Weak perturbations of the p-Laplacian
aa r X i v : . [ m a t h . A P ] F e b WEAK PERTURBATIONS OF THE P–LAPLACIAN
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVA ˇR´IK
Abstract.
We consider the p-Laplacian in R d perturbed by a weakly coupled potential.We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in theweak coupling limit separately for p > d and p = d and discuss the connection with Sobolevinterpolation inequalities. AMS Mathematics Subject Classification:
Keywords: p-Laplacian, weak coupling, Sobolev inequalities1.
Introduction
In this paper we consider the functional Q V [ u ] = Z R d ( |∇ u | p − V | u | p ) dx, u ∈ W ,p ( R d ) , p > , (1.1)with a given function V : R d → R which is assumed to vanish at infinity in a sense to bemade precise. We are interested in the minimization problem λ ( V ) = inf u ∈ W ,p ( R d ) Q V [ u ] R R d | u | p dx . (1.2)If (1.2) admits a minimizer u , then the latter satisfies in the weak sense the non-lineareigenvalue equation − ∆ p ( u ) − V | u | p − u = λ ( V ) | u | p − u , (1.3)where − ∆ p ( u ) := −∇ · ( |∇ u | p − ∇ u ) is the p -Laplacian. Equation (1.3) is a particularcase of a quasilinear differential problem and we refer to the monographs [LU, PS] andto [S1, S2, Tr] for the general theory of such equations. The p -Laplacian equation with azero-th order term V has attracted particular attention. Existence of positive solutions tothe equation − ∆ p ( u ) = V | u | p − u and related regularity questions were studied in [PoSh,PT2, TT, To, PT1]. For the discussion of maximum and comparison principles and positiveLiouville theorems, see [GS, PTT].In the present paper we are going to study the behaviour of λ ( αV ) for small values of α .It is not difficult to see that λ ( αV ) → α → V .Our goal here is to find the correct asymptotic order and the correct asymptotic coefficient. Key words and phrases. p-Laplacian, weak coupling, Sobolev inequality.c (cid:13)
It turns out that the asymptotic order depends essentially on the relation between the valuesof the exponent p and the dimension d . If p < d , then by the Hardy inequality [OK] we have Z R d |∇ u | p dx ≥ (cid:16) d − pp (cid:17) p Z R d | u | p | x | p dx, u ∈ W ,p ( R d ) , d > p . Therefore, if | V ( x ) | ≤ C | x | − p for some C >
0, then λ ( αV ) = 0 for all α small enough.However, if p ≥ d and R R d V >
0, then we have λ ( αV ) < α >
0. The latter is easilyverified by a suitable choice of test functions. Moreover, if V is bounded and compactlysupported, then λ ( αV ) < α > R R d V = 0, see [PT1, Prop. 4.5].Consequently, we will always assume that p ≥ p .The question about the asymptotic behavior of λ ( αV ) for small α was intensively studiedin the linear case p = 2 (see, e.g., [BGS, Kl1, KS, Si]), where equation (1.3) defines theground state energy of the Schr¨odinger operator − ∆ − V . In particular, it turns out thatfor sufficiently fast decaying V we have p − λ ( αV ) = 12 α Z R V dx − c α + o ( α ) , α → , d = 1 , p = 2 , (1.4)with an explicit constant c depending on V , see [Si]. The proof of (1.4) is based on theBirman-Schwinger principle and on the explicit knowledge of the unperturbed Green func-tion. With suitable modifications, this method was applied also to Schr¨odinger operatorswith long-range potentials, [BGS, Kl2], and even to higher order and fractional Schr¨odingeroperators [AZ1, AZ2, Ha].Much less is known about the non-linear case p = 2 where the operator-theoretic methodsdeveloped for p = 2 cannot be used. We will therefore apply a different, purely variationaltechnique which allows us to analyze the asymptotic behaviour of λ ( αV ) for all p >
1. Asimilar variational approach has already been used in a linear problem in [FMV], but here wetake it much further into the quasi-linear realm (where, for instance the symmetry reductionthat we crucial in [FMV] is no longer available).We will present our main results separately for p > d , see Theorem 2.1, and for p = d , seeTheorem 2.2. In the case p > d we shall show, in particular, that there is a close relationbetween the asymptotic behaviour of λ ( αV ) and the Sobolev interpolation inequality (see,e.g., [Ad, Thm 5.9]) k u k p ∞ ≤ S d,p k∇ u k dp k u k p − dp , u ∈ W ,p ( R d ) , d < p . (1.5)By convention S d,p will always denote the optimal (that is, smallest possible) constant in(1.5). On one hand, the constant S d,p enters into the asymptotic coefficient in the expansionof λ ( αV ), see equation (2.1). On the other hand, minimizers of problem (1.2), when suitablyrescaled and normalised, converge (up to a subsequence) locally uniformly to a minimizer ofthe Sobolev inequality (1.5) as α →
0, see Proposition 3.7.The case p = d is much more delicate and requires (slightly) more regularity of thepotential V since functions in W ,d ( R d ), which appear in (1.2), are not necessarily bounded.While the case p > d can be dealt with by energy methods (i.e. on the W ,p ( R d ) levelof regularity), heavier PDE technics (Harnack’s inequality, H¨older continuity bounds) arenecessary to deal with p = d . The subtly of the case p = d can also be seen in the asymptotic EAK PERTURBATIONS OF THE P–LAPLACIAN 3 order: while λ ( αV ) vanishes algebraically as α → p > d , it vanishes exponentially fastfor p = d , see equation (2.2).As we shall see, the asymptotic coefficient will depend on V only through R R d V dx . Weemphasize here that we do not impose a sign condition on V . Thus, the positive andthe negative parts of V contribute both to the asymptotic coefficient and there will becancellations. This is one of main difficulties that we overcome. In fact, if V is non-negative,then the proof is considerably simpler.A common feature of both Theorems 2.1 and Theorem 2.2 is that their proofs rely, amongother things, on the fact that minimizers u α of (1.2), suitably normalized, converge locallyuniformly to a constant. While in the case d < p this follows from Morrey’s Sobolev inequal-ity and energy considerations, for d = p we have to employ a regularity argument related tothe H¨older continuity of u α , see Lemma 4.6, with explicit dependence on the coefficients ofthe equation. 2. Main results
Our main results describe the asymptotics of the infimum λ ( αV ) of the functional Q αV [ u ]as α →
0, see (1.1) and (1.2). Our first theorem concerns the subcritical case p > d . Theorem 2.1.
Let p > d ≥ . Let V ∈ L ( R d ) be such that R R d V ( x ) dx > . Then lim α → α − pp − d λ ( αV ) = − p − dp (cid:18) dp (cid:19) dp − d (cid:18) S d,p Z R d V ( x ) dx (cid:19) pp − d , (2.1) where S d,p is the sharp constant in the Sobolev inequality (1.5) . We also have a theorem that describes the asymptotics of the minimizers of the functional Q αV [ u ]; see Proposition 3.7.In the endpoint case d = p we have Theorem 2.2.
Let p = d > . Suppose that V ∈ L q ( R d ) ∩ L ( R d ) for some q > and that R R d V ( x ) dx > . Then lim α → α d − log 1 | λ ( αV ) | = d ω d − d (cid:18)Z R d V ( x ) dx (cid:19) − d − , (2.2) where ω d denotes the surface area of the unit sphere in R d . Remark 2.3.
Let us compare the assumptions on V in Theorems 2.1 and 2.2. If p > d and V + / ∈ L ( R d ), V − ∈ L ( R d ), then Theorem 2.1 easily implies thatlim α → α − pp − d λ ( αV ) = −∞ . Thus, at least under the additional hypothesis V − ∈ L ( R d ), the condition V + ∈ L ( R d )is necessary and sufficient for finite asymptotics of α − pp − d λ ( αV ). This is not true for theasymptotics of α d − log | λ ( αV ) | − in the case p = d , and this is the reason for the additionalassumption V ∈ L q ( R d ) for some q >
1. Indeed, we claim that there are 0 ≤ V ∈ L ( R d )such that λ ( αV ) = −∞ for any α >
0. To see this, choose σ ∈ (1 , d ) and consider V ( x ) = | x | − d | log | x || − σ for | x | ≤ e − and V ( x ) = 0 for | x | > e − . Then σ > V ∈ L ( R d ).Since σ < d we can choose a ρ ∈ [( σ − /d, ( d − /d ) and define u ( x ) = | ln | x || ρ ζ ( x ), where TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAˇR´IK the function ζ ∈ C ∞ ( R d ) equals one in a neighborhood of the origin. Then ρ < ( d − /d implies that u ∈ W ,d ( R d ), whereas ρ ≥ ( σ − /d implies that R R d V | u | d dx = ∞ . Thus, Q αV [ u ] = −∞ for any α > Remark 2.4.
In the quadratic case p = 2, Theorems 2.1 and 2.2 recover the asymptoticsoriginally found in [Si] using a different, operator theoretic approach. Both (2.1) and (2.2)were originally proved in [Si] under more restictive conditions on V . For d = 1 theserestrictions were later removed in [Kl1, Sec.4]; note also that according to Lemma 3.3 belowwe have S , = 1 for p = 2 and d = 1.While our theorems give a complete answer in the case V ∈ L ( R d ) (plus additional as-sumptions if p = d ) with R R d V dx >
0, the following questions, which we consider interesting,remain open:(1) What happens if V ∈ L ( R d ) (plus some additional assumptions), but R R d V dx = 0?For results in the case p = 2, see [Si, Kl1, BCEZ].(2) What happens if V / ∈ L ( R d ), but V ( x ) = | x | − σ (1+ o (1)) as | x | → ∞ with 0 < σ ≤ d ?For results in the case p = 2, see [Kl2].The proofs of Theorems 2.1 and 2.2 are given in Sections 3 and 4 respectively. Notation.
Given r > x ∈ R d we denote by B ( r, x ) ⊂ R d the open ball withradius r centred in x . If x = 0, then we write B r instead of B ( r, ⊂ R d we denote by Ω c its complement in R d . The L q norm of a function u in Ω willbe denoted by k u k L q (Ω) if Ω = R d and by k u k q if Ω = R d .3. Case d < p
Before we proceed with the proof of Theorem 2.1 we give some preliminary results con-cerning Sobolev inequality (1.5) and the properties of the functional Q V [ u ].3.1. Sobolev inequality.
We recall that S d,p denotes the optimal constant in the Sobolevinterpolation inequality (1.5). In this subsection we discuss a closely related (and, in fact,equivalent, as we shall show) minimization problem which depends on a parameter v > q > d ≥
1. We define E ( v ) = inf k u k p =1 (cid:0) k∇ u k pp − v | u (0) | p (cid:1) . (3.1)(Note that by the Sobolev embedding theorem any function in W ,q ( R d ), q > d , has acontinuous representative and therefore u (0) is unambiguously defined. The following lemmashows, in particular, that E ( v ) > −∞ . Lemma 3.1.
Let p > d ≥ and v > . Then E ( v ) = − p − dp (cid:18) dp (cid:19) dp − d ( S d,p v ) pp − d . Moreover, the infimum is attained by a non-negative, symmetric decreasing function. Finally,any minimizing sequence is relatively compact in W ,p ( R d ) . We include a proof of this lemma for the sake of completeness.
EAK PERTURBATIONS OF THE P–LAPLACIAN 5
Proof.
By the Sobolev inequality (1.5) we have | u (0) | p ≤ k u k p ∞ ≤ S d,p k∇ u k dp k u k p − dp and, therefore, if k u k p = 1, k∇ u k pp − v | u (0) | p ≥ k∇ u k pp − v S d,p k∇ u k dp ≥ inf X ≥ (cid:16) X p − v S d,p X d (cid:17) = − p − dp (cid:18) dp (cid:19) dp − d ( S d,p v ) pp − d . This shows that E ( v ) ≥ − p − dp (cid:16) dp (cid:17) dp − d ( S d,p v ) pp − d . In particular, E ( v ) > −∞ .To prove the reverse inequality, we first note that, by scaling, E ( v ) = E (1) v pp − d . (To see this, write u in the form u ( x ) = v dp ( p − d ) w ( v p − d x ).) We note also that E ( v ) < u ∈ W ,p ( R d ) with k u k p = 1 and u (0) = 0 we clearly have k∇ u k pp − v | u (0) | p → −∞ as v → ∞ and therefore E ( v ) < v . By the scalinglaw, this implies that E ( v ) < v .)Now let u ∈ W ,p ( R d ). Then, by the Sobolev embedding theorem u can be assumed tobe continuous and vanishing at infinity, so there is an a ∈ R d such that | u ( a ) | = k u k ∞ . Let˜ u ( x ) = u ( x + a ) / k u k p . Then, by the definition of E ( v ), k∇ ˜ u k pp − v | ˜ u (0) | p ≥ E ( v ) , i.e., k∇ u k pp ≥ v k u k p ∞ + E ( v ) k u k pp = v k u k p ∞ + E (1) v pp − d k u k pp . Since this is true for any v > k∇ u k pp ≥ v k u k p ∞ + E ( v ) k u k pp ≥ sup v> (cid:16) v k u k p ∞ + E (1) v pp − d k u k pp (cid:17) = k u k p d ∞ k u k − p ( p − d ) d p | E (1) | − p − dd (cid:18) p − dp (cid:19) p − dd dp . This proves that S d,p ≤ | E (1) | p − dp (cid:16) p − dp (cid:17) − p − dp (cid:16) dp (cid:17) dp .We next prove that any minimizing sequence is relatively compact in W ,p ( R d ). Let( u n ) ⊂ W ,p ( R d ) be a minimizing sequence for E ( v ). Using the bounds in the first part ofthe proof it is easy to see that ( u n ) is bounded in W ,p ( R d ) and therefore, after passing toa subsequence if necessary, we may assume that u n converges weakly in W ,p ( R d ) to some u ∈ W ,p ( R d ). By weak convergence,lim inf n →∞ k∇ u n k pp ≥ k∇ u k pp , ≥ lim inf n →∞ k u n k pp ≥ k u k pp , (3.2)and, by the Rellich–Kondrashov theorem (see, e.g., [LL, Thm. 8.9]), u n (0) → u (0). Weconclude that0 > E ( v ) = lim n →∞ (cid:0) k∇ u n k pp − v | u n (0) | p (cid:1) ≥ k∇ u k pp − v | u (0) | p ≥ E ( v ) k u k pp . TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAˇR´IK
This, together with the second assertion in (3.2) implies that k u k p = 1. Together with thefirst assertion in (3.2) and the convergence of u n (0) it also implies that k∇ u n k p → k∇ u k p .Thus, u n converges in fact strongly to u in W ,p ( R d ).Thus, we have shown that there is a minimizer. In view of the rearrangement inequalities k∇ u ∗ k p ≤ k∇ u k p , k u ∗ k p = k u k p and | u ∗ (0) | ≥ | u (0) | (see, e.g., [Ta] and [LL, Thm. 3.4]) wesee that among the minimizers there is a non-negative, symmetric decreasing function. Thisconcludes the proof. (cid:3) Remark 3.2.
It is easy to see that E ( v ) = inf k u k p =1 (cid:0) k∇ u k pp − v k u k p ∞ (cid:1) . This will be useful in the following.In one dimension we can compute the value of the sharp constant S d,p in (1.5). Lemma 3.3. If d = 1 , then S ,p = p for any p > .Proof. Let u be the (symmetric decreasing) optimizer for E ( v ). The Euler–Lagrange equa-tion reads ( p − u ′′ ( x ) ( − u ′ ( x )) p − = λu ( x ) p − in (0 , ∞ ) , (3.3)together with the boundary condition2( − u ′ (0+)) p − = vu (0) p − . Multiplying (3.3) by u ′ we obtain (cid:0) ( p − − u ′ ) p − λu p (cid:1) ′ = 0 in (0 , ∞ ) . Since u ∈ W ,p ( R d ) we have u ( x ) → x → ∞ . Since ( p − − u ′ ) p − λu p is constant,lim x →∞ u ′ ( x ) exists as well and, therefore, needs to be zero. Thus( p − − u ′ ) p − λu p = 0 in (0 , ∞ ) . (3.4)Note that this shows that λ >
0. Moreover, we obtain − u ′ = (cid:18) λp − (cid:19) p u in (0 , ∞ ) , and, thus, u ( x ) = u (0) exp − (cid:18) λp − (cid:19) p x ! in (0 , ∞ ) . The boundary condition implies that λ = ( p − v/ p/ ( p − . We conclude that E ( v ) = 2 R ∞ | u ′ | p dx − vu (0) p R ∞ u p dx = − ( p − (cid:16) v (cid:17) pp − . By Lemma 3.1 this implies the assertion. (cid:3)
EAK PERTURBATIONS OF THE P–LAPLACIAN 7
Preliminaries.Lemma 3.4.
Let p > d and assume that V ∈ L ( R d ) . Then for any u ∈ W ,p ( R d ) , Q V [ u ] ≥ − p − dp (cid:18) dp (cid:19) dp − d (cid:18) S d,p Z R d V + dx (cid:19) pp − d k u k pp . (3.5) Moreover, Q V [ u ] is weakly lower semi-continuous in W ,p ( R d ) .Proof. For any u ∈ W ,p ( R d ), Q V [ u ] ≥ k∇ u k pp − Z R d V + dx k u k p ∞ ≥ E (cid:18)Z R d V + dx (cid:19) . The second inequality used Remark 3.2. The first assertion now follows from Lemma 3.1.To prove weak lower semi-continuity assume that ( u j ) converges weakly in W ,p ( R d ) tosome u . Then the sequence ( u j ) is bounded in W ,p ( R d ) and hence, by (1.5), in L ∞ ( R d ).We have (cid:12)(cid:12)(cid:12) Z R d V ( | u j | p − | u | p ) dx (cid:12)(cid:12)(cid:12) ≤ k u j − u k L ∞ ( B R ) k f j k ∞ k V k + 2 sup j k u j k p ∞ ! k V k L ( B cR ) , (3.6)where f j := ( | u j | p − | u | p ) / ( | u j | − | u | ) satisfies | f j | ≤ p max {| u j | p − , | u | p − } and is thereforebounded. Since the sequence ( u j ) is bounded in W ,p ( R d ), inequality (1.5) implies that k f j k ∞ is bounded uniformly with respect to j . On the other hand, the Rellich-Kondrashovtheorem (see, e.g., [LL, Thm.8.9]) says that ( u j ) converges to u uniformly on compactsubsets of R d . Hence, sending first j → ∞ and then R → ∞ in (3.6) shows that thefunctional R R d V | u | p dx is weakly continuous on W ,p ( R d ). Since k∇ u k pp is weakly lowersemi-continuous, due to the fact that p >
1, the same is true for Q V [ u ]. (cid:3) Remark 3.5.
Note that inequality (3.5) yields the lower bound in (2.1) in the case V ≥ Corollary 3.6.
Let V ∈ L ( R d ) and p > d . Assume that λ ( V ) < . Then there is anon-negative function u ∈ W ,p ( R d ) such that λ ( V ) = Q V [ u ] k u k pp . (3.7) Proof.
Let ( u j ) be a minimizing sequence for Q V , normalized such that k u j k p = 1 for any j ∈ N . Since λ ( V ) <
0, we may assume without loss of generality that Q V [ u j ] < j ∈ N . Hence with the help of (1.5) we get k∇ u j k pp < Z R d V + | u j | p dx ≤ k V + k k u j k p ∞ ≤ S d,p k V + k k∇ u j k dp . (3.8)Since p > d , it follows that the sequence ( u j ) is bounded in W ,p ( R d ) and, after passing toa subsequence if necessary, we may assume that ( u j ) converges weakly in W ,p ( R d ) to some u ∈ W ,p ( R d ). The weak convergence implies k u k p ≤ lim inf j →∞ k u j k p = 1 . Since Q V [ u ] is weakly lower semicontinuous by Lemma 3.4, the above inequality implies0 > λ ( V ) = lim j →∞ Q V [ u j ] ≥ Q V [ u ] ≥ λ ( V ) k u k pp ≥ λ ( V ) . TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAˇR´IK
This implies that Q V [ u ] = λ ( V ) and k u k p = 1, i.e., u is a minimizer for the problem (1.2).Since u ∈ W ,p ( R d ) implies | u | ∈ W ,p ( R d ) with |∇| u || = |∇ u | almost everywhere (see,e.g., [LL, Thm. 6.17]), we may choose u non-negative. (cid:3) Proof of Theorem 2.1. Upper bound.
For any fixed function ϕ ∈ W ,p ( R d ) with k ϕ k p = 1 we define v α ( x ) := α dp ( p − d ) ϕ ( α p − d x ) , α > . Then k v α k p = 1 for all α > λ ( αV ) ≤ Q αV [ v α ] = α pp − d (cid:16) k∇ ϕ k pp − Z R d V ( x ) | ϕ ( α p − d x ) | p dx (cid:17) . Since ϕ ∈ W ,p ( R d ), the Sobolev embedding implies that ϕ ∈ C ( R d ) ∩ L ∞ ( R d ) and therefore,by dominated convergence, Z R d V ( x ) | ϕ ( α p − d x ) | p dx → Z R d V dx | ϕ (0) | p as α → . Since ϕ is arbitrary, we have shown thatlim sup α → α pd − p λ ( αV ) = inf k ϕ k p =1 (cid:18) k∇ ϕ k pp − Z R d V dx | ϕ (0) | p (cid:19) = E (cid:18)Z R d V dx (cid:19) . The upper bound in Theorem 2.1 now follows from Lemma 3.1.3.4.
Proof of Theorem 2.1. Lower bound.
It follows from the proof of the upper boundthat λ ( αV ) < α > α there is a non-negative minimizer u α of the problem (1.2). (It is easy to see that, in fact, λ ( αV ) < all α >
0. Indeed, α − Q αV [ u ] is non-increasing for every u ∈ W ,p ( R d ) andtherefore α − λ ( αV ) is non-increasing. Thus, if it is negative for some α >
0, it is negativefor all larger α ’s.)We normalize u α so that k u α k p = 1. The key step in the proof is to show thatlim α → α − dp − d Z R d V ( x ) ( u α ( x ) p − u α (0) p ) dx = 0 . (3.9)Assuming this for the moment, let us complete the proof. We define f α ( x ) = α − dp ( p − d ) u α (cid:16) x α − p − d (cid:17) (3.10)and observe that k f α k p = 1 and k∇ f α k pp − Z R d V α ( x ) f α ( x ) p dx = α − pp − d Q αV [ u α ] , where V α ( x ) = α − d/ ( p − d ) V ( x α − / ( p − d ) ). Since (3.9) can be rewritten aslim α → (cid:18)Z R d V α ( x ) f α ( x ) p dx − Z R d V dx f α (0) p (cid:19) = 0 , EAK PERTURBATIONS OF THE P–LAPLACIAN 9 we obtain lim inf α → α − pp − d λ ( αV ) = lim inf α → α − pp − d Q αV [ u α ]= lim inf α → (cid:18) k∇ f α k pp − Z R d V dx f α (0) p (cid:19) ≥ E (cid:18)Z R d V dx (cid:19) = − p − dp (cid:18) dp (cid:19) dp − d (cid:18) S d,p Z R d V ( x ) dx (cid:19) pp − d . (3.11)The last equality comes from Lemma 3.1. This is the lower bound claimed in Theorem 2.1.It remains to prove (3.9). Arguing as in (3.8) we obtain k∇ u α k pp ≤ α S d,p k V + k k∇ u α k dp ,and therefore k∇ u α k p ≤ Cα p − d . (3.12)According to (1.5) this also implies k u α k p ∞ ≤ C ′ α dp − d . (3.13)By Morrey’s Sobolev inequality there is a constant M = M d,p such that for all v ∈ W ,p ( R d )and all x, y ∈ R d one has | v ( x ) − v ( y ) | ≤ M| x − y | ( p − d ) /p k∇ v k p . (3.14)We now fix R > x ∈ B R | u α ( x ) − u α (0) | ≤ M R p − dp k∇ u α k p ≤ C R α p − d This, together with (3.13), yields for all x ∈ B R | u α ( x ) p − u α (0) p | ≤ p | u α ( x ) − u α (0) | max { u α ( x ) p − , u α (0) p − } ≤ C ′ R α p + d ( p − p ( p − d ) Thus, α − dp − d (cid:12)(cid:12)(cid:12)(cid:12)Z R d V ( x ) ( u α ( x ) p − u α (0) p ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ α − dp − d k V k sup B R | u pα − u α (0) p | + α − dp − d k u α k p ∞ Z B cR | V | dx ≤ α p C ′ R k V k + 2 C ′ Z B cR | V | dx . Letting first α → R → ∞ we obtain (3.9). This completes the proof.3.5. Convergence of minimizers.
The following theorem about the behavior of the u α isan (almost) immediate consequence of Lemma 3.1 and Theorem 2.1 and its proof. Proposition 3.7.
Let p > d and let V ∈ L ( R d ) with R R d V ( x ) dx > . For α > let u α bea non-negative minimizer of Q αV [ · ] with k u α k p = 1 and define f α by (3.10) . Then for anysequence ( α n ) ⊂ (0 , ∞ ) converging to zero there is a subsequence ( α n k ) and an f ∈ W ,p ( R d ) such that f α nk → f in W ,p ( R d ) . Moreover, f is a minimizer of (3.1) with v = R R d V dx . We recall that, by the Sobolev embedding theorem and the Rellich–Kondrachov theorem,convergence in W ,p ( R d ) for p > d implies convergence in L ∞ ( R d ) and in C , ( p − d ) /p ( R d ).We also note that if the minimizer of the Sobolev inequality (1.5) is unique (up to trans-lations, dilations and multiplication by constants), then Proposition 3.7 implies that f α converges as α → Proof.
It follows from (3.11) together with the upper bound in Theorem 2.1 that ( f α ) is aminimizing sequence for problem (3.1) with v = R R d V dx . Therefore, the assertion followsfrom the relative compactness asserted in Lemma 3.1. (cid:3) Case d = p Throughout this section we suppose that p = d . Similarly as in the case d < p we startwith a couple of preliminary lemmas which which will be used to ensure existence of aminimizer of problem (1.2).4.1. Preliminary results.Lemma 4.1.
Assume that V ∈ L q ( R d ) with some q > . Then Q V [ u ] / k u k dd is bounded frombelow and Q V [ · ] is weakly lower semi-continuous in W ,p ( R d ) . Recall that by Sobolev inequalities, see, e.g., [Ad], for every r ∈ [ d, ∞ ) there is a constant˜ S d,r such that k u k r ≤ ˜ S d,r k∇ u k θd k u k − θd , for all u ∈ W ,d ( R d ) . (4.1)Here 0 ≤ θ < dr = 1 − θ . Proof.
H¨older’s inequality and (4.1) with r = dq/ ( q −
1) imply that Z R d V | u | d dx ≤ k V + k q k u k dr ≤ k V + k q ˜ S d,r k∇ u k dθd k u k d (1 − θ ) d . Thus, Q V [ u ] ≥ k∇ u k dd − k V + k q ˜ S d,r k∇ u k dθd k u k d (1 − θ ) d ≥ inf X ≥ (cid:16) X − k V + k q ˜ S d,r X θ k u k d (1 − θ ) d (cid:17) ≥ − C k V + k − θ q k u k dd where C > d and q (through r ). This proves lower boundedness.Now let us prove weak lower semi-continuity of Q V [ u ]. As in the proof of Lemma 3.4it suffices to show that R R d V | u | p dx is weakly continuous on W ,d ( R d ). Assume that ( u j )converges weakly in W ,d ( R d ) to some u . Given δ > δ = { x ∈ R d : | V ( x ) | > δ } .Since ( u j ) is bounded in L d ( R d ), we have (cid:12)(cid:12)(cid:12) Z Ω cδ V ( | u | d − | u j | d ) dx (cid:12)(cid:12)(cid:12) ≤ C δ (4.2)
EAK PERTURBATIONS OF THE P–LAPLACIAN 11 with C independent of j . Moreover, the Sobolev inequality (4.1) implies that u j is uniformlybounded in L r ( R d ) for every r ∈ [ d, ∞ ). Hence by H¨older inequality (cid:12)(cid:12)(cid:12) Z Ω δ V ( | u | d − | u j | d ) dx (cid:12)(cid:12)(cid:12) ≤ k V k q (cid:18)Z Ω δ || u | d − | u j | d | qq − dx (cid:19) q − q = k V k q (cid:18)Z Ω δ | ( | u | − | u j | ) ϕ j | qq − dx (cid:19) q − q , where for every r ∈ [ d, ∞ ) there is a C r such that k ϕ j k r ≤ C r for all j . Since Ω δ hasfinite measure, u j → u in L r (Ω δ ) for any r < ∞ by the Rellich–Kondrashov theorem. (Forinstance, in [LL, Thm. 8.9], the Rellich–Kondrashov theorem is only stated for boundedsets. However, for any ε > ω ⊂ Ω δ such that | Ω δ \ ω | < ε . Then u j → u in L r ( ω ) by the bounded Rellich-Kondrashov theorem and, since ( u j ) is boundedin L s (Ω δ ) for some s > r , by H¨older k u j k L r (Ω δ \ ω ) ≤ k u j k L s (Ω δ ) ε ( s − r ) /s . Thus, u j → u in L r (Ω δ ), as claimed.)We thus conclude, again with r = 2 q/ ( q − Z Ω δ | ( | u | − | u j | ) ϕ j | qq − dx ≤ C qq − r (cid:18)Z Ω δ | u − u j | qq − dx (cid:19) / → j → ∞ . This in combination with (4.2) proves the claimed weak continuity. (cid:3)
Proof of Theorem 2.2. Upper bound.Proposition 4.2.
Let V ∈ L ( R d ) be such that R R d V ( x ) dx > . Then lim sup α → α d − log 1 | λ ( αV ) | ≤ d ω d − d (cid:16) Z R d V ( x ) dx (cid:17) − d − . (4.3) Proof.
Let β > v β defined by v β ( x ) = 1 if | x | ≤ , v β ( x ) = (cid:18) − log | x | log β (cid:19) + if | x | > . (4.4)Then v β ∈ W ,d ( R d ) and, since 0 ≤ v β ≤ χ {|·| <β } , we have k v β k dd ≤ c β d for all β > c > d . Moreover, Q αV [ v β ] ≤ ω d (log β ) − d − α Z R d V ( x ) dx + αR β with R β = Z {| x | > } V + (cid:18) − (cid:18) − log | x | log β (cid:19) + (cid:19) dx . By dominated convergence, R β → β → ∞ .Let ε > β ε > R β ≤ ε Z R d V dx for all β ≥ β ε . Now, for any α , define β ( α ) = exp (cid:18) ω d α (1 − ε ) R R d V dx (cid:19) / ( d − ! . Note that β ( α ) > ω d (log β ( α )) d − − α (1 − ε ) Z R d V dx = 0 . Define α ε > β ( α ε ) = β ε . Then for α ≤ α ε our upper bound on Q αV [ v β ] is non-positiveand therefore λ ( αV ) ≤ Q αV [ v β ( α ) ] k u β ( α ) k dd ≤ c − β ( α ) − d (cid:18) ω d (log β ( α )) − d − α Z R d V ( x ) dx + αR β (cid:19) = − c − α (cid:18) ε Z R d V dx − R β ( α ) (cid:19) exp − d (cid:18) ω d α (1 − ε ) R R d V dx (cid:19) / ( d − ! . (4.5)This implies lim sup α → α d − log 1 | λ ( αV ) | ≤ d ω d − d (cid:16) (1 − ε ) Z R d V ( x ) dx (cid:17) − d − . By letting ε → (cid:3) Corollary 4.3.
Let V satisfy assumptions of Lemma 4.1. Then for every α > there existsa locally bounded positive function u α ∈ W ,d ( R d ) such that λ ( αV ) k u α k dd = Q αV [ u α ] .Proof. Inequality (4.5) with β large enough shows that λ ( αV ) < α >
0. Hence theexistence of a non-negative minimizer u α follows from Lemma 4.1 in the same way as in thecase d < p . Since u α is a non-negative weak solution of (1.3), the Harnack inequality [S1,Thm. 6] implies that u α is locally bounded and positive. (cid:3) Proof of Theorem 2.2. Lower bound.
The case of positive V . Proposition 4.4.
Assume that ≤ V ∈ L q ( R d ) ∩ L ( R d ) for some q > with V . Thenthere are α > and C > such that for all < α ≤ α we have λ ( αV ) ≥ − C α − exp " − (cid:18) d d − ω d α R R d V dx (cid:19) d − . (4.6) Proof.
Let V ∗ be the symmetric decreasing rearrangement of V . Since R R d V dx = R R d V ∗ dx , R R d V q dx = R R d ( V ∗ ) q dx and, by rearrangement inequalities (see, e.g., [Ta] and [LL, Thm.3.4]), λ ( αV ) ≥ λ ( αV ∗ ) , we may and will assume in the following that V = V ∗ .By Corollary 4.3 there is a minimizer u α of Q αV [ u ] / k u k dd . Again, by rearrangementinequalities, we may assume that u α is a radially symmetric function which is non-increasingwith respect to the radius. Let ρ > EAK PERTURBATIONS OF THE P–LAPLACIAN 13 loss in assuming that ρ = 1, but in the proof of Proposition 4.5 we will repeat the argumentwith a general ρ .) We normalize u α such that u α ( x ) = u α ( | x | ) = 1 , for all x ∈ R d with | x | = ρ . Let R ≥ ρ be a parameter to be specified later and let χ be defined by χ ( r ) = 1 if 0 ≤ r ≤ ρ, χ ( r ) = (cid:16) − r − ρR − ρ (cid:17) + if r > ρ . Then for any ε ∈ (0 ,
1] we have k∇ ( χ u α ) k dd ≤ (1 + ε ) k χ ∇ u α k dd + c ε − d k u α ∇ χ k dd ≤ (1 + ε ) k∇ u α k dd + c ′ ε − d R − d k u α k dd , and therefore k∇ u α k dd ≥ k∇ ( χ u α ) k dd / (1 + ε ) − c ′′ ε − d R − d k u α k dd . (4.7)Since χ u α has support in the ball of radius of radius R and is bounded from below by oneon the ball of radius ρ , the formula for the capacity of two nested balls [M, Sec. 2.2.4] gives k∇ u α k dd ≥ ω d (log( R/ρ )) − d ε − c ′′ ε − d R − d k u α k dd . (4.8)Moreover, since | u α ( x ) | ≤ | x | >
1, we obtain λ ( αV ) ≥ ω d (log( R/ρ )) − d − (1 + ε ) α (cid:16)R B V u dα dx + R B c V dx (cid:17) (1 + ε ) k u α k dd − c ′′ ε d − R d . (4.9)We next claim that there are constants C > and α > < α ≤ α ,sup B ρ (cid:16) u dα − (cid:17) ≤ Cα d − . (4.10)Accepting this for the moment and returning to (4.9) we obtain λ ( αV ) ≥ ω d (log( R/ρ )) − d − (1 + ε ) (cid:16) Cα d − (cid:17) α R R d V dx (1 + ε ) k u α k dd − c ′′ ε d − R d . For given 0 < ε ≤ < α ≤ α we now choose R = ρ exp ω d (1 + ε ) (cid:16) Cα d − (cid:17) α R R d V dx d − so that λ ( αV ) ≥ − c ′′ ε d − ρ d exp − d ω d (1 + ε ) (cid:16) Cα d − (cid:17) α R R d V dx d − . Finally, we choose ε = Cα d − to obtain λ ( αV ) ≥ − c ′′′ α exp − d ω d (cid:16) C ′ α d − (cid:17) α R R d V dx d − . (4.11)Up to increasing c ′′′ this implies the statement of the proposition. Thus, it remains to prove (4.10). For simplicity we give the proof only for ρ = 1 (which isenough for the proof of the proposition). We apply Alvino’s version of the Moser–Trudingerinequality [Al] to the function u α − < u α ( r ) − ≤ C k∇ u α k L d ( B ) | log r | d − d , r ≤ . (4.12)Using this upper bound on u α we arrive at k∇ u α k dL d ( B ) ≤ k∇ u α k dd ≤ α Z R d V | u α | d dx ≤ α d − (cid:18) k V k L ( B ) + C k∇ u α k dL d ( B ) ω d Z V ( r ) | log r | d − r d − dr (cid:19) . The assumption V ∈ L q ( R d ) for some q > V ∈ L ( B , | log | x || d − dx ), andtherefore there is a C ′ > α > < α ≤ α k∇ u α k dL d ( B ) ≤ C ′ α /d . Re-inserting this into (4.12), we find for all 0 < α ≤ α < u α ( r ) − ≤ C ′′ α /d | log r | d − d , r ≤ . (4.13)Hence the minimizer u α satisfies for all 0 < r ≤ − r u ′ α ( r )) d − ) ′ = α V ( r ) u α ( r ) d − r d − + λ ( α ) u α ( r ) d − r d − (4.14) ≤ α V ( r ) r d − (cid:0) C ′′ α d | log r | d − d (cid:1) d − and (( − r u ′ α ( r )) d − ) ′ = α V ( r ) u α ( r ) d − r d − + λ ( α ) u α ( r ) d − r d − (4.15) ≥ λ ( α ) r d − (cid:0) C ′′ α d | log r | d − d (cid:1) d − . Since the right hand sides of (4.14) and (4.15) are integrable with respect to r (for (4.14)we use here again the assumption that V ∈ L ( R d ) ∩ L q ( R d ) for some q > − r u ′ α ( r )) d − has a finite limit as r →
0. Since u α ∈ W ,d ( R d ), it follows that this limitmust be zero. Thus, from (4.14) we get for all 0 < r ≤ − r u ′ α ( r )) d − ≤ α Z r V ( s ) s d − (cid:0) C ′′ α d | log s | d − d (cid:1) d − ds ≤ α k V k L q ( B ) (cid:18)Z r s d − (cid:0) C ′′ α d | log s | d − d (cid:1) q ′ ( d − ds (cid:19) /q ′ ≤ C ′′′ α k V k L q ( B ) r d/q ′ (1 + | log r | ) ( d − d . Finally, this implies that u α ( r ) − − Z r u ′ α ( s ) ds ≤ (cid:0) C ′′′ α k V k L q ( B ) (cid:1) d − Z r s dq ′ ( d − (1 + | log s | ) ( d − d dss . EAK PERTURBATIONS OF THE P–LAPLACIAN 15
Since the integral on the right side converges, we have shown (4.10). This completes theproof of the lemma. (cid:3)
The case of compactly supported V . Proposition 4.5.
Let V be a function with compact support, R R d V ( x ) dx > and V ∈ L q ( R d ) for some q > . Then there are α > and C > such that for all < α ≤ α wehave λ ( αV ) ≥ − exp − d d − ω d α R R d V dx (1 + Cα d ) ! d − . (4.16)Similarly as in the case d < p a key ingredient in the proof is to show that minimizers,when suitably normalised, converge locally to a constant function. In the case d < p wededuced this from Morrey’s inequality. Here the argument is considerably more complicatedand based on Harnack’s inequality for quasi-linear equations. We shall prove Lemma 4.6.
For each d ∈ N , q > and M > there are constants C > and β ∈ (0 , with the following property. Let ρ > and assume that W ∈ L q l oc ( R d ) with W ≤ in B c ρ and ρ d − dq k W k L q ( B ρ ) ≤ M . Then, if u ∈ W ,d ( R d ) is a positive, weak solution of theequation − ∆ d ( u ) = W u d − in R d satisfying inf B ρ u ≤ and if y ∈ R d and r > are so that B (3 r, y ) ⊂ B ρ , we have sup B ( r,y ) u − inf B ( r,y ) u ≤ C k W k /dL q ( B ρ ) ρ − q − β r β . (4.17)The point of this lemma is that the dependence of W enters explicitly on the right sideof (4.17). In our application, we will have k W k L q ( B ρ ) →
0, and therefore Lemma 4.6 showsthat the oscillations of u vanish with an explicit rate.We recall that u is a weak solution of − ∆ d ( u ) = W | u | d − u in R d if Z R d |∇ u | d − ∇ u · ∇ ϕ dx = Z R d W | u | d − u ϕ dx (4.18)for any ϕ ∈ W ,d ( R d ).The following lemma, whose proof can be found, for instance, in [Mo1, Mo2] or [LU, Lem.2.4.1], plays a key role in the proof of Lemma 4.6. Lemma 4.7.
Let Ω ⊆ R d be open and assume that u ∈ W ,d (Ω) is such that there areconstants K > and β > such that for all y ∈ Ω and r > with B ( r, y ) ⊂ Ω one has Z B ( r,y ) |∇ u | d dx ≤ K r βd . (4.19) Then for all y ∈ Ω and r > such that B (3 r/ , y ) ⊂ Ω we have sup B ( r/ ,y ) u − inf B ( r/ ,y ) u ≤ β (cid:18) Kω d (cid:19) d r β . (4.20) Proof of Lemma 4.6.
By the Harnack inequality [S1, Thm.6] there is a constant C , whichdepends only on d , q and an upper bound on ρ d − dq k W k L q ( B ρ ) such thatsup B ρ u ≤ C inf B ρ u . Since inf B ρ u ( x ) ≤
1, we conclude thatsup B ρ u ( x ) ≤ C . (4.21)Our goal is to apply Lemma 4.7 with Ω = B ρ . We have to verify condition (4.19) forsome K and β . First, note that Z R d |∇ u | d dx = Z R d W u d dx ≤ Z B ρ W u d dx ≤ ω − q d (5 ρ ) d − dq k W k L q ( B ρ ) C d = c N , (4.22)where we have set c = ω − q d d − dq and N = ρ d − dq k W k L q ( B ρ ) C d . (4.23)Hence, for any β >
0, (4.19) holds for any ball B ( r, y ) ⊂ B ρ with r ≥ ρ provided we choosethe constan K at least as big as c N ρ − βd .Thus, it remains to verify (4.19) for r < ρ . Let 0 ≤ ζ ≤ B which is ≡ B and satisfies |∇ ζ | ≤
1. Let y and s be such that B (2 s, y ) ⊂ B ρ .We choose the test function ϕ ( x ) = ζ ( | x − y | /s )( u ( x ) − a ) in (4.18), where the parameter a will be specified later. This gives the inequality Z B ( s,y ) |∇ u | d dx ≤ Z R d ζ ( | x − y | /s ) |∇ u | d dx ≤ Z B (2 s,y ) | W | u d − | u − a | dx + s − Z A ( s,y ) |∇ u | d − | u − a | dx . (4.24)with A ( s, y ) = B (2 s, y ) \ B ( s, y ). Now we set a = | A ( s,y ) | R A ( s,y ) u dx , where | A ( s, y ) | denotesthe Lebesgue measure of A ( s, y ). By the H¨older and Poincar´e inequalities, Z A ( s,y ) |∇ u | d − | u − a | dx ≤ (cid:16) Z A ( s,y ) |∇ u | d dx (cid:17) d − d (cid:16) Z A ( s,y ) | u − a | d dx (cid:17) d ≤ C P s Z A ( s,y ) |∇ u | d dx , where C P is the constant in the Poincar´e inequality in A (1 , A ( s, y ) is given by C P s . This fact was used in the previousbound.Let us bound the first term on the right side of (4.24). Since both u and | a | are boundedfrom above by C on B (2 s, y ), see (4.21), we have Z B (2 s,y ) | W | u d − | u − a | dx ≤ k W k L ( B (2 s,y )) C p ≤ c N ( s/ρ ) d − dq , where c = ω − q d d +1 − dq .Thus, (4.24) implies Z B ( s,y ) |∇ u | d dx ≤ c N ( s/ρ ) d − dq + C P Z A ( s,y ) |∇ u | d dx, EAK PERTURBATIONS OF THE P–LAPLACIAN 17 where c = 2 d +1 − dq ω − q d . Adding C P R B ( s,y ) |∇ u | d dx to both sides of the above inequalitywe arrive at Z B ( s,y ) |∇ u | d dx ≤ c N ( s/ρ ) d − dq + κ Z B (2 s,y ) |∇ u | d dx, (4.25)with c = c / (1 + C P ) and κ = C P C P < . To simplify the notation, we introduce the shorthand D ( s ) = R B ( s,y ) |∇ u | d dx . Iteratinginequality (4.25) gives D (2 − n s ) ≤ c N ( s/ρ ) d − dq n ( dq − d ) n − X j =0 (cid:0) κ d − dq (cid:1) j + κ n D ( s )for all n ∈ N and every s > B ( s, y ) ⊂ B ρ . Next, we sum the geometric serieson the right side and obtain a c and a µ < d and q ) such that2 n ( dq − d ) n − X j =0 (cid:0) κ d − dq (cid:1) j ≤ c µ n for all n ∈ N . Thus, recalling (4.22), D (2 − n s ) ≤ (cid:16) c c ( s/ρ ) d − dq + c (cid:17) N max { µ n , κ n } (4.26)for all n ∈ N and all s such that B ( s, y ) ⊂ B ρ .Now let B ( r, y ) ⊂ B ρ with r < ρ . There are k ∈ N and t ∈ [1 ,
2) such that 2 − k − tρ
The beginning of the proof is identical to that of Proposition 4.4.Let ρ > V is contained in B ρ . We let again u α be a minimizerof Q αV [ u ] / k u k dd . From Corollary 4.3 we know that u α can be chosen strictly positive andtherefore we may normalize u α by inf B ρ u α = 1. Arguing exactly as before we arrive at thefollowing variant of (4.9), λ ( αV ) ≥ ω d (log( R/ρ )) − d − (1 + ε ) α R R d V | u α | d dx (1 + ε ) k u α k dd − c ′′ ε − d R − d . (4.27) We now claim that there is a constant
C > d , q , V , but not on α ) suchthat | u α ( x ) − | ≤ C α d for all x ∈ B ρ . (4.28)Indeed, this follows from Lemma 4.6 applied to W = αV + λ ( αV ) and u = u α with B ( r, y ) = B ρ . Note that we indeed have inf B ρ u α ≤ inf B ρ u α = 1. Moreover, we use the fact that λ ( αV ) ≥ − Cα , which follows easily from the bounds in Lemma 4.1.With a similar choice as in Lemma 4.4 for R we obtain λ ( αV ) ≥ − c ′′ ε d − ρ d exp − d ω d (1 + ε ) (cid:16) C α d (cid:17) α R R d V dx d − . Choosing ε = Cα d we obtain λ ( αV ) ≥ − c ′′′ α d − d exp − d ω d (cid:16) C ′ α d (cid:17) α R R d V dx d − . This implies the statement of the proposition. (cid:3)
The general case.
We can finally give the
Proof of Theorem 2.2.
We use an approximation argument and fix ε ∈ (0 ,
1) and
R > V < = V χ {|·|
0. It then follows from Proposition4.5 thatlim inf α → α d − log 1(1 − ε ) | λ ((1 − ε ) − αV < ) | ≥ (1 − ε ) d − d ω d − d (cid:18)Z B R V ( x ) dx (cid:19) − d − . On the other hand, we recall from Proposition 4.6 that there are constants
C > α > < α ≤ α ε , λ ( ε − αV > ) ≥ − Cεα − exp − εd d − ω d α R B cR V + dx ! d − EAK PERTURBATIONS OF THE P–LAPLACIAN 19
Moreover, we recall from Proposition 4.2 that for every δ ∈ (0 ,
1) there are constants C δ > α δ such that for all 0 < α ≤ α δ (1 − ε ), λ ((1 − ε ) − αV < ) ≤ − (1 − ε ) − α C δ exp − (1 − ε ) d d − ω d α (1 − δ ) R B R V dx ! d − . (4.29)Thus, for α ≤ min { α ε, α δ (1 − ε ) } , | λ ( ε − αV > ) || λ ((1 − ε ) − αV < ) | ≤ Cε (1 − ε ) C δ α exp − εd d − ω d α R B cR V + dx ! d − + (1 − ε ) d d − ω d α (1 − δ ) R B R V dx ! d − For every fixed ε and δ there is an R > R > R , ε R B cR V + dx > − ε (1 − δ ) R B R V dx .
Thus, for all
R > R we have lim α → | λ ( ε − αV > ) || λ ((1 − ε ) − αV < ) | = 0 . To summarize, we have shown that for all ε ∈ (0 ,
1) and for all
R > R ,lim inf α → α d − log 1 | λ ( αV ) | ≥ (1 − ε ) d − d ω d − d (cid:18)Z B R V ( x ) dx (cid:19) − d − . Letting ε → R → ∞ we obtain the theorem. (cid:3) Acknowledgements
Partial financial support through Swedish research council grant FS-2009-493 (T. E.),U.S. National Science Foundation grant PHY-1347399 (R. F.) and grant MIUR-PRIN08grant for the project “Trasporto ottimo di massa, disuguaglianze geometriche e funzionali eapplicazioni” (H. K.) is acknowlegded.
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E-mail address : [email protected] Rupert L. Frank, Mathematics 253-37, Caltech, Pasadena, CA 91125, USA
E-mail address : [email protected] Hynek Kovaˇr´ık, DICATAM, Sezione di Matematica, Universit`a degli studi di Brescia, ViaBranze, 38 - 25123 Brescia, Italy
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