Critical functions and blow-up asymptotics for the fractional Brezis–Nirenberg problem in low dimension

. For s ∈ (0 , 1) and a bounded open set Ω ⊂ R N with N > 2 s , we study the fractional Brezis(cid:21)Nirenberg type minimization problem of (cid:28)nding


Introduction and main results
Let N ∈ N and 0 < 2s < N for some s ∈ (0, 1), and let Ω ⊂ R N be a bounded open set.The goal of the present paper is to analyze the variational problem of minimizing, for a given a ∈ C(Ω), the quotient functional S a [u] := R N |(−∆) s/2 u| 2 dy + Ω a(y)u(y) 2 dy u 2 over functions in the space where u ∈ H s (R and the fractional Laplacian operator (−∆) s u is dened for any u ∈ H s (R N ) through the Fourier representation (−∆) s u = F −1 (|ξ| 2s Fu). (1.4) We also recall the singular integral representation of the fractional Laplacian (see [11,25]): (−∆) s u(x) := C N,s P.V.
The associated inmum, S(a) := inf S a [u] : u ∈ H s (Ω) , (1.7) is to be compared with the number S := S N,s := S(0), which is equal to the best constant in the fractional Sobolev embedding given by S N,s := 2 2s π s Γ( N +2s 2 ) Γ( N −2s 2 ) Γ(N/2) Γ(N ) . (1.9) We note that the embedding H s (Ω) → L p+1 (Ω) and the associated best constant are in fact independent of Ω and equal to the best full-space Sobolev constant S N,s (see [19]).In the classical case s = 1, problem (1.7) has been rst studied in the famous paper [9] by Brezis and Nirenberg, who were interested in obtaining positive solutions to the associated elliptic equation.One of the main ndings in that paper is that the behavior of (1.7) depends on the space dimension N in a rather striking way.Indeed, when N ≥ 4, then S(a) < S if and only if a(x) < 0 for some x ∈ Ω.On the other hand, when N = 3, then S(a) = S whenever a ∞ is small enough, leaving open the question of characterizing the cases S(a) < S in terms of a.In [20], Druet proved that, for N = 3, the following equivalence holds: S(a) < S ⇐⇒ φ a (x) < 0 for some x ∈ Ω, where φ a (x) denotes the Robin function associated to a (see (1.11) below).This answered positively a conjecture previously formulated by Brezis in [8].
In this paper, we shall exclusively be concerned with the low-dimensional range 2s < N < 4s.This is the natural replacement of the classical case N = 3. s = 1, as indicated by the results above.
One may also notice that when 2s < N , the Green's function for (−∆) s on R N behaves like G(x, y) ∼ |x−y| −N +2s near the diagonal and thus fails to be in L 2 loc (R N ) precisely if N ≤ 4s, compare [29].
A central notion to what follows is that of a critical function a, which was introduced by Hebey and Vaugon in [28] for for s = 1 and readily generalizes to the fractional situation.Indeed, the following denition is naturally suggested by the behavior of S(a) just described.
When N ≥ 4s, the result of [38] implies that the only critical potential is a ≡ 0. For this case, or more generally for N > 2s with a ≡ 0, the recent literature is rather rich in rened results going beyond [38].
Notably, in [15] and [14], the authors prove the fractional counterpart of some conjectures by Brezis and Peletier [10] concerning the blow-up asymptotics of minimizers to the problem S(−ε) and a related problem with subcritical exponent p − ε as ε → 0. In the classical case s = 1, these results are due to Han [27] and Rey [33,34].Corresponding existence results, also for non-minimizing multi-bubble solutions, are also given in [15,14], as well as in [17,26].
In contrast to this, in the more challenging setting of dimension 2s < N < 4s, critical functions can have all possible shapes and are necessarily non-zero, compare [20] and Corollary 1.3 below.In this setting, and notably in the presence of a critical function, results of HanRey type as just discussed are much more scarce in the literature.Even in the local case s = 1 and N = 3, the conjecture of Brezis and Peletier (see [10,Conjecture 3.(ii)]) which involves a (constant) critical function has only been proved recently in [23].For the fractional case 2s < N < 4s, we are not aware of any results going beyond [36].The purpose of the present paper is to start lling this gap.
1.1.Main results.For all of our results, a crucial role is played by the Green's function of (−∆) s + a, which we introduce now.For a function a ∈ C(Ω) such that (−∆) s + a is coercive, i.e. for some c > 0, dene G a : Ω × R N → R as the unique function such that for every xed x ∈ Ω Here, we set γ N,s = 2 2s π N/2 Γ(s) , so that (−∆) s |y| −N +2s = γ N,s δ 0 on R N .Thus this choice of γ N,s ensures that we can write G a as a sum of its singular part and its regular part H a (x, y) as follows: The function H a is continuous up to the diagonal, see e.g.Lemma A.3.Therefore, we may dene the Robin function φ a (x) := H a (x, x), x ∈ Ω. ( We prove several properties of the Green's functions G a in Appendix A.
Our rst main result is the following extension of Druet's theorem from [20] to the fractional case.
Theorem 1.2 (Characterization of criticality).Let 2s < N < 4s and let a ∈ C(Ω) be such that (−∆) s + a is coercive.The following properties are equivalent.
(iii) S(a) is achieved by some function u ∈ H s (Ω).
As an immediate corollary, we can characterize critical functions in terms of their Robin function.
The implications (i) ⇒ (ii) and (ii) ⇒ (iii) in Theorem 1.2 are well-known: indeed, (i) ⇒ (ii) easily follows by the proper choice of test functions thanks to Theorem 2.1 below; the implication (ii) ⇒ (iii) is the fractional version of the seminal observation in [9] (see [38,Theorem 2]).
Our proof of (iii) ⇒ (ii) is the content of Proposition 3.1 below and follows [20, Step 1].The most involved proof is that implication is (ii) ⇒ (i), which we give in Section 4. We adapt the strategy developed by Esposito in [21], who gave an alternative proof of that implication.His approach is based on an expansion of the energy functional S a−ε [u ε ] as ε → 0, where a is critical as in Denition 1.1 and u ε is a minimizer of S(a − ε).
In fact, by using the techniques applied in the recent work [24] for s = 1, we are even able to push this expansion of S a−ε [u ε ] further by one order of ε and derive precise asymptotics of the energy S(a − ε) and of the sequence (u ε ).
To give a precise statement of our results, let us x some more assumptions and notations.We denote the zero set of the Robin function φ a by N a := {x ∈ Ω : φ a (x) = 0}.
It follows from Theorem 1.2 that inf Ω φ a (x) = 0 if and only if a is critical.In particular, N a is not empty if a is critical.
We will consider perturbations of a of the form a + εV , with non-constant V ∈ L ∞ (Ω).For such V , following [24], we let Finally, we shall assume that Ω has C 2 boundary and that (1.12)By Corollary 2.2, we have a priori that a(x) ≤ 0 on N a .Therefore assumption (1.12) is not severe.
We point out that with our methods we are able to prove the following theorems only for the restricted dimensional range 8 3 s < N < 4s, which enters in Section 5. We discuss this assumption in some more detail after the statement of Theorem 1.6 below.
The following theorem describes the asymptotics of the perturbed minimal energy S(a + εV ) as ε → 0+.It shows in particular the non-obvious fact that the condition N a (V ) = ∅ is sucient to have S(a + εV ) < S.
Theorem 1.4 (Energy asymptotics).Let 8 3 s < N < 4s.Let us assume that N a (V ) = ∅.Then S(a + εV ) < S for all ε > 0 and where σ N,s > 0 is a dimensional constant given explicitly by The constants A N,s , α N,s , c N,s , d N,s and b N,s are given explicitly in Lemma B.5 below.
On the other hand, when N a (V ) = ∅, the next theorem shows that the asymptotics become trivial provided Q V > 0 on N a .Only in the case when min Na Q V = 0 we do not obtain the precise leading term of S(a + εV ) − S.
For a potential V such that N a (V ) = ∅, and thus S(a + εV ) < S by Theorem 1.4, a minimizer u ε of S(a + εV ) exists by Theorem 1.2.We now turn to studying the asymptotic behavior of the sequence (u ε ).In fact, since our methods are purely variational, we do not need to require that the u ε satisfy a corresponding equation and we can equally well treat a sequence of almost minimizers in the sense of (1.17) below.
Since the functional S a is merely a perturbation of the standard Sobolev quotient functional, it is not surprising that to leading order, the sequence u ε approaches the family of functions (1.13) The U x,λ are precisely the optimizers of the fractional Sobolev inequality on R N . (1.14) and satisfy the equation with c N,s > 0 given explicitly in Lemma B.5.
Since we are working on the bounded set Ω, the rst renement of the approximation consists in 'projecting' the functions U x,λ to H s (Ω).That is, we consider the unique function P U x,λ ∈ H s (Ω) in the weak sense, that is, x,λ η dy for every η ∈ H s (Ω).
Finally, we introduce the space and denote by T ⊥ x,λ ⊂ H s (Ω) its orthogonal complement in H s (Ω) with respect to the scalar product Moreover, let us denote by Π x,λ and Π ⊥ x,λ the projections onto T x,λ and T ⊥ x,λ respectively.
Then we have the following result.
Theorem 1.6 (Concentration of almost-minimizers).Let 8 3 s < N < 4s.Suppose that (1.17) Then there exist sequences , and (α ε ) ⊂ R such that, up to extraction of a subsequence, Moreover, as ε → 0, we have The constants α N,s , c N,s , d N,s and b N,s are given explicitly in Lemma B.5.
Theorem 1.6 should be seen as the low-dimensional counterpart of [15, Theorems 1.1 and 1.2], which concerns N > 4s.The decisive additional complication to be overcome in our case is the presence of a non-zero critical function a.More concretely, the coecient φ a (x) of the subleading term of the energy expansion vanishes due to criticality of a (compare Theorem 2.1 and Lemma 5.5).As a consequence, it is only after further rening the expansion that we are able to conclude the desired information about the concentration behavior of the sequence u ε .
In the same vein, the energy expansions from Theorem 1.4 are harder to obtain than their analogues in higher dimensions N ≥ 4s.Indeed, for N > 4s we have where cN,s > 0 is some dimensional constant.In this case, a sharp upper bound on S(εV ) can already be derived from testing S εV against the family of functions P U x,λ .In contrast, for 2s < N < 4s this family needs to be modied by a lower order term in order give the sharp upper bound for Theorem 1.4 (see (2.1) and Theorem 2.1 below).For details of the computations in case N ≥ 4s, we refer to the forthcoming work [18].It is noteworthy that the auxiliary minimization problem giving the subleading coecient in (1.19) is local in V in the sense that it only involves the pointwise value V (x), whereas that of Theorem 1.4 contains the non-local quantity Q V .
Let us now describe in more detail the approach we use in the proofs of Theorems 1.4, 1.5 and 1.6, which are in fact intimately linked.Firstly, the family of functions ψ x,λ dened in (2.1) below yields an upper bound for S(a + εV ), which will turn out to be sharp.The strategy we use to prove the corresponding lower bound on S a+εV [u ε ], for a sequence (u ε ) as in (1.17), can be traced back at least to work of Rey [33,34] and BahriCoron [3] on the classical BrezisNirenberg problem for s = 1; it was adapted to treat problems with a critical potential a when s = 1, N = 3 in [21] and, more recently, in [24,23].This strategy features two characteristic steps, namely (i) supplementing the initial asymptotic expansion u ε = α ε (P U xε,λε + w ε ), obtained by a concentration-compactness argument, by the orthogonality condition w ε ∈ T ⊥ xε,λε and (ii) using a certain coercivity inequality, valid for functions in T ⊥ xε,λε , to improve the bound on the remainder w ε .The basic instance of this strategy is carried out in Section 3. Indeed, after performing steps (i) and (ii), in Proposition 3.6 below we are able to exclude concentration near ∂Ω and obtain a quantitative bound on w ε = α −1 ε u ε − P U xε,λε .As in [33] and [22], this information is enough to arrive at (1.19) and similar conclusions when N > 4s; see the forthcoming paper [18] for details.
On the other hand, when 2s < N < 4s, the bound that Proposition 3.6 provides for the modied dierence u ε − ψ xε,λε is still insucient.For s = 1, it was however observed in [24] that one can rene the expansion of u ε by reiterating steps (i) and (ii).Here, we carry out their strategy in a streamlined version (compare Remark 5.1) and for fractional s ∈ (0, 1).That is, one writes w ε = ψ xε,λε − P U xε,λε + q ε , decomposes q ε = t ε + r ε with r ε ∈ T ⊥ xε,λε and applies the coercivity inequality a second time.We are able to conclude as long as the technical condition 8s/3 < N is met (which is equivalent to λ −3N +6s = o(λ −2s )).Indeed, in that case the leading contributions of t ε to the energy, which enter to orders λ −N +2s and λ −2N +4s , cancel precisely; see Lemma 5.8.If 8s/3 ≤ N , a plethora of additional terms in t ε , which contribute to orders λ −k(N −2s) with 3 ≤ k ≤ 2s N −2s , will become relevant, and we were not able to treat those in a systemized way.It is natural to expect that the cancellation phenomenon that occurs for k = 1, 2 still persists for k ≥ 3.This would allow to prove Theorems 1.4, 1.5, and 1.6 for general N > 2s.For further details of the argument and the diculties just discussed, we refer to Section 5.
As far as we know, the role of the threshold congurations given by k(N − 2s) = 2s for k ≥ 1 in the fractional BrezisNirenberg problem (1.7) has only been investigated in the literature for k = 1 corresponding to N = 4s, below which the problem is known to behave dierently by the results quoted above.It would be exciting to exhibit some similar, possibly rened, qualitative change in the behavior of (1.7) at one or each of the following thresholds N = 3s, N = 8s/3, N = 10s/4, etc.
To conclude this introduction, let us mention that several works in the literature (see [41,4,6]) treat the problem corresponding to (1.7) for a dierent notion of Dirichlet boundary conditions for (−∆) s on Ω, namely the spectral fractional Laplacian, dened by classical spectral theory using the L 2 (Ω)-ONB of Dirichlet eigenfunctions for −∆.In contrast to this, the notion of (−∆) s we use in this paper, as dened in (1.4) or (1.5) on H s (Ω) given by (1.2), usually goes in the literature by the name of restricted fractional Laplacian.A nice discussion of these two operators, as well as a method of unied treatment for both, can be found in [17] (see also [37]).
Our method of proof just described is rather dierent from most of the previous contributions to the fractional BrezisNirenberg problem.Namely, we do not need to pass through the extension formulation for (−∆) s due to either [13] for the restricted or to [12,40] for the spectral fractional Laplacian.On the other hand, using the properties of P U x,λ (as given in Lemma A.2) allows us to avoid lengthy calculations with singular integrals, appearing e.g. in [38], while at the same time optimizing the cuto procedure with respect to [38].
For a set M and functions f, g : M → R + , we shall write f (m) g(m) if there exists a constant C > 0, independent of m, such that f (m) ≤ Cg(m) for all m ∈ M , and accordingly for .If f g and g f , we write f ∼ g.
The various constants appearing throughout the paper and their numerical values are collected in Lemma B.5 in the appendix.

Proof of the upper bound
The following theorem gives the asymptotics of S a+εV [ψ x,λ ], for the test function   Theorem 2.1 (Expansion of S a+εV [ψ x,λ ]).As λ → ∞, uniformly for x in compact subsets of Ω and for ε ≥ 0, and In particular, (2.4) Here, T i (φ, λ) are (possibly empty) sums of the form for suitable coecients γ i (k) ∈ R, where K = 2s N −2s is the largest integer less than or equal to 2s N −2s .
Theorem 2.1 is valid irrespective of the criticality of a.The following corollary states two consequences of Theorem 2.1, which concern in particular critical potentials.
Based on Theorem 2.1, we can now easily derive the following upper bound for S(a + εV ) provided that N a (V ) is not empty.
, are strictly positive by our assumptions, we are in the situation of Lemma B.6.
Proof of Theorem 2.1.
Step 1: Expansion of the numerator.
Therefore, recalling Lemma A.2, We now treat the four terms on the right side separately.
A simple computation shows that Thus the rst term is given by c N,s The second term is, by Lemma A.4, The third term will be combined with a term coming from Ω (a + εV )ψ 2 x,λ dy, see below.
The fourth term can be bounded, by Lemma B.1 and recalling f Now we treat the potential term.We have Similarly to the above, the terms containing f x,λ are bounded by and Finally, we combine the main term with the third term in the expansion of (−∆) s/2 ψ x,λ 2 2 from above.
Recalling that Since Moreover, again by (2.10), and using that h This completes the proof of the claimed expansion (2.2).
Firstly, writing ψ , we deduce that the remainder term is o(λ −2s ).To evaluate the main term, from Taylor's formula for t → t p , we have Here, By Lemma A.4, the claimed expansion (2.3) follows.
Step 3: Expansion of the quotient.For α = 2/p ∈ (0, 1), and xed a > 0, we again use the Taylor expansion By Step 2, we may apply this with a = A N,s and b and thus (2.11) for some term T 3 (φ, λ) as in (2.5).Multiplying this expansion with (2.2), we obtain By integrating the equation (−∆) s U 0,1 = c N,s U p−1 0,1 and using the fact that U 0,1 minimizes the Sobolev quotient on R N (or by a computation on the numerical values of the constants given in Lemma B.5), we have c N,s A 2s N N,s = S. Hence, this is the expansion claimed in (2.4).

Proof of the lower bound I: a first expansion
3.1.Non-existence of a minimizer for S(a).In this section, we prove that for a critical potential a, the inmum S(a) is not attained.As we will see in Section 3.2, this implies the important basic fact that the minimizers for S(a + εV ) must blow up as ε → 0.
The following is the main result of this section.Proposition 3.1 (Non-existence of a minimizer for S(a)).Suppose that a ∈ C(Ω) is a critical potential.Then is not achieved.
For s = 1, Proposition 3.1 was proved by Druet [20] and we follow his strategy.The feature that makes the generalization of [20] to s ∈ (0, 1) not completely straightforward is its use of the product rule for ordinary derivatives.Instead, we shall use the identity where Proof of Proposition 3.1.For the sake of nding a contradiction, we suppose that there exists u which achieves S(a), normalized so that Then u satises the equation with Lagrange multiplier S = S N,s equal to the Sobolev constant.(Indeed, this value is determined by integrating the equation against u and using (3.2).) Since S(a) = S, we have, for every ϕ ∈ C ∞ (R N ) and ε > 0, and abbreviating p = 2N We shall expand both sides of (3.4) in powers of ε.For the left side, a simple Taylor expansion together with (3.2) gives Expanding the right side is harder and we need to invoke the fractional product rule (3.1).Firstly, integrating by parts we have By (3.1), we can write where we write Writing out (−∆) s ϕ as the singular integral given by (1.5), we obtain (we drop the principal value for simplicity) The last equality follows by symmetrizing in the x and y variables.
Thus we can write the right side of (3.4) as where we used equation (3.3).After expanding the square (1 + εϕ) 2 , the terms of orders 1 and ε on both sides of (3.4) cancel precisely.For the coecients of ε 2 , we thus recover the inequality We now make a suitable choice of ϕ, which turns (3.8) into the desired contradiction.As in [20], we choose ϕ i (y) := (S(y)) i , i = 1, ..., N + 1, where S : R N → S N is the (inverse) stereographic projection, i.e. [31,Sec. 4.4] (3.9) Moreover we may assume (up to scaling and translating Ω if necessary) that the balancing condition is satised.Since [20] is rather brief on this point, we include some details in Lemma 3.2 below for the convenience of the reader.
By denition, we have N +1 i=1 ϕ 2 i = 1.Testing (3.8) with ϕ i and summing over i thus yields, by (3.10), To obtain a contradiction and nish the proof, we now show that N +1 i=1 I(ϕ i ) < S(p−2).By denition of ϕ i , we have (3.12) To evaluate this integral further, we pass to S N .Set J S −1 (η) := det DS −1 (η) and dene , changing variables in (3.12) gives By applying rst CauchySchwarz and then Hölder's inequality, we estimate where the last inequality is strict.Indeed, U vanishes near the south pole of S N , hence there cannot be equality in Hölder's inequality applied with the functions U 2 and 1.Moreover, in the above we abbreviated δ N,s := (note that this number is independent of η ∈ S N ).By transforming back to R n and evaluating a Beta function integral, the explicit value of δ N,s can be computed explicitly to be .
Inserting this into estimate (3.14), as well as the explicit values of C N,s given in (1.6) and of S N,s given in (1.9), a direct computation then gives 2s N .
It can be easily shown by induction over N that for every N ∈ N, and hence This is the desired contradiction to (3.11).
Here is the lemma that we referred to in the previous proof.It expands an argument sketched in [20, Step 1].To emphasize its generality, instead of u p we state it for a general nonnegative function h satises the property stated in the lemma.

Prole decomposition.
The following proposition gives an asymptotic decomposition of a general sequence of normalized (almost) minimizers of S(a + εV ).
Proposition 3.3 (Prole decomposition).Let a ∈ C(Ω) be critical and let V ∈ C(Ω) be such that U p 0,1 dy. (3.17) Then there are sequences , and (α ε ) ⊂ R such that, up to extraction of a subsequence, Moreover, as ε → 0, we have In all of the following, we shall always work with a sequence u ε that satises the assumptions of Proposition 3.3.For readability, we shall often drop the index ε from α ε , x ε , λ ε and w ε , and write d := d ε := d(x ε ).Moreover, we make the convention that we always assume the strict inequality S(a + εV ) < S.
(3. 19) In Theorems 1.4 and 1.6 we assume N a (V ) = ∅, so assumption (3.19) is certainly justied in view of Corollary 2.3.For Theorem 1.5, where we assume N a (V ) = ∅, we discuss the role of assumption (3.19) in the proof of that theorem in Section 6.
Proof.Step 1.We derive a preliminary decomposition in terms of the Sobolev optimizers U z,λ and without orthogonality condition on the remainder, see (3.20) below.
The assumptions imply that the sequence (u ε ) is bounded in H s (Ω), hence up to a subsequence we may assume u ε u 0 for some u 0 ∈ H s (Ω).By the argument given in [24, Proof of Proposition 3.1, Step 1], the fact that S a+εV [u ε ] → S(a) = S implies that either u 0 is a minimizer for S(a), unless u 0 ≡ 0. Since such a minimizer does not exist by Proposition 3.1, we conclude that in fact u ε 0 in H s (Ω).
. The assumption (1.17) therefore implies that (u ε ) is a minimizing sequence for the Sobolev quotient . Therefore the assumptions of [32,Theorem 1.3] are satised, and we may conclude by that theorem that there are sequences in Ḣs (R N ), for some β ∈ R. By the normalization condition from (1.17), β ∈ {±1}.Now, a change of where still σ ε → 0 in Ḣs (R N ), since the Ḣs (R N )-norm is invariant under this change of variable.
Moreover, since Ω is smooth, the fact that µεΩ+zε Step 2. We make the necessary modications to derive (3.18) for some η > 0. Then if η is small enough, the minimization problem has a unique solution.
By the decomposition from Step 1 and Lemma A.2, we have ) is satised by u ε for all ε small enough, with a constant η ε tending to zero.
We thus obtain the desired decomposition by taking (x ε , λ ε , α ε ) to be the solution to (3.22) and w ε := α −1 ε u ε − P U xε,λε .To verify the claimed asymptotic behavior of the parameters, note that since η ε → 0, by denition of the minimization problem (3.22), we have Since Ω is bounded, the convergence x ε → x 0 ∈ Ω is ensured by passing to a suitable subsequence.Finally, using (1.17  Proposition 3.4 (Coercivity inequality).For all x ∈ R n and λ > 0, we have for all v ∈ T ⊥ x,λ .
As a corollary, we can include the lower order term Ω av 2 , at least when d(x)λ is large enough and at the price of having a non-explicit constant on the right side.This is the form of the inequality which we shall use below to rene our error bounds in Sections 3.4 and 5.2.
Proposition 3.5 (Coercivity inequality with potential a).Let (x n ) ⊂ Ω and (λ n ) ⊂ (0, ∞) be sequences such that dist(x n , ∂Ω)λ n → ∞.Then there is ρ > 0 such that for all n large enough, for all v ∈ T ⊥ xn,λn .(3.24) Proof.Abbreviate U n := U xn,λn and T n := T xn,λn .We follow the proof of [21] and dene Then C n is bounded from below, uniformly in n.We rst claim that C n is achieved whenever C n < 1.Indeed, x n and let v k be a minimizing sequence.Up to a subsequence, On the other hand, the left hand side of the above inequality must itself be non-negative, for otherwise ) yields a contradiction to the denition of C n as an inmum.Thus the above inequality must be in fact an equality, whence (−∆) s/2 v ∞ = 1.
We have thus proved that C n is achieved if C n < 1.
Now, assume for contradiction, up to passing to a subsequence, that lim n→∞ C n =: L ≤ 0. By the rst part of the proof, let v n be a minimizer satisfying for all w ∈ T ⊥ n .Up to passing to a subsequence, we may assume v n v ∈ H s (Ω).We claim that Assuming (3.26) for the moment, we obtain a contradiction as follows.Testing (3.26) against v ∈ H s (Ω) gives On the other hand, by coercivity of (−∆) s + a, the left hand side must be nonnegative and hence v ≡ 0. By compact embedding, we deduce v n → 0 strongly in L 2 (Ω) and thus This is the desired contradiction to lim n→∞ C n ≤ 0.
At last, we prove (3.26).Let ϕ ∈ H s (Ω) be given and write ϕ = u n + w n , with u n ∈ T n and w n ∈ T ⊥ n .
By (3.25) and using (3.28) On the one hand, we have and similarly Here we used again that by weak convergence.
From the convergence to zero of these scalar products, one can conclude u n → 0 by using the fact that the P U Let us now prepare the proof of Proposition 3.4.We recall that S : R N → S N \ {S} (where S = −e N +1 is the southpole) denotes the inverse stereographic projection dened in (3.9), with Jacobian J S (x Given a function v on R N , we may dene a function u on S N by setting The inverse of this map is of course given by v(y) := u S (y) := u(S(y))J S (y) The exponent in the determinantal factor is chosen such that v L p (R N ) = u L p (S N ) .
For a basis (Y l,m ) of L 2 (S N ) consisting of L 2 -normalized spherical harmonics, write u ∈ L 2 (S N ) as u = l,m u l,m Y l,m with coecients u l,m ∈ R. With the Paneitz operator P 2s is dened by It is well-known (see [5]) that, for every v ∈ C ∞ 0 (R N ), we have, (−∆) s v(x) = J S (x) N +2s 2N P 2s u(S(x)), (3.30)where u = v S .Thus we have [7]), the equality extends to all v ∈ D s,2 (R N ).In particular it holds for v ∈ H s (Ω).
Since the eigenvalues λ l of P 2s are increasing in l, changing back variables to R N , we deduce from By an explicit computation using the numerical values of λ 2 given by (3.29) and c N,s given in Lemma B.5, this is equivalent to which is the desired inequality.
3.4.Improved a priori bounds.The main section of this section is the following proposition, which improves Proposition 3.3.It states that the concentration point x 0 does not lie on the boundary of Ω and gives an optimal quantitative bound on w.
The proposition will readily follow from the following expansion of S a+εV [u ε ] with respect to the decomposition u ε = α(P U x,λ + w) obtained in the previous section.
Lemma 3.7.As ε → 0, we have Proof of Proposition 3.6.Using the almost minimality assumption (1.17) and the coercivity inequality from Proposition 3.5, the expansion from Lemma 3.7 yields the inequality for some c > 0. By Lemma A.1, we have the lower bound φ 0 (x) d −N +2s .Together with the estimate we obtain by taking δ small enough Since all three terms on the right side are nonnegative, the proposition follows.
Proof of Lemma 3.7.
Step 1: Expansion of the numerator.By orthogonality, we have The main term can be written as where we used P U Next, we have Finally, using that H 0 (x, y) Since H 0 (x, y) d −N +2s by Lemma A.1, the last term is Similarly, (where one needs to distinguish the cases where 1 − 2s is positive, negative or zero because the dr- Collecting all the previous estimates, we have proved The potential term splits as Ω (a + εV )(P U x,λ + w) as well as Ω (a + εV )P U x,λ w + εV w 2 dy P U x,λ p w p +ε w In summary we have, for the numerator of S a+εV [u ε ], Step 2: Expansion of the denominator.By Taylor's formula, Note that, strictly speaking, we use this formula if p ≥ 3.If 2 < p ≤ 3, the same is true without the remainder term P U p−3 |w| 3 , which does not aect the rest of the proof.To evaluate the main term, write P U where we used that by Lemmas A.2 and B.1 Next, the integral of the remainder term is controlled by The term linear in w is Now by orthogonality of w, we have Moreover, using ϕ x,λ p (dλ) − N −2s 2 by Lemma A.2, we get In summary we have, for the denominator of S a+εV [u ε ], Step 3: Expansion of the quotient.Using Taylor's formula, we nd, for the denominator, Multiplying this with the expansion for the denominator found above, we obtain Expressing the various constants using Lemma B.5, we nd c N,s U 0,1 This yields the expansion claimed in the lemma.
We will now use this information to prove the desired expansion (4.2) of the energy S(a − ε) = S a−ε [P U x,λ + w].Indeed, using the already established bound By integrating the equation for w against w and recalling S(a−ε) , we easily nd the asymptotic identity (compare [21, eq. ( 8)] for s = 1) Inserting this in (4.6), together with the expansion of (−∆) s/2 P U x,λ The numerator of S a−ε [P U x,λ + w], by the computations in the proof of Lemma 3.7, is given by x,λ w 2 dy . ).
Now we are ready to return to our ndings about w0 .Indeed, by (4.4), and observing that G 0 (x, •) is an admissible test function in (4.5), we get Ω a (λ By inserting this into (4.9) and observing that γ N,s = c N,s a N,s by the numerical values given in Lemma B.5, the proof is complete.
Now we have all the ingredients to give a quick proof of our rst main result.
Proof of Theorem 1.2.As explained after the statement of the theorem, it only remains to prove the implication (ii) ⇒ (i).Suppose thus S(a) < S and let c > 0 be the smallest number such that ā := a+c satises S(ā) = S.For ε > 0, let u ε be the sequence of minimizers S(ā − ε), normalized to satisfy (4.1).By Lemma 4.1, we have Letting ε → 0, this shows φ ā(x 0 ) ≤ 0. By the resolvent identity, we have for every x ∈ Ω and hence φ a (x 0 ) is strictly monotone in a.Thus φ a (x 0 ) < φ a+c (x 0 ) = 0, and the proof is complete.

Proof of the lower bound II: a refined expansion
This section is the most technical of the paper.It is devoted to extracting the leading term of the remainder w and to obtaining suciently good bounds on the new error term.In Section 5.2 we will need to work under the additional assumption 8s/3 < N in order to obtain the required precision.
Concretely, we write and decompose the remainder further into a tangential and an orthogonal part x,λ .(We keep omitting the subscript ε.)A rened expansion of S a+εV [u ε ] then yields an error term in r which can be controlled using the coercivity inequality of Proposition 3.5.The rened expansion is derived in Section 5.2 below.
On the other hand, since t is an element of the (N + 2)-dimensional space T x,λ , it can be bounded by essentially explicit computations.This is achieved in Section 5.1.
Remark 5.1.The present Section 5 thus constitutes the analogon of [24,Section 6], where the same analysis is carried out for the case s = 1 and N = 3.We emphasize that, despite these similarities, our approach is conceptually somewhat simpler than that of [24].Indeed, the argument in [24] relies on an intermediate step involving a spectral cuto construction, through which the apriori bound On the contrary, we are able to conduct the following analysis with only the weaker bound ∇q = O(λ − N −2s

2
) at hand (which follows from Proposition 3.6).This comes at the price of some additional explicit error terms in r, which can however be conveniently absorbed (see Lemmas 5.7 and 5.9).Since N > 8s/3 is fullled when N = 3, s = 1, this simplied proof of course also works in the particular situation of [24].5.1.A precise description of t.For λ large enough, the functions P U x,λ , ∂ λ P U x,λ and ∂ xi P U x,λ , i = 1, ..., N are linearly independent.There are therefore uniquely determined coecients β, γ, δ i , i = 1, ..., N , such that Here the choice of the dierent powers of λ multiplying the coecients is justied by the following result.
As a corollary, we obtain estimates on t in various norms.Lemma  Proof of Lemma 5.2.Step 1.We introduce the normalized basis functions

As
and prove that x,λ , and r ∈ T ⊥ x,λ , we have Thus, where we used that by Lemma B.2, (−∆) s/2 P U x,λ −1 1.The bound for a 2 follows similarly.To obtain the claimed improved bound for a j , j = 3, ..., N + 2, we write Here we wrote H a (x, y) − H 0 (x, y) = φ a (x) − φ 0 (x) + O(|x − y|) and used that by oddness of This concludes the proof of (5.3).
Step 2. We write Our goal is to show that b j = a j + O(λ −N +2s ) sup k a k , j = 1, .., N + 2. (5.4) From (5.4) we conclude by the estimates on the a j from (5.3) and Lemma B.2.
Remark 5.4.By treating the terms in the above proof more carefully, it can be shown in fact that λ N −2s β, λ N −2s−1 γ and λ N −2s+2 δ i have a limit as λ → ∞.Indeed, for instance, the leading orders of the expressions Ω U N +2s N −2s x,λ (H a (x, •) − H 0 (x, •)) dy and (−∆) s/2 P U x,λ going into the leading behavior of β can be explicitly evaluated, see Lemma A.4 and the proof of Lemma B.2 respectively.We do not need the behavior of the coecients β, γ, δ i to that precision in the following, so we do not state them explicitly.

The new expansion of S a+εV [u].
Our goal is now to expand the value of the energy functional S a+εV [u ε ] with respect to the rened decomposition introduced above, namely In all that follows, we work under the important assumption that − 3N + 6s < −2s, i.e. 8  3 s < N (5.7) so that λ −3N +6s = o(λ −2s ).Assumption (5.7) has the consequence that, using the available bounds on t and r, we can expand the energy S a+εV [u] up to o(λ −2s ) errors in a way that does not depend on t.This is the content of the next lemma.
We emphasize that the contribution of t enters only into the remainders o(λ −2s ) + o(ελ −N +2s ) + o(φ a (x)λ −N +2s ).This is remarkable because t enters to orders λ −N +2s >> λ −2s and λ −2N +4s >> λ −2s (if N < 3s) into both the numerator and the denominator of S a+εV [u ε ], see Lemmas 5.6 and 5.7 below.When calculating the quotients, these contributions cancel precisely, as we verify in Lemma 5.8 below.Heuristically, such a phenomenon is to be expected because (up to projection onto H s (Ω) and perturbation by a + εV ) by denition t represents the directions along which the quotient functional is invariant.As already pointed out in the introduction, we suspect, but cannot prove, that in the absence of assumption (5.7) the contributions of t to the higher order coecients λ −kN +2ks for 3 ≤ k ≤ 2N N −2s would continue to cancel.
We prove Lemma 5.5 by separately expanding the numerator and the denominator of S a+εV [u ε ].We abbreviate and write E ε [u, v] for the associated bilinear form.
The remaining term Ω ψ p−2 x,λ rt dy needs to be expanded more carefully.Using and using assumption (5.7), the remainder is bounded by Now using orthogonality of r and the expansion (5.1) of s, by some standard calculations, whose details we omit, one obtains where we used again assumption (5.7) for the last equality.
It remains only to treat the t-independent term Ω (ψ x,λ + r) p dy.We nd Using orthogonality of r, we get that Ω U p−1 x,λ r dy = 0 and hence Collecting all the estimates gives the claim of the lemma.
We can now prove the claimed expansion of the energy functional.
Proof of Lemma 5.5.We write the expansions of the numerator and the denominator as where and and Taylor expanding up to and including second order, we nd We now observe I Hence we can simplify the expression of the denominator to Multiplying this with the expansion of the numerator from above, we nd We show in Lemma 5.8 below that the bracket involving the terms N 1 and D 1 involving s vanishes up to order o(λ −2s ), due to cancellations.Noting that D −2/p 0 N 0 is nothing but S a+εV [ψ], the expansion claimed in Lemma 5.5 follows.
Lemma 5.8.Assume (5.7) and let N 0 , N 1 , D 0 , D 1 be dened as in the proof of Lemma 5.5.Then and where we abbreviated B N,s := In particular, Proof.We start with expanding where we also used assumption (5.7).Next, recalling ((−∆) (Observe that the leading order term with γ vanishes because (Observe that the leading order term with γ vanishes because Putting together the above, we end up with the claimed expansion for D 1 .
The last assertion of the lemma follows from the expansions of N 0 , D 0 , N 1 and D 1 by an explicit calculation whose details we omit.
Based on the rened expansion of S a+εV [u ε ] obtained in Lemma 5.5, we are now in a position to give the proofs of our main results.
We rst use the coercivity inequality from Proposition 3.4 to control the terms involving r that appear in Lemma 5.5.
Lemma 5.9 (Coercivity result).There is ρ > 0 such that, as ε → 0, Proof.Recalling the denition (5.10) of I[r] and observing that N 0 /D 0 = c N,s , we nd by Proposition for some ρ > 0. The remaining error term can be bounded as follows.
where we used Lemma A.4.By choosing δ > 0 small enough, we obtain the conclusion.
Proof.By Lemma 6.1 we only need to prove that Q V (x 0 ) < 0.
Inserting the upper bound from Corollary 2.3 on S − S(a + εV ) into (6.1), and using R ≥ 0, we obtain Using Lemma 6.1 and the assumption a < 0 on N a , we have that C 2 is strictly positive and remains bounded away from zero by assumption.Since N a (V ) is not empty, the same is clearly true for C 1 .
Proof of Theorem 1.6.We may rst observe that the upper and lower bounds on S(a + εV ) already discussed in the proof of Theorem 1.This concludes the proof of (A.4).Here the constant d N,s > 0 is given by (B.2).The asymptotics are uniform for x in compact subsets of Ω.

3. 3 .
Coercivity.In the following sections, our goal is to improve the bounds from Proposition 3.3 step by step.The following inequality, and its improvement in Proposition 3.5 below, will be central.For s = 1, these inequalities are due to Rey [34, Eq. (D.1)] and Esposito[21, Lemma 2.1], respectively, whose proofs inspired those given below.
[20]a 20.2]), it leads to more complicated terms than those arising in Druet's proof.To be more precise, the term Ω u 2 |∇ϕ| 2 from[20]is replaced by the term I(ϕ) dened in(3.6), which is more involved to evaluate for the right choice of ϕ.
+2sdy, with C N,s as in(1.6).While the relation (3.1) can be veried by a simple computation (see e.g.[25, [3,m(3.20).The crucial argument is furnished by[2, Proposition 4.3], which generalizes the corresponding statement by Bahri and Coron[3, Proposition 7]to fractional s ∈ (0, 1).It states the following.Suppose that u ∈ H s (Ω) with u H s desired.In view of(3.28), the proof of (3.26) is thus complete if we can show (−∆) s/2 u n → 0. This is again a consequence of weak convergence to zero of the U n .Indeed, by Lemmas B.1 and B.2 we have s/2 ϕ dy + Ω avϕ dy as s/2 t 2 + s/2 t 2 .From Lemma B.2 and the expansion (5.1) for t, one easily sees that