Advanced Nonlinear Studies 3 (2003), **–**
A Note on Bifurcation from the Essential Spectrum Marino Badiale
∗
Dipartimento di Matematica Universit` a di Torino Via Carlo Alberto 10, 10123 Torino, Italy e-mail:
[email protected] Received 10 February 2003 Communicated by Antonio Ambrosetti
Abstract In this paper we study a semilinear elliptic equation in all IRN . This equation depends on a parameter λ, and we obtain, for small λ < 0, solutins which are small in H 1 (IRN ). In this sense we have solutions bifurcating from the origin and, as the differential operator involved is the laplacian, we say that we have solutions bifurcating from the bottom of the essential spectrum of the laplacian. By a change of variables we transform the original bifurcation problem into a perturbation one. We adopt a variational procedure, looking for critical points of a suitable functional. We apply a recently developped reduction method, which allows to reduce the original variational problem in H 1 (IRN ) to a variational problem in a finite-dimensional manifold, and then we solve this last problem. In this way we are also able to manage the presence of critical nonlinearities, in the sense of Sobolev embedding. 1991 Mathematics Subject Classification. 35J60. Key words. Nonlinear elliptic equation, bifurcation, reduction method ∗ This research was supported by MURST “Variational Methods and Nonlinear Differential Equations”
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M. Badiale
Introduction and main results
We consider the following equation −∆ψ − λψ = a(x)|ψ|p−1 ψ + b(x)|ψ|q−1 ψ, lim|x|→∞ ψ(x) = 0,
x ∈ IRN ,
(1)
N +2 where N ≥ 1, 1 < p < q ≤ N −2 if N ≥ 3 (and q < +∞ if N = 1, 2), p < 1 + 4/N . λ is a negative parameter, and we look for families of solutions bifurcating from the origin in H 1 (IRN ), that is for couples (λ, ψλ ) such that ψλ ∈ H 1 (IRN ), ψλ is a solution of (1) in the weak sense of H 1 (IRN ), λ ∈ (λ0 , 0) for some λ0 < 0 and ψλ → 0 in H 1 (IRN ) as λ → 0. Hence these families of solutions bifurcate not from an eigenvalue but from a point of the essential spectrum of −∆ in IRN . This kind of bifurcation has been studied in several recent papers, see for example [14], [15], [16]. In this paper we continue the work initiated in [2] and [3], where problem (1) was studied in the case N = 1, and in [9], where many results of [2] where extended to any dimension N . In [2] and [9] it is assumed that the function a has, in some sense, a positive limit A at infinity, and through a change of variables the equation (1) is reduced to a perturbation problem (see below). The hypotheses used in [9] are, roughly speaking, of two R different kinds: it is assumed either that the function a(x) − A is integrable and IRN (a(x) − A)dx 6= 0, ot that it is asymptotic, at infinity, to 1/|x|γ with γ ∈ ]0, N [. So two R interesting limit cases are left out: the case in which a(x)−A is integrable but IRN (a(x)−A)dx = 0, and the case in which a(x)−A is asymptotic to 1/|x|N . In this paper we will see that also in these cases, adding some more hypotheses, we obtain existence of solutions bifurcating from the origin R in H 1 (IRN ). Furthemore, in the case IRN (a(x) − A)dx = 0 we obtain bifurcation of two families of solutions. Now let us state precisely our results. For our first result (theorem 1.1 below) we will use the following hypotheses.
(h1 ) a is continuous and bounded. (h2 ) There exists A > 0 such that the function x → (a(x) − A)|x| is in L1 (IRN ). R R (h3 ) IRN (a(x) − A)dx = 0 and IRN (a(x) − A)xi dx 6= 0 for some i = 1, . . . , N . (h4 ) b is continuous and bounded. 2N
q−p − 1 we also assume that there exists β ∈ [1, β ∗ [ (h5 ) b ∈ L N +2 (IRN ). If N ≥ 2 p−1 N such that b ∈ Lβ (IR ), where
N (p − 1) q−p if N > 2 − 1, (N + 1)(p − 1) − 2(q − p) p−1 q−p β ∗ = +∞ if N = 2 − 1. p−1 β∗ =
For our next result (theorem 1.2 below) we will use the following hypotheses.
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Bifurcation from the essential spectrum
(h6 ) a is continuous. (h7 ) There are A > 0, A1 6= 0 sucht that |x|N (a(x) − A) → A1 as |x| → ∞. (h8 ) The function a(x) − A −
A1 1+|x|N
belongs to L1 (IRN ).
(h9 ) b is continuous and bounded. 2N
q−p (h10 ) b ∈ L N +2 (IRN ). If N > 2 p−1 , we also assume that there exists β ∈ [1, β ∗ ] N such that b ∈ Lβ (IR ), where
β∗ =
N (p − 1) . N (p − 1) − 2(q − p)
We can now state our main results. N +2 Theorem 1.1 Assume 1 < p < q ≤ N −2 if N ≥ 3, and q < +∞ if N = 1, 2. 4 Assume p < 1 + N . Suppose that (h1 ), . . . , (h5 ) hold. Then (1) has two families of solutions bifurcating from the origin in H 1 (IRN ). N +2 Theorem 1.2 Assume 1 < p < q ≤ N −2 if N ≥ 3, and q < +∞ if N = 1, 2. 4 Assume p < 1 + N . Suppose that (h6 ), . . . , (h10 ) hold. Then (1) has a family of solutions bifurcating from the origin in H 1 (IRN ).
Remark 1.3 When p ≥ 1 + 4/N , the families of solutions that we find still bifurcate from the origin in L∞ (IRN ), but in H 1 (IRN ) they can bifurcate from infinity or can be bounded away both from zero and infinity. To prove our results, we need a change of variables, which tranforms the original problem in a perturbation one. Let us set u(x) = ε2/(1−p) ψ(x/ε), λ = −ε2 , so that equation (1) changes to q−p
− ∆u + u = A|u|p−1 u + (a(x/ε) − A)|u|p−1 u + ε2 p−1 b(x/ε)|u|q−1 u.
(2)
∈ H 1 (IRN ) is a family of solutions of (2), bounded as ε → 0, then ψε (x) = ε uε (εx) is a family of solutions of (1) and ψε (x) → 0 in H 1 (IRN ), as ε → 0 (because p < 1 + 4/N ). Hence, to find solutions of (1) bifurcating from the origin, we will look for bounded families of H 1 -solutions of (2). The paper is organized as follows: after the introduction (section 1) we give in section 2 a brief sketch of the critical point theory for perturbed functionals that we use to prove theorems 1.1 and 1.2. In section 3 we prove theorem 1.1 and in section 4 we prove theorem 1.2. If uε 2/(p−1)
Notations We collect below a list of the main notation used throughout the paper.
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M. Badiale
• By B(x, r) we mean the open ball in IRN of center x and radius r. • x · y is the usual scalar product of x, y ∈ IRN . • ωN is the (N − 1)-surface area of the sphere ∂B(0, 1). • 2∗ =
2N N −2
is the critical exponent for the Sobolev embedding, when N ≥ 3.
• We will use C to denote any positive constant, that can change from line to line.
2
Abstract theory for perturbed functionals
In the proof of our results we use a variational method to study critical points of perturbed functionals. The method has been developed in [4], [1] , [2] and then has been applied to many different problems, see [5], [6], [7], [8], [12]. We do not repeat here the main idea of the method. We just state the results that we need, referring to the quoted papers, in particular to [4], [1] , [2], forR the proofs. Let us consider the Hilbert space E = H 1 (IRN ) with norm ||u||2 = IRN |∇u|2 + u2 dx and the family of functionals fε (u) =
1 ||u||2 − F (u) + G(ε, u), 2
where F (u) = and G = G1 + G2 where G1 (ε, u) = and
( G2 (ε, u) =
−1 p+1
0 −1 q+1
R
IRN
A p+1
Z
|u|p+1 dx,
IRN
(a(x/ε) − A)|u|p+1 dx if ε 6= 0, if ε = 0
q−p
ε2 p−1
0
R
IRN
b(x/ε)|u|q+1 dx if ε 6= 0, if ε = 0
We need the following properties: (G0 ) G is continuous in (ε, u) ∈ IR × E and G(0, u) = 0 for all u ∈ E; (G1 ) G is of class C 2 with respect to u ∈ E. (G2 ) The maps (ε, u) 7→ G0 (ε, u), (ε, u) 7→ G00 (ε, u) are continuous. If DF (u), Du G(ε, u) are the first differentials of F and G with respect to u, and 2 D2 F (u), Duu G(ε, u) are the second differentials and L(E, E) is the space of linear endomorphism of E, we define by F 0 (u), respectively G0 (ε, u), the functions defined by setting (F 0 (u)|v) = DF (u)[v], ∀ v ∈ E,
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Bifurcation from the essential spectrum
and, respectively, (G0 (ε, u)|v) = Du G(ε, u)[v],
∀ v ∈ E, 00
where (· | ·) is the scalar product in E. Similarly, F (u), resp. G00 (ε, u), denotes the maps in L(E, E) defined by (F 00 (u)v|w) = D2 F (u)[v, w]
2 (G00 (ε, u)v|w) = Duu G(ε, u)[v, w].
Let us recall that the equation f00 (u) = 0 is the following nonlinear equation −∆u + u = A|u|p−1 u, x ∈ IRN , N 1 u ∈ H (IR ).
(3)
It is well known (see [10], [11], [13]) that there exists a unique positive radial solution z of (3), that z is strictly radial decreasing, has an exponential decay at infinity together with its derivatives, and that f0 possesses a N −dimensional manifold of critical points Z = {zθ (x) = z(x + θ) | θ ∈ IRN }. Of course z0 = z. We make the following further assumptions on G. (G3 ) There exists a function ϕ : (−ε0 , ε0 ) → IR, such that ϕ(ε) → 0 as ε → 0, and a continuous function Γ : Z → IR such that, for all zθ ∈ Z, Γ(zθ ) = lim
ε→0
G(ε, zθ ) ϕ(ε)
and G0 (ε, z) = o(|ϕ(ε)|1/2 ). With the same arguments of [2] (see also [1], [4]) we can prove the following theorem. Theorem 2.1 Suppose that (G0 − G3 ) hold and assume there exist an open bounded set U ⊂ IRN and θ∗ ∈ U such that, setting z ∗ = zθ∗ ∈ Z, we have either min Γ(zθ ) > Γ(z ∗ ), θ∈U
or max Γ(zθ ) < Γ(z ∗ ). θ∈U
(4)
Then, for ε small, fε has a critical point uε . Notice that in [2] only the case U = B(θ∗ , δ) and ϕ(ε) = εα (α > 0) is treated, but it is easy to see that the same arguments hold for general ϕ and U . To apply teorem 2.1 we have to verify hypotheses (G0 − G3 ). The verification of (G0 − G2 ) is the same as in [9], so we state this fact, without proof, in the following lemma. Lemma 2.2 Assume that fε , F , G are defined as before. Assume that either the hypotheses of theorem 1.1 or those of theorem 1.2 are satisfied. Then also (G0 ), (G1 ), (G2 ) are satisfied. We are left with the verification of (G3 ), and this will be done in the next two sections.
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M. Badiale
3
Proof of theorem 1.1
In this section we prove theorem 1.1. As we pointed out above, we only have to verify that hypothesis (G3 ) holds. LemmaR 3.1 Let us assume (h1 ), . . . .(h5 ). Let us define γ = (γ1 , . . . , γN ) ∈ IRN by γi = IRN (a(x) − A) xi dx. For θ ∈ IRN we also define Γ(θ) = −
1 ∇z p+1 (θ) · γ. p+1
(5)
Then G(ε, zθ ) = Γ(θ) εN +1
lim
ε→0
(6)
and N
G0 (ε, zθ ) = O(ε 2 +1 ).
(7)
Notice that in this case ϕ(ε) = εN +1 . Proof. The argument to prove (7) is the same as in [9] and we don’t repeat it here. So let us prove (6). We will prove lim
ε→0
G1 (ε, zθ ) = Γ(θ) εN +1
(8)
G2 (ε, zθ ) = 0, (9) εN +1 which of course imply (6). Let us begin by proving (8). By the change of variables y = x/ε we have lim
ε→0
Z −(p + 1)G1 (ε, zθ )
x a − A z p+1 (x + θ)dx ε
= IRN
= εN
Z
(a(y) − A) z p+1 (εy + θ)dy.
IRN
By Lagrange’s theorem, we get z p+1 (θ + εy) = z p+1 (θ) + ε∇z p+1 (θ + hεy) · y, where h = h(y, θ, ε) ∈ [0, 1]. Hence
−(p + 1)G1 (ε, zθ )
= εN
Z
(a(y) − A)z p+1 (θ)dy +
IRN
ε
N +1
Z IRN
(a(y) − A)∇z p+1 (θ + hεy) · ydy =
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Bifurcation from the essential spectrum
εN +1
N Z X i=1
(a(y) − A)Di z p+1 (θ + hεy)yi dy.
IRN
N
1
As (a(y) − A)yi ∈ L (IR ) and Di z p+1 is bounded, by dominated convergence, we obtain Z (a(y)−A)Di z
p+1
Z
(a(y)−A)Di z p+1 (θ)yi dy = Di z p+1 (θ)γi ,
(θ +hεy)yi dy →
IRN
IRN
hence N
X −(p + 1)G1 (ε, zθ ) Di z p+1 (θ)γi = ∇z p+1 (θ) · γ, → N +1 ε i=1 q−p so (8) is proved. To prove (9), we distinguish two cases. Assume first N < 2 p−1 − 1. In this case Z x q−p 1 ε−N −1 G2 (ε, zθ ) = − ε2 p−1 −N −1 b z q+1 (x + θ)dx, q+1 ε IRN
and this expression of course vanishes as ε → 0, uniformly with respect to θ, because q−p the integral is bounded. Hence, let us assume N ≥ 2 p−1 − 1. We have −N −1 ε G2 (ε, zθ ) = −
Cε
2 q−p p−1 −N −1
Z IRN
Cε
q−p 1 ε2 p−1 −N −1 q+1
Z b
x
IRN
ε
z0q+1 (x + θ)dx ≤
β−1 x β β1 Z β(q+1) β β−1 (x + θ) dx ≤ z0 b dx ε IRN
N 2 q−p p−1 −N −1+ β
Z
β
|b(y)| dy
β1 ,
IRN q−p where β is given by (h5 ). This term goes to zero as, by (h5 ), 2 p−1 −N −1+ N β > 0. .
We can then conclude the proof of theorem 1.1. We know that z is radial and strictly radial decreasing. Hence we can write z0 (x) = η(|x|) where η : IR+ → IR+ , η 0 (r) < 0 if r > 0, η 0 (0) = 0 and η 0 (r) → 0 as r → ∞. This implies that ζ(r) = d p+1 (r) has a negative absolute minimum, that is, there are r0 > 0 and m0 < 0 dr η 1 such that m0 = ζ(r0 ) ≤ ζ(r) < 0 for all r > 0. It is ∇z p+1 (θ) = ζ(|θ|) |θ| θ, hence Γ(θ) = −
1 1 1 ∇z0p+1 (θ) · γ = − ζ(|θ|) θ · γ. p+1 p+1 |θ|
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M. Badiale
1 It is then easy to see that θ∗ = r0 |γ| γ is an absolute maximum point for Γ, and −θ∗ is an absolute minimum point. It is also possible to take R > 0 such that for all θ with |θ| ≥ R or |θ · γ| < R1 we have |Γ(θ)| < 12 |Γ(θ∗ )| = 12 |Γ(−θ∗ )| (recall that ζ(r)/r is bounded in (0, +∞)). Let us define
U1 = {θ ∈ IRN | |θ| < R, θ · γ > 1/R},
U2 = {θ ∈ IRN | |θ| < R, θ · γ < −1/R}.
It is then easy to see that Γ(θ∗ ) > inf Γ(θ), θ∈U1
Γ(−θ∗ ) < sup Γ(θ). θ∈U2
We can then apply theorem 2.1. We obtain two families {(ε, u1ε )}, {(ε, u2ε )} of solutions of (2) such that {uiε } is a bounded set in H 1 (IRN ). To be precise, we have (see [2] and [9]) uiε (x) = z0 (x + θεi ) + w(ε, θεi )(x) where θεi ∈ Ui and w(ε, θεi ) → 0 as ε → 0. Hence θεi → θi ∈ U i , as ε → 0, so uiε (x) → z0 (x + θi ) and θ1 6= θ2 . This implies that u1ε 6= u2ε , so the two families are distinct (at least for small ε’s) and they give rise to two distinct families of solutions of (1) bifurcating from the origin.
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Proof of theorem 1.2
To prove theorem 1.2 we have, as before, to prove only that hypothesis (G3 ) holds. Let us define ϕ(ε) = εN |log ε|. Lemma 4.1 Let us assume (h6 ), . . . .(h10 ). Define 1 A1 ωN z0p+1 (θ). p+1
(10)
G(ε, zθ ) = Γ(θ) ϕ(ε)
(11)
Γ(θ) = − Then we have lim
ε→0
and G0 (ε, zθ ) = o(|ϕ(ε)|1/2 ).
(12)
Proof. The proof of (12) is the same as in [9], and we do not repeat it. To prove (11), we prove the following two statements. lim
ε→0
G1 (ε, zθ ) = Γ(θ) ϕ(ε)
(13)
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Bifurcation from the essential spectrum
G2 (ε, zθ ) = 0. ε→0 ϕ(ε) lim
(14)
Again, the proof of (14) is the same as in [9], so we only have to prove (13). For this, let us write Z − (p + 1)G1 (ε, zθ )
(a(x/ε) − A) z p+1 (x + θ)dx
=
(15)
IRN
Z
a(x/ε) − A − A1 (1 + |x/ε|N )−1 z p+1 (x + θ)dx +
= ZIR
N
A1 (1 + |x/ε|N )−1 z p+1 (x + θ)dx.
IRN
By the change of variables y = x/ε, and using the hypothesis a(x/ε) − A − A1 (1 + |x/ε|N )−1 ∈ L1 (IRN ), it is easy to get Z a(x/ε) − A − A1 (1 + |x/ε|N )−1 z p+1 (x + θ)dx = IRN
εN
Z
a(y) − A − A1 (1 + |y|N )−1 z p+1 (εy + θ)dy =
IRN
εN
Z
a(y) − A − A1 (1 + |y|N )−1 z p+1 (θ)dy + o(εN ) = o(ϕ(ε)).
IRN
Let us study the second integral on the right-hand side of (15). We have Z
(1 + |x/ε|N )−1 z p+1 (x + θ)dx
IRN
Z
1 z p+1 (x + θ)dx εN + |x|N # Z +∞ "Z 1 N p+1 = ε z (x + θ)dσx dρ N N 0 ∂B(0,ρ) ε + |x| "Z # Z +∞ 1 N p+1 = ε z (x + θ)dσx dρ εN + ρN 0 ∂B(0,ρ) "Z # Z +∞ ρN −1 N p+1 z (ρy + θ)dσy dρ. = ε εN + ρN ∂B(0,1) 0 = ε
N
IRN
Here by dσx or dσy we mean the surface measures of ∂B(0, ρ) or ∂B(0, 1). By an integration by parts, this is equal to ! ρ=+∞ Z 1 − εN log(ρN + εN ) z p+1 (θ + ρy)dσy N |y|=1 ρ=0
(16)
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M. Badiale
ε
N
Z
+∞
0
1 log(ρN + εN ) N
!
Z ∇z
p+1
(θ + ρy) · ydσy
dρ.
|y|=1
Let us now study these two terms. We know that, for some K, δ > 0, it is z p+1 (x) ≤ Ke−δ|x| . Hence we easily get Z z p+1 (θ + ρy)dσy ≤ K1 e−δ1 ρ , |y|=1
for suitable K1 , δ1 > 0, independent from θ when θ is in a fixed bounded set. From this we get easily Z log(ρN + εN ) z p+1 (θ + ρy)dσy → 0 as ρ → +∞, |y|=1
and hence, for small ’s, ! # " Z ρ=+∞ 1 log(ρN + εN ) z p+1 (θ + ρy)dσy = εN N ρ=0 |y|=1 −ε
N
Z
z p+1 (θ)dσy = εN | logε|z p+1 (θ)ωN = ωN ϕ(ε)z p+1 (θ).
logε |y|=1
We estimate the other integral in (16) in the following way: for small ε’s, ! Z +∞ Z 1 N N N p+1 log(ρ + ε ) ∇z (θ + ρy) · ydσy dρ ≤ −ε N 0 |y|=1 ! Z +∞ Z 1 εN | log(ρN + εN )| |∇z p+1 (θ + ρy)|dσy dρ ≤ N 0 |y|=1 ! Z +∞ Z 1 log(ρN + 1) + | log(ρN )| |∇z p+1 (θ + ρy)|dσy dρ. εN N 0 |y|=1 This last integral converges, because the integrand function f (ρ) =
N
log(ρ + 1) + log(ρN )
!
Z |∇z
p+1
(θ + ρy)|dσy
|y|=1
is continuous in (0, +∞), in ρ = 0 is dominated by the integrable function | log(ρ)| and has an exponential decay at infinity, uniformly in θ when θ is in a bounded set. Hence we can write Z ! Z +∞ 1 N N N p+1 log(ρ + ε ) ∇z (θ + ρy) · ydσy dρ ≤ CεN = o(ϕ(ε)). ε N |y|=1 0 Collecting all these results we obtain
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Bifurcation from the essential spectrum
G1 (ε, zθ ) = Γ(θ)ϕ(ε) + o(ϕ(ε)), hence (11) is proved. Let us now prove (12). We give the arguments in the case N ≥ 3; the cases N = 1, 2 are similar and in fact easier. We have
||G01 (ε, zθ )||
Z 0 p (a(x/ε) − A)z (x + θ)vdx ≤ = sup (G1 (ε, zθ ) | v) = sup N ||v||≤1 ||v||≤1 IR
Z |a(x/ε) − A|
sup ||v||≤1
2N N +2
p N2N +2
|z(x + θ)|
+2 Z N2N dx
IRN
|v|
2N N −2
−2 N2N ≤ dx
IRN
Z |a(x/ε) − A|
C
2N N +2
+2 N2N
dx
.
IRN 2N
By hypothesis (h7 ), the function |a(y) − A| N +2 is asymptotic, at infinity, to the 2N 2
functions |y|− N +2 , so of course it belongs to L1 (IRN ). Hence, by the change of variables x/ε = y, we obtain ||G01 (ε, zθ )|| ≤ Cε1+N/2 so (12) is proved. To conclude the proof of theorem 1.2 we notice that z(θ) has a strict maximum at θ = 0, so the same holds for z p+1 (θ). Hence we can apply theorem 2.1, taking U = B(0, 1) and θ∗ = 0. We obtain a family {uε } of solutions of (2), from which we come back, in the usual way, to a family (λ, ψλ ) of solutions of (1) bifurcating from the origin. Remark 4.2 Let us try to explain the meaning of hypothesis (h8 ). From (h7 ) we deduce that g(x) = |x|N (a(x) − A) − A1 → 0 as |x| → ∞. We can write a(x) − A −
A1 g(x) = |x|N |x|N
and A1 a(x) − A − = A1 1 + |x|N
1 1 − |x|N 1 + |x|N
+
g(x) . |x|N
A1 The function a(x) − A − 1+|x| N is of course locally integrable, so hypothesis (h8 ) 1 deals with its behavior at infinity. It is obvious that |x|1N − 1+|x| N is integrable at g(x) infinity, so (h8 ) is equivalent to the hypothesis that |x| N is integrable at infinity. We know that g(x) → 0, so what we need is that g(x) decays fast enough at infinity: for example if g(x) is asymptotic to 1/|x|α , for any α > 0, or to 1/(log|x|)α , for any α > 1, then (h8 ) is satisfied, while it is not satisfied if g(x) is asymptotic to 1/log|x|.
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M. Badiale
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