx| if e−α < |x| ≤ 1   η (x) if x > 1, α  | |  where ηα ∈ D(R4) and satisfies the following boundary conditions: ∂η 1 α η = 0, = , α|∂B1 ∂ν ∂B1 √8π2α (cid:12) 1 with B is the unit ball in R4. In addition,(cid:12)η , η , ∆η are all equal to O (1) 1 (cid:12) α α α ∇ √α as α tends to infinity. (cid:16) (cid:17) By a simple calculation (see Appendix A), we obtain that 1 1 1 f 2 = O , f 2 = O and ∆f 2 = 1+O as α + . k αkL2 α k∇ αkL2 α k αkL2 α → ∞ Also, we can s(cid:16)ee t(cid:17)hat f ⇀ 0 in(cid:16)H2(cid:17)(R4). (cid:16) (cid:17) α α The lack of compactness i→n∞the Orlicz space (R4) displayed by the sequence (f ) α L when α goes to infinity can be stated qualitatively as follows: Proposition 1.5. The sequence (f ) defined by (1.6) satisfies: α 1 f , as α + . α k kL → √32π2 → ∞ 1 Proof. Firstly, we shall prove that liminf f . For that purpose, let us α α k kL ≥ √32π2 →∞ consider λ > 0 such that |fα(x)|2 e λ2 1 dx κ. − ≤ ZR4(cid:18) (cid:19) Then |fα(x)|2 e λ2 1 dx κ. − ≤ Zx e−α(cid:18) (cid:19) | |≤ But for x e α, we have − | | ≤ α 1 x 2e2α α f (x) = + −| | . α 8π2 √32π2α ≥ 8π2 r r So we deduce that e−α 2π2 e8πα2λ2 1 r3 dr κ. − ≤ Z0 (cid:16) (cid:17) 1The notation g(α) = O(h(α)) as α + , where g and h are two functions defined on some → ∞ neighborhood of infinity, means the existence of positive numbers α0 and C such that for any α>α0 we have g(α) C h(α). | |≤ | | CHARACTERIZATION OF THE LACK OF... 5 Consequently, e 4α 2π2 e8πα2λ2 1 − κ, − 4 ≤ which implies that (cid:16) (cid:17) 1 1 λ2 . ≥ 32π2 + 8π2 log(2κ +e 4α) α−→ 32π2 α π2 − →∞ This ensures that 1 liminf f . α α k kL ≥ √32π2 →∞ 1 To conclude, it suffices to show that limsup f . To go to this end, let α α k kL ≤ √32π2 us fix ε > 0 and use Inequality (1.4) wi→th∞β = 32π2 ε. Thus, there exists C > 0 ε − such that (32π2 ε) |fα(x)|2 f 2 e − k∆fαk2L2 −1 dx ≤ Cε k∆fαkL22 . ZR4 ! k αkL2 The fact that lim f = 0 leads to α L2 α k k →∞ 1 limsup f 2 , α k k ≤ 32π2 ε α L →∞ − which ends the proof of the result. (cid:3) The following result specifies the concentration effect revealed by the family (f ): α Proposition 1.6. With the above notation, we have π2 ∆f 2 δ(x = 0) and e32π2fα2 1 (e4 +3)δ(x = 0) as α in (R4). α | | ′ | | → − → 16 → ∞ D Proof. For any smooth compactly supported function ϕ, let us write ∆f (x) 2ϕ(x) dx = I +J +K , α α α α | | R4 Z with I = ∆f (x) 2ϕ(x) dx, α α | | Zx e−α | |≤ J = ∆f (x) 2ϕ(x) dx and α α | | Ze−α x 1 ≤| |≤ K = ∆f (x) 2ϕ(x) dx. α α | | Zx 1 | |≥ Noticing that ∆f (x) = 8e2α if x e α, we get α √−32π2α | | ≤ − ϕ L∞ I k k 0. α | | ≤ α α−→ →∞ This ends the proof of the first assertion. 6 INES BEN AYEDAND MOHAMED KHALILZGHAL On the other hand, as ∆f = 2 if e α x 1, we get α x2√−8π2α − ≤ | | ≤ | | 1 1 1 1 J = ϕ(0) dx+ ϕ(x) ϕ(0) dx α 2π2α x 4 2π2α x 4 − Ze−α≤|x|≤1 | | Ze−α≤|x|≤1 | | 1 1 (cid:0) (cid:1) = ϕ(0)+ ϕ(x) ϕ(0) dx. 2π2α x 4 − Ze−α≤|x|≤1 | | (cid:0) (cid:1) Using the fact that ϕ(x) ϕ(0) x ϕ we obtain that L∞ | − | ≤ | |k∇ k ϕ J ϕ(0) k∇ kL∞(1 e α) 0. α − | − | ≤ α − α−→ →∞ Finally, taking advantage of the existence of a positive constant C such that ∆η C and as ϕ is a smooth compactly supported function, we deduce that k αkL∞ ≤ √α K 0. α | | α−→ →∞ This ends the proof of the first assertion. For the second assertion, we write e32π2fα(x)2 1 ϕ(x) dx = L +M +N , | | α α α − R4 Z (cid:16) (cid:17) where L = e32π2fα(x)2 1 ϕ(x) dx, α | | − Zx e−α | |≤ (cid:16) (cid:17) M = e32π2fα(x)2 1 ϕ(x) dx and α | | − Ze−α≤|x|≤1(cid:16) (cid:17) N = e32π2fα(x)2 1 ϕ(x) dx. α | | − Z|x|≥1(cid:16) (cid:17) We have L = e32π2 fα(x)2 1 ϕ(x) ϕ(0) dx+ e32π2 fα(x)2 1 ϕ(0) dx. α | | | | − − − Z|x|≤e−α(cid:16) (cid:17) Z|x|≤e−α(cid:16) (cid:17) (cid:0) (cid:1) Arguing as above, we infer that 2 Lα e32π2|fα(x)|2 1 ϕ(0) dx 2π2 ϕ L∞ e32π2(cid:18)√8πα2+√321π2α(cid:19) 1 e−5α. − − ≤ k∇ k − 5 (cid:12) Zx e−α (cid:12) ! (cid:12) | |≤ (cid:16) (cid:17) (cid:12) A(cid:12) s the right hand side of the last inequ(cid:12)ality goes to zero when α tends to infinity, (cid:12) (cid:12) we find that L e32π2fα(x)2 1 ϕ(0) dx 0. α | | (cid:12) −Zx e−α − (cid:12) α−→→∞ (cid:12) | |≤ (cid:16) (cid:17) (cid:12) Besides, (cid:12) (cid:12) (cid:12) (cid:12) e−α e32π2|fα(x)|2 1 ϕ(0) dx = 2π2e4(α+1)eα1ϕ(0) ee4ααr4−2e2α(2+α1)r2r3 dr − Z|x|≤e−α(cid:16) (cid:17) Z0 π2 ϕ(0)e 4α. − − 2 CHARACTERIZATION OF THE LACK OF... 7 Now, performing the change of variable s = reα, we get e32π2|fα(x)|2 1 ϕ(0) dx = 2π2eα1+4ϕ(0) 1s3esα4−2(2+α1)s2 ds π2ϕ(0)e−4α, − − 2 Z|x|≤e−α(cid:16) (cid:17) Z0 which implies, in view of Lebesgue’s theorem, that 1 π2 lim L = 2π2e4ϕ(0) s3e 4s2 ds = (e4 5)ϕ(0). α − α 16 − →∞ Z0 Also, writing 4(log|x|)2 4(log|x|)2 Mα = ϕ(x) ϕ(0) e α 1 dx+ ϕ(0) e α 1 dx, − − − Ze−α≤|x|≤1 (cid:16) (cid:17) Ze−α≤|x|≤1 (cid:16) (cid:17) (cid:0) (cid:1) π2 we infer that M converges to ϕ(0) by using the following lemma the proof of α 2 which is similar to that of Lemma 1.9 in [5]. (cid:3) Lemma 1.7. When α goes to infinity, 1 1 1 1 r4eα4 log2r dr and r3eα4 log2r dr . −→ 5 −→ 2 Ze−α Ze−α Finally, in view of the existence of a positive constant C such taht η C k αkL∞ ≤ √α and as ϕ is a smooth compactly supported function, we get N 0, α α−→ →∞ which achieves the proof of the proposition. 1.4. Statement of the results. Before entering into the details, let us introduce some definitions as in [5] and [8]. Definition 1.8. We shall designate by a scale any sequence α := (α ) of positive n real numbers going to infinity. Two scales α and β are said orthogonal if β n log . α → ∞ n (cid:12) (cid:16) (cid:17)(cid:12) The set of profiles is (cid:12) (cid:12) (cid:12) (cid:12) := ψ L2(R,e 4sds); ψ L2(R), ψ = 0 . − ′ ] ,0] P ∈ ∈ |−∞ n o Remark 1.9. The profiles belong to the H¨older space C1. Indeed, for any profile ψ 2 and real numbers s and t, we have by Cauchy-Schwarz inequality t 1 ψ(s) ψ(t) = ψ′(τ) dτ ψ′ L2(R) s t 2. | − | ≤ k k | − | (cid:12)Zs (cid:12) (cid:12) (cid:12) Our main goal is to establish th(cid:12)at the chara(cid:12)cterization of the lack of compactness (cid:12) (cid:12) of critical Sobolev embedding H2 (R4) ֒ (R4) rad → L 8 INES BEN AYEDAND MOHAMED KHALILZGHAL can be reduced to the example (1.6). In fact, we can decompose the function f as α follows: α log x f (x) = L | | +r (x), α α 8π2 − α r (cid:16) (cid:17) where 1 if t 1 ≥ L(t) = t if 0 t < 1  ≤ 0 if t < 0  and 1√−3|x2|π22e2αα if |x| ≤ e−α rα(x) =  0 if e−α < |x| ≤ 1  ηα(x) if x > 1. | | The sequence α is a scale, the function L is a profile and the function r is called  α the remainder term. We can easily see that r 0 in . Indeed, for all λ > 0, we have α α−→ L →∞ e|rαλ(2x)|2 1 dx 2π2 e−α e11+6πr24αe4λα2 1 r3 dr − ≤ − Z|x|≤e−α(cid:16) (cid:17) Z0 (cid:16) (cid:17) π2e 4α 8π4λ2e16π21αλ2αe−4α e16π21αλ2 1 − 0. ≤ − − 2 α−→ →∞ h (cid:16) (cid:17) i Moreover, since η belongs to (R4) and satisfies η C for some C > 0, we D k αkL∞ ≤ √α get |rα(x)|2 e λ2 1 dx 0. − α−→ Z|x|>1(cid:16) (cid:17) →∞ α log x Let us observe that h (x) := L | | does not belong to H2(R4). To α 8π2 − α r overcome this difficulty, we shall convo(cid:16)late the p(cid:17)rofile L with an approximation to the identity ρ where ρ (s) = α ρ(α s) with ρ is a positive smooth compactly n n n n supported function satisfying (1.7) suppρ [ 1,1] and ⊂ − 1 (1.8) ρ(s) ds = 1. Z 1 − More precisely, we shall prove that the lack of compactness can be described in terms of an asymptotic decomposition as follows: Theorem 1.10. Let (u ) be a bounded sequence in H2 (R4) such that n n rad (1.9) u ⇀ 0, n n →∞ (1.10) limsup u = A > 0, and n 0 n k kL →∞ (1.11) lim limsup u (x) 2 dx = 0. n R→∞ n→∞ Z|x|>R| | CHARACTERIZATION OF THE LACK OF... 9 Then, there exists a sequence (α(j)) of pairwise orthogonal scales and a sequence of profiles (ψ(j)) in such that up to a subsequence extraction, we have for all ℓ 1 P ≥ ℓ α(j) log x (1.12) u (x) = n ψ(j) ρ(j) − | | +r(ℓ)(x), n s8π2 ∗ n α(j) n j=1 (cid:18) n (cid:19) X (cid:0) (cid:1) where ρn(j)(s) = αn(j)ρ(αn(j)s) and limsup rn(ℓ) ℓ−→→∞ 0. n L →∞ (cid:13) (cid:13) Remarks 1.11. a) As in [8], the decom(cid:13)posi(cid:13)tion (1.12) is not unique. b) The assumption (1.11) means that there is no lack of compactness at infinity. It is in particularly satisfied when the sequence (u ) is supported in a fixed compact of n R4 and also by the sequences α(j) log x (1.13) g(j)(x) := n ψ(j) ρ(j) − | | n s8π2 ∗ n α(j) (cid:18) n (cid:19) (cid:0) (cid:1) involved in the decomposition (1.12). c) As it is mentioned above, the functions h(nj)(x) := α8π(nj2)ψ(j) −αlo(ngj)|x| do not belong to H2(R4). However, we have q (cid:16) (cid:17) (1.14) g(j) h(j) 0, n − n (R4) n−→ L →∞ (cid:13) (cid:13) where the functions g(j) are d(cid:13)efined by (cid:13)(1.13). Indeed, by the change of variable n s = −loαg(nj|x)| and using the fact that, for any integer number j, ψ(j)∗ρ(nj) is supported in [ 1 , [ and ψ(j) is supported in [0, [, we infer that for all λ > 0 −α(nj) ∞ ∞ (j) (j) 2 (j) 2 e gn (x)−λhn (x) −1 dx = 2π2αn(j) ∞ e8απn2λ2 ψ(j)∗ρ(nj) (s)−ψ(j)(s) −1 e−4α(nj)s ds. R4 1 Z (cid:16) (cid:12)(cid:12) (cid:12)(cid:12) (cid:17) Z−α(nj) (cid:16) (cid:12)(cid:12)(cid:0) (cid:1) (cid:12)(cid:12) (cid:17) Since 1 t ψ(j) ρ(j) (s) ψ(j)(s) ψ(j) s ψ(j)(s) ρ(t) dt, ∗ n − ≤ − α(j) − (cid:12)(cid:0) (cid:1) (cid:12) Z−1(cid:12) (cid:16) n (cid:17) (cid:12) (cid:12) (cid:12) we obtain(cid:12), according to Cauchy-Sc(cid:12)hwarz inequality, (cid:12) (cid:12) 1 t 2 ψ(j) ρ(j) (s) ψ(j)(s) 2 . ψ(j) s ψ(j)(s) dt ∗ n − − α(j) − (cid:12)(cid:0) (cid:1) (cid:12) Z−1(cid:12)(cid:12) 1(cid:16) n (cid:17) (cid:12)(cid:12) (cid:12) (cid:12) . α(j)(cid:12) α(nj) ψ(j)(s τ) ψ(j)((cid:12)s) 2 dτ n − − 1 Z−α(nj) (cid:12) (cid:12) (1j) (cid:12) s (cid:12)2 . α(j) αn ψ(j) ′(u) du dτ. n Z−α(n1j) (cid:16)Zs−τ (cid:12)(cid:0) (cid:1) (cid:12) (cid:17) (cid:12) (cid:12) 10 INES BEN AYEDAND MOHAMED KHALILZGHAL Applying again Cauchy-Schwarz inequality, we get 1 s ψ(j) ρ(j) (s) ψ(j)(s) 2 . α(j) α(nj) ψ(j) ′(u) 2 du τ dτ ∗ n − n | | (cid:12)(cid:0) (cid:1) (cid:12) Z−α(n1j) (cid:16)Zs−τ (cid:12)(cid:0) (cid:1) (cid:12) (cid:17) (cid:12) (cid:12) 1 s (cid:12) (cid:12) . sup ψ(j) ′(u) 2 du. αn(j) |τ|≤α(n1j) Zs−τ (cid:12)(cid:0) (cid:1) (cid:12) (cid:12) (cid:12) Then, there exists a positive constant C such that C sup s (ψ(j))′(u)2du e gn(j)(x)−λh(nj)(x) 2 −1 dx . αn(j) ∞ eλ2|τ|≤α(n1j) Rs−τ| | −1e−4α(nj)s ds R4 1 Z (cid:16) (cid:12)(cid:12) (cid:12)(cid:12) (cid:17) Z−α(nj)   . I +J ,   n n where C sup s (ψ(j))′(u)2du In = αn(j) ∞eλ2s∈[s0,∞[,|τ|≤α(n1j) Rs−τ| | −1e−4α(nj)s ds and Zs0    C sup s (ψ(j))′(u)2du Jn = αn(j) s0 eλ2s∈[−α(n1j),s0],|τ|≤α(n1j) Rs−τ| | −1e−4α(nj)s ds, 1 Z−α(nj)     for some positive real s . 0 Noticing that C(cid:13)(cid:13)(cid:13)(cid:13)(ψ(j))′(cid:13)(cid:13)(cid:13)(cid:13)2L2(R) e−4α(nj)s0 In . e λ2 1 ,  −  4 we infer that   lim I = 0. n n →∞ Moreover, the fact that s Cn := C sup ψ(j) ′(u) 2 du 0, s∈[−α(n1j),s0],|τ|≤α(n1j) Zs−τ (cid:12)(cid:0) (cid:1) (cid:12) n−→→∞ (cid:12) (cid:12) implies that Cn e4 e−4α(nj)s0 lim Jn = lim eλ2 1 − = 0. n n − 4 →∞ →∞ (cid:16) (cid:17) This leads to (1.14) as desired. d) Similarly to the proof of Proposition 1.15 in [5], we get by using (1.14) 1 ψ(j)(s) lim g(j) = lim h(j) = max . n n (R4) n n (R4) √32π2 s>0 √s →∞ L →∞ L (cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13)