Abstract
For \(s \in (0,1)\), \(N > 2s\), and a bounded open set \(\Omega \subset {\mathbb {R}}^N\) with \(C^2\) boundary, we study the fractional Brezis–Nirenberg type minimization problem of finding
where the infimum is taken over all functions \(u \in H^s({\mathbb {R}}^N)\) that vanish outside \(\Omega \). The function a is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions \(N \in (2\,s, 4\,s)\), we prove that the Robin function \(\phi _a\) satisfies \(\inf _{x \in \Omega } \phi _a(x) = 0\), which extends a result obtained by Druet for \(s = 1\). In dimensions \(N \in (8s/3, 4s)\), we then study the asymptotics of the fractional Brezis–Nirenberg energy \(S(a + \varepsilon V)\) for some \(V \in L^\infty (\Omega )\) as \(\varepsilon \rightarrow 0+\). We give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the concentration speed and the location of concentration points.
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1 Introduction and main results
Let \(N \in {\mathbb {N}}\) and \(0< 2s < N\), for some \(s \in (0,1)\), and let \(\Omega \subset {\mathbb {R}}^N\) be a bounded open set with \(C^2\) boundary. The goal of the present paper is to analyze the variational problem of minimizing, for a given \(a \in C({\overline{\Omega }})\), the quotient functional
over functions in the space
where \(u \in H^s({\mathbb {R}}^N)\) iff
and the fractional Laplacian operator \({(-\Delta )^{s}}u\) is defined for any \(u \in H^s({\mathbb {R}}^N)\) through the Fourier representation
We also recall the singular integral representation of the fractional Laplacian (see [12, 26]):
where
The associated infimum,
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
where \(\Gamma (x):= \int _0^\infty t^{x-1} e^{-t} \mathop {}\!\textrm{d}t\) denotes the Euler Gamma function.
We note that the embedding \({{\widetilde{H}}^s(\Omega )}\hookrightarrow L^{p+1}(\Omega )\) and the associated best constant are in fact independent of \(\Omega \) and equal to the best full-space Sobolev constant \(S_{N,s}\). This follows, e.g., from the computations in [42]. An alternative proof of this fact is provided by Theorem 2.1 below.
In the classical case \(s = 1\), problem (1.7) has been first studied in the famous paper [10] by Brezis and Nirenberg, who were interested in obtaining positive solutions to the associated elliptic equation. One of the main findings in that paper is that the behavior of (1.7) depends on the space dimension N in a rather striking way. Indeed, when \(N \ge 4\), then \(S(a) < S\) if and only if \(a(x) < 0\) for some \(x \in \Omega \). On the other hand, when \(N = 3\), then \(S(a) = S\) whenever \(\Vert a\Vert _\infty \) 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:
where \(\phi _a(x)\) denotes the Robin function associated to a (see (1.11) below). This answered positively a conjecture previously formulated by Brezis in [9].
For a fractional power \(s \in (0,1)\) and assuming \(a = - \lambda \) for some constant \(\lambda >0\), Brezis–Nirenberg type results have been obtained by Servadei and Valdinoci:
-
(i)
In [42], they proved that, for \(N \ge 4s\), \(S(-\lambda ) < S\) whenever \(\lambda > 0\);
-
(ii)
In [40], they proved that, for \(2s< N < 4s\), there is \(\lambda _{s} \in (0, \lambda _{1,s})\) (where \(\lambda _{1,s}\) is the first Dirichlet eigenvalue of \((-\Delta )^s\)) such that for every \(\lambda \in (\lambda _{s}, \lambda _{1,s})\), one has \(S(-\lambda ) < S\).
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 \((-\Delta )^s\) on \({\mathbb {R}}^N\) behaves like \(G(x,y) \sim |x-y|^{-N+2s}\) near the diagonal and thus fails to be in \(L^2_\text {loc}({\mathbb {R}}^N)\) precisely if \(N \le 4s\), compare [31].
A central notion to what follows is that of a critical function a, which was introduced by Hebey and Vaugon in [30] for \(s = 1\) and readily generalizes to the fractional setting. Indeed, the following definition is naturally suggested by the behavior of S(a) just described.
Definition 1.1
(Critical function) Let \(a \in C({\overline{\Omega }})\). We say that a is critical if \(S(a) = S\) and \(S({\tilde{a}}) < S(a)\) for every \({\tilde{a}} \in C({\overline{\Omega }})\) with \({\tilde{a}} \le a\) and \({\tilde{a}} \not \equiv a\).
When \(N \ge 4s\), the result of [42] implies that the only critical potential is \(a \equiv 0\). For this case, or more generally for \(N > 2s\) with \(a \equiv 0\), the recent literature is rather rich in refined results going beyond [42]. Notably, in [15, 16], the authors proved the fractional counterpart of some conjectures by Brezis and Peletier [11] concerning the blow-up asymptotics of minimizers to the problem \(S(-\varepsilon )\) and a related problem with subcritical exponent \(p-\varepsilon \) as \(\varepsilon \rightarrow 0\). For the fractional subcritical problem, we also mention the result on Gamma convergence from [36]. In the classical case \(s = 1\), such results are due to Han [29] and Rey [37, 38]. Corresponding existence results, also for non-minimizing multi-bubble solutions, are also given in [15, 16], as well as in [18, 28].
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 Han–Rey 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 [11, Conjecture 3.(ii)]) which involves a (constant) critical function has only been proved recently in [24]. The purpose of the present paper is to treat the analogous question for low dimensions \(2s< N < 4s\) in the fractional setting.
1.1 Main results
For all of our results, a crucial role is played by the Green’s function of \({(-\Delta )^{s}}+ a\), which we introduce now. For a function \(a \in C({\overline{\Omega }})\) such that \((-\Delta )^s + a\) is coercive, i.e.
for some \(c > 0\), we define \(G_a: \Omega \times {\mathbb {R}}^N \rightarrow {\mathbb {R}}\) as the unique function such that for every fixed \(x \in \Omega \)
Here, we set \(\gamma _{N,s} = \frac{2^{2\,s} \pi ^{N/2} \Gamma (s)}{\Gamma (\frac{N-2\,s}{2})}\), so that \({(-\Delta )^{s}}|y|^{-N+2\,s} = \gamma _{N,s} \delta _0\) on \({\mathbb {R}}^N\). Thus, this choice of \(\gamma _{N,s}\) 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 define the Robin function
We prove several properties of the Green’s functions \(G_a\) in Appendix A.
Our first 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 \(a \in C({\overline{\Omega }})\) be such that \((-\Delta )^s + a\) is coercive. The following properties are equivalent.
-
(i)
There is \(x \in \Omega \) such that \(\phi _a(x) < 0\).
-
(ii)
\(S(a) < S\).
-
(iii)
S(a) is achieved by some function \(u \in {{\widetilde{H}}^s(\Omega )}\).
As an immediate corollary, we can characterize critical functions in terms of their Robin function.
Corollary 1.3
Let a be critical. Then \(\inf _{x \in \Omega } \phi _a(x) = 0\).
The implications \((i) \Rightarrow (ii)\) and \((ii) \Rightarrow (iii)\) in Theorem 1.2 are well-known: indeed, \((i) \Rightarrow (ii)\) easily follows by the proper choice of test functions thanks to Theorem 2.1 below; the implication \((ii) \Rightarrow (iii)\) is the fractional version of the seminal observation in [10] (see [42, Theorem 2]).
Our proof of \((iii) \Rightarrow (ii)\) is the content of Proposition 3.1 below and follows [20, Step 1]. The most involved proof is that of the implication \((ii) \Rightarrow (i)\), which we give in Sect. 4. We adapt the strategy developed by Esposito in [22], who gave an alternative proof of that implication for \(s=1\). His approach is based on an expansion of the energy functional \({\mathcal {S}}_{a - \varepsilon }[u_\varepsilon ]\) as \(\varepsilon \rightarrow 0\), where a is critical as in Definition 1.1 and \(u_\varepsilon \) is a minimizer of \(S(a - \varepsilon )\).
In fact, by using the techniques applied in the recent work [25] for \(s = 1\), we are even able to push this expansion of \({\mathcal {S}}_{a - \varepsilon }[u_\varepsilon ]\) further by one order of \(\varepsilon \) and derive precise asymptotics of the energy \(S(a -\varepsilon )\) and of the sequence \((u_\varepsilon )\).
To give a precise statement of our results, let us fix some more assumptions and notations. We denote the zero set of the Robin function \(\phi _a\) by
It follows from Theorem 1.2 that \(\inf _\Omega \phi _a(x) = 0\) if and only if a is critical. In particular, \({\mathcal {N}}_a\) is not empty if a is critical.
We will consider perturbations of a of the form \(a + \varepsilon V\), with non-constant \(V \in L^\infty (\Omega )\). For such V, following [25], we let
and
Finally, we shall assume that \(\Omega \) has \(C^2\) boundary and that
By Corollary 2.2, we have a priori that \(a(x) \le 0\) on \({\mathcal {N}}_a\). Also, (1.12) is clearly satisfied for the canonical case of a being constant, since a critical forces \(a < 0\) in that case. In this sense we may say that assumption (1.12) is not severe. The \(C^2\) assumption on \(\Omega \) ensures that the concentration point \(x_0\) of \(u_\varepsilon \) does not lie on the boundary; see Proposition 3.6 and Lemma A.1 below.
We point out that with our methods we are able to prove the following theorems only for the restricted dimensional range \(\tfrac{8}{3}s< N < 4s\), which enters in Sect. 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 \(S(a + \varepsilon V)\) as \(\varepsilon \rightarrow 0+\). It shows, in particular, the non-obvious fact that the condition \({\mathcal {N}}_a(V) \ne \emptyset \) is sufficient to have \(S(a+ \varepsilon V) < S\).
Theorem 1.4
(Energy asymptotics) Let \( \tfrac{8}{3} s< N < 4s\). Let us assume that \({\mathcal {N}}_a(V) \ne \emptyset \). Then, \(S(a+\varepsilon V) < S\) for all \(\varepsilon >0\) and
where \(\sigma _{N,s} > 0\) is a dimensional constant given explicitly by
The constants \(A_{N,s}\), \(\alpha _{N,s}\), \(c_{N,s}\), \(d_{N,s}\), and \(b_{N,s}\) are given explicitly in Lemma B.5 below.
On the other hand, when \({\mathcal {N}}_a(V) = \emptyset \), the next theorem shows that the asymptotics become trivial provided \(Q_V > 0\) on \({\mathcal {N}}_a\). Only in the case when \(\min _{{\mathcal {N}}_a} Q_V = 0\) we do not obtain the precise leading term of \(S(a + \varepsilon V) - S\).
Theorem 1.5
(Energy asymptotics, degenerate case) Let \( \tfrac{8}{3} s< N < 4\,s\). Let us assume that \({\mathcal {N}}_a(V) = \emptyset \). Then \(S(a+\varepsilon V) = S + o(\varepsilon ^2)\) as \(\varepsilon \rightarrow 0^+\). If, in addition, \(Q_V(x) >0\) for all \(x \in {\mathcal {N}}_a\) then \(S(a+\varepsilon V) = S\) for sufficiently small \(\varepsilon >0\).
For a potential V such that \({\mathcal {N}}_a(V) \ne \emptyset \), and thus \(S(a + \varepsilon V) < S\) by Theorem 1.4, a minimizer \(u_\varepsilon \) of \(S(a + \varepsilon V)\) exists by Theorem 1.2. We now turn to studying the asymptotic behavior of the sequence \((u_\varepsilon )\). In fact, since our methods are purely variational, we do not need to require that the \(u_\varepsilon \) 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 \({\mathcal {S}}_{a}\) is merely a perturbation of the standard Sobolev quotient functional, it is not surprising that, to leading order, the sequence \(u_\varepsilon \) approaches the family of functions
The \(U_{x, \lambda }\) are precisely the optimizers of the fractional Sobolev inequality on \({\mathbb {R}}^N\)
and satisfy the equation
with \(c_{N,s} > 0\) given explicitly in Lemma B.5.
Since we are working on the bounded set \(\Omega \), the first refinement of the approximation consists in ‘projecting’ the functions \(U_{x, \lambda }\) to \({{\widetilde{H}}^s(\Omega )}\). That is, we consider the unique function \(PU_{x, \lambda }\in {{\widetilde{H}}^s(\Omega )}\) satisfying
in the weak sense, that is,
for every \(\eta \in {{\widetilde{H}}^s(\Omega )}\).
Finally, we introduce the space
and denote by \(T_{x, \lambda }^\perp \subset {{\widetilde{H}}^s(\Omega )}\) its orthogonal complement in \({{\widetilde{H}}^s(\Omega )}\) with respect to the scalar product \((u,v):= \int _{{\mathbb {R}}^N} {(-\Delta )^{s/2}}u {(-\Delta )^{s/2}}v \mathop {}\!\textrm{d}y\). Moreover, let us denote by \(\Pi _{x, \lambda }\) and \(\Pi _{x, \lambda }^\bot \) the projections onto \(T_{x, \lambda }\) and \(T_{x, \lambda }^\bot \) respectively.
Then we have the following result.
Theorem 1.6
(Concentration of almost-minimizers) Let \( \tfrac{8}{3} s< N < 4\,s\). Suppose that \((u_\varepsilon ) \subset {{\widetilde{H}}^s(\Omega )}\) is a sequence such that
Then there exist sequences \((x_\varepsilon ) \subset \Omega \), \((\lambda _\varepsilon ) \subset (0, \infty )\), \((w_\varepsilon ) \subset T_{x_\varepsilon , \lambda _\varepsilon }^\bot \), and \((\alpha _\varepsilon ) \subset {\mathbb {R}}\) such that, up to extraction of a subsequence,
Moreover, as \(\varepsilon \rightarrow 0\), we have
Finally, \(r_\varepsilon \in T^\perp _{x_\varepsilon ,\lambda _\varepsilon }\) and \(\Vert (-\Delta )^{\frac{s}{2}} r \Vert _{L^2({\mathbb {R}}^N)}^2 = o\left( \varepsilon ^{\frac{2\,s}{4\,s-N}}\right) \).
The constants \(\alpha _{N,s}\), \(c_{N,s}\), \(d_{N,s}\), and \(b_{N,s}\) are given explicitly in Lemma B.5.
Remark 1.7
Since a is critical and \(\phi _a(x_0) = 0\), \(x_0\) is in particular a global minimum of \(\phi _a\). Thus, we obtain the commonly found necessary condition \(\nabla \phi _a(x_0) = 0\), provided that \(\phi _a\) is differentiable. We strongly expect this to be the case for smooth enough a, with a proof along the lines of Lemma A.3 (see also [24, Sect. B.2]), but prefer to not go into these details here.
Theorem 1.6 should be seen as the low-dimensional counterpart of [16, 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 coefficient \(\phi _a(x)\) 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 refining the expansion that we are able to conclude the desired information about the concentration behavior of the sequence \(u_\varepsilon \).
In the same vein, the energy expansions from Theorem 1.4 are harder to obtain than their analogues in higher dimensions \(N \ge 4\,s\). Indeed, for \(N > 4s\) we have
where \({\tilde{c}}_{N,s} > 0\) is some dimensional constant. In this case, a sharp upper bound on \(S(\varepsilon V)\) can already be derived from testing \({\mathcal {S}}_{\varepsilon V}\) against the family of functions \(PU_{x, \lambda }\). In contrast, for \(2s< N < 4s\) this family needs to be modified 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 \(N \ge 4s\), we refer to the forthcoming work [19]. It is noteworthy that the auxiliary minimization problem giving the subleading coefficient 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 \(\psi _{x, \lambda }\) defined in (2.1) below yields an upper bound for \(S(a + \varepsilon V)\), which will turn out to be sharp. The strategy we use to prove the corresponding lower bound on \({\mathcal {S}}_{a + \varepsilon V}[u_\varepsilon ]\), for a sequence \((u_\varepsilon )\) as in (1.17), can be traced back at least to work of Rey [37, 38] and Bahri–Coron [3] on the classical Brezis–Nirenberg problem for \(s =1\); it was adapted to treat problems with a critical potential a when \(s=1\), \(N = 3\) in [22] and, more recently, in [24, 25]. This strategy features two characteristic steps, namely (i) supplementing the initial asymptotic expansion \(u_\varepsilon = \alpha _\varepsilon (PU_{x_\varepsilon , \lambda _\varepsilon } + w_\varepsilon )\), obtained by a concentration-compactness argument, by the orthogonality condition \(w_\varepsilon \in T_{x_\varepsilon , \lambda _\varepsilon }^\bot \) and (ii) using a certain coercivity inequality, valid for functions in \(T_{x_\varepsilon , \lambda _\varepsilon }^\bot \), to improve the bound on the remainder \(w_\varepsilon \). The basic instance of this strategy is carried out in Sect. 3. Indeed, after performing steps (i) and (ii), in Proposition 3.6 below we are able to exclude concentration near \(\partial \Omega \) and obtain a quantitative bound on \(w_\varepsilon = \alpha _\varepsilon ^{-1} u_\varepsilon - PU_{x_\varepsilon , \lambda _\varepsilon }\). As in and [23, 37], this piece of information is enough to arrive at (1.19) and similar conclusions when \(N > 4s\); see the forthcoming paper [19] for details.
On the other hand, when \(2s< N < 4s\), the bound that Proposition 3.6 provides for the modified difference \(u_\varepsilon - \psi _{x_\varepsilon , \lambda _\varepsilon }\) is still insufficient. For \(s = 1\), it was however observed in [25] that one can refine the expansion of \(u_\varepsilon \) by reiterating steps (i) and (ii). Here, we carry out their strategy in a streamlined version (compare Remark 5.1) and for fractional \(s \in (0,1)\). That is, one writes \(w_\varepsilon = \psi _{x_\varepsilon , \lambda _\varepsilon } - PU_{x_\varepsilon , \lambda _\varepsilon } + q_\varepsilon \), decomposes \(q_\varepsilon = t_\varepsilon + r_\varepsilon \) with \(r_\varepsilon \in T_{x_\varepsilon , \lambda _\varepsilon }^\bot \) 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 \(\lambda ^{-3N+6s} = o(\lambda ^{-2s})\)). Indeed, in that case the leading contributions of \(t_\varepsilon \) to the energy, which enter to orders \(\lambda ^{-N+2s}\) and \(\lambda ^{-2N+4s}\), cancel precisely; see Lemma 5.8. If \(8s/3 \le N\), a plethora of additional terms in \(t_\varepsilon \), which contribute to orders \(\lambda ^{-k(N -2\,s)}\) with \(3 \le k \le \tfrac{2\,s}{N-2\,s}\), 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 \(k \ge 3\). 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 difficulties just discussed, we refer to Sect. 5.
One may note that the method of proof just described makes no use of two common techniques used to treat similar fractional-order problems. Firstly, we do not employ the extension formulation for \((-\Delta )^s\) due to either [14] for the restricted or to [13, 44] for the spectral fractional Laplacian, differently from, e.g., [5, 15, 16, 18, 28]. Secondly, using the properties of \(PU_{x, \lambda }\) (as given in Lemma A.2) we avoid lengthy calculations with singular integrals, appearing e.g. in [42], while at the same time optimizing the cutoff procedure with respect to [42]. We do use the singular integral formulation of \((-\Delta )^s\) in the proof of Proposition 3.1, but not in the main line of the asymptotic analysis argument.
To conclude this introduction, let us mention that several works in the literature (see [5, 7, 45]) treat the problem corresponding to (1.7) for a different notion of Dirichlet boundary conditions for \((-\Delta )^s\) on \(\Omega \), namely the spectral fractional Laplacian, defined by classical spectral theory using the \(L^2(\Omega )\)-ONB of Dirichlet eigenfunctions for \(-\Delta \). In contrast to this, the notion of \((-\Delta )^s\) we use in this paper, as defined in (1.4) or (1.5) on \({{\widetilde{H}}^s(\Omega )}\) 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 unified treatment for both, can be found in [18] (see also [41]).
1.2 Further perspectives
As far as we know, the role of the threshold configurations given by \(k(N -2s) = 2s\) for \(k \ge 1\) in the fractional Brezis–Nirenberg 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 differently by the results quoted above. It would be exciting to exhibit some similar, possibly refined, 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.
For the fractional Brezis-Nirenberg problem in general, and the low-dimensional range \(2s< N < 4s\), many intriguing questions around the blow-up analysis remain open. For instance, one may consider PDE solutions \((u_\varepsilon )\) which are not necessarily energy-minizing and even possibly admit several concentration points, as done for \(s = 1\) in [4, 32].
Another possible extension includes the case of higher-order derivatives \(s > 1\), for which the maximum principle may fail and additional boundary conditions need to be supplemented. For instance, to the best of our knowledge, even for \(s = 2\) the analogue of Theorem 1.2 is not known (we refer to [27, Chapters 7.5–7.9] for some polyharmonic problems with critical growth). Finally, it could be of interest to study the limit case \(2s = N\) corresponding to the Moser–Trudinger inequality, see [21] for a very general blow-up result in this case when \(s = 1\).
1.3 Notation
We will often abbreviate the fractional critical Sobolev exponent by \(p:= \frac{2N}{N-2s}\). For any \(q \ge 1\), we abbreviate \(\Vert \cdot \Vert _q:= \Vert \cdot \Vert _{L^q({\mathbb {R}}^N)}\). When \(q=2\), we sometimes write \(\Vert \cdot \Vert := \Vert \cdot \Vert _2\).
Unless stated otherwise, we shall always assume \(s\in (0,1)\) and \(N \in (2\,s, 4\,s)\).
For \(x \in \Omega \), we use the shorthand \(d(x) = \text {dist}(x, \partial \Omega )\).
For a set M and functions \(f,g: M \rightarrow {\mathbb {R}}_+\), we shall write \(f(m) \lesssim g(m)\) if there exists a constant \(C > 0\), independent of m, such that \(f(m) \le C g(m)\) for all \(m \in M\), and accordingly for \(\gtrsim \). If \(f \lesssim g\) and \(g \lesssim f\), we write \(f \sim g\).
The various constants appearing throughout the paper and their numerical values are collected in Lemma B.5 in the Appendix.
2 Proof of the upper bound
The following theorem gives the asymptotics of \({\mathcal {S}}_{a + \varepsilon V}[\psi _{x, \lambda }]\), for the test function
as \(\lambda \rightarrow \infty \).
Theorem 2.1
(Expansion of \({\mathcal {S}}_{a + \varepsilon V}{[\psi _{x, \lambda }]}\)) As \(\lambda \rightarrow \infty \), uniformly for x in compact subsets of \(\Omega \) and for \(\varepsilon \ge 0\),
and
In particular,
Here, \({\mathcal {T}}_i(\phi , \lambda )\) are (possibly empty) sums of the form
for suitable coefficients \(\gamma _i(k) \in {\mathbb {R}}\), where \(K = \lfloor \frac{2\,s}{N-2\,s} \rfloor \) is the largest integer less than or equal to \(\frac{2s}{N-2s}\).
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.
Corollary 2.2
(Properties of critical potentials)
-
(i)
If \(S(a) = S\), then \(\phi _a(x) \ge 0\) for all \(x \in \Omega \).
-
(ii)
If \(S(a) = S\) and \(\phi _a(x) = 0\) for some \(x \in \Omega \), then \(a(x) \le 0\).
Proof
Both statements follow from Theorem 2.1 applied with \(\varepsilon = 0\). Indeed, suppose that either \(\phi _a(x) < 0\) or \(\phi _a(x) = 0 < a(x)\) for some \(x \in \Omega \). In both cases, (2.4) gives \(S(a) \le {\mathcal {S}}_{a}[\psi _{x, \lambda }] < S\) for \(\lambda > 0\) large enough, contradiction. \(\square \)
Based on Theorem 2.1, we can now derive the following upper bound for \(S(a+\varepsilon V)\) provided that \({\mathcal {N}}_a(V)\) is not empty.
Corollary 2.3
Suppose that \({\mathcal {N}}_a(V) \ne \emptyset \). Then \(S(a+\varepsilon V) < S\) for all \(\varepsilon > 0\) and, as \(\varepsilon \rightarrow 0+\),
where
Proof
Let us fix \(\varepsilon > 0\) and \(x \in {\mathcal {N}}_a(V)\). Then, by (2.4),
We first optimize the right side over \(\lambda > 0\). Since \(A_\varepsilon := (-a(x) +o(1)) (\alpha _{N,s} + c_{N,s} d_{N,s} b_{N,s} )\) and \(B_\varepsilon := -Q_V(x)+o(1)\), are strictly positive by our assumptions, we are in the situation of Lemma B.6. Picking \(\lambda = \lambda _0(\varepsilon )\) given by (B.5), we have \(o(\lambda ^{-2s}) = o(\varepsilon ^\frac{2s}{4s-N})\). Thus, by (B.6), we get, as \(\varepsilon \rightarrow 0+\),
Optimizing over \(x\in {\mathcal {N}}_a(V)\) completes the proof of (2.6). In particular, \(S(a + \varepsilon V) < S\) for small enough \(\varepsilon > 0\). Since \(S(a + \varepsilon V)\) is a concave function of \(\varepsilon \) (being the infimum over u of functions \({\mathcal {S}}_{a + \varepsilon V}[u]\) which are linear in \(\varepsilon \)) and \(S(a) = S\), this implies that \(S(a + \varepsilon V) < S\) for every \(\varepsilon > 0\). \(\square \)
Proof of Theorem 2.1
Step 1: Expansion of the numerator. Since \((-\Delta )^s H_a(x, \cdot ) = a G_a(x, \cdot )\), the function \(\psi _{x, \lambda }\) satisfies
Therefore, recalling Lemma A.2,
We now treat the four terms on the right side separately.
A simple computation shows that \(\int _{{\mathbb {R}}^N {\setminus } B_{d(x)}(x)} U_{x, \lambda }^p \mathop {}\!\textrm{d}y = {\mathcal {O}}(\lambda ^{-N})\). Thus the first term is given by
The second term is, by Lemma A.4,
The third term will be combined with a term coming from \(\int _\Omega (a + \varepsilon V) \psi _{x, \lambda }^2 \mathop {}\!\textrm{d}y\), see below.
The fourth term can be bounded, by Lemma B.1 and recalling \(\Vert f_{x, \lambda }\Vert _\infty \lesssim \lambda ^{-\frac{N+4-2s}{2}}\) from Lemma A.2, by
Now we treat the potential term. We have
Similarly to the computations above, the terms containing \(f_{x, \lambda }\) are bounded by
and
Finally, we combine the main term with the third term in the expansion of \(\Vert {(-\Delta )^{s/2}}\psi _{x, \lambda }\Vert _2^2\) from above. Recalling that
with h as in Lemma B.3, we get
Since
by Lemma B.4, we have
Moreover, again by (2.10), and using the fact that \(h \in L^2({\mathbb {R}}^N)\) by Lemma B.3,
with
and
This completes the proof of the claimed expansion (2.2).
Step 2: Expansion of the denominator. Recall \(p = \frac{2N}{N-2\,s}\). Firstly, writing \(\psi _{x, \lambda }= U_{x, \lambda }- \lambda ^{-\frac{N-2s}{2}} H_a(x, \cdot ) - f_{x, \lambda }\), we have
Using Lemma B.1 and the bound \(\Vert f_{x, \lambda }\Vert _\infty \lesssim \lambda ^{-\frac{N+4-2\,s}{2}}\), we deduce that the remainder term is \(o(\lambda ^{-2s})\). To evaluate the main term, from Taylor’s formula for \(t \mapsto t^p\), we have
Here, \({p \atopwithdelims ()k}:= \frac{\Gamma (p+1)}{\Gamma (p-k+1) \Gamma (k+1)}\) is the generalized binomial coefficient and \(K = \lfloor \frac{2\,s}{N-2\,s} \rfloor \) as in the statement of the theorem. Applying this with \(a = U_{x, \lambda }(y)\) and \(b = - \lambda ^{-\frac{N-2s}{2}} H_a(x, \cdot )\), we find
By Lemma A.4, the claimed expansion (2.3) follows.
Step 3: Expansion of the quotient. For \(\alpha = 2/p \in (0,1)\), and fixed \(a > 0\), we again use the Taylor expansion
By Step 2, we may apply this with \(a = A_{N,s}\) and \(b = - p a_{N,s} \phi _a(x) \lambda ^{-N+2\,s} + {\mathcal {T}}_1 (\phi _a(x), \lambda ) + p d_{N,s} b_{N,s} a(x) \lambda ^{-2\,s} + o(\lambda ^{-2\,s})\). Since \(b = {\mathcal {O}}(\lambda ^{-N+2\,s})\) and \(K+1 > \frac{2\,s}{N-2\,s}\), we have \({\mathcal {O}}(b^{K+1}) = o(\lambda ^{-2s})\) and thus
for some term \({\mathcal {T}}_3(\phi , \lambda )\) as in (2.5). Multiplying this expansion with (2.2), we obtain
By integrating the equation \((-\Delta )^s U_{0,1} = c_{N,s} U_{0,1}^{p-1}\) and using the fact that \(U_{0,1}\) minimizes the Sobolev quotient on \({\mathbb {R}}^N\) (or by a computation on the numerical values of the constants given in Lemma B.5), we have \(c_{N,s} A_{N,s}^\frac{2s}{N} = S\). Hence, this is the expansion claimed in (2.4). \(\square \)
3 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 infimum S(a) is not attained. As we will see in Sect. 3.2, this implies the important basic fact that the minimizers for \(S(a+ \varepsilon V)\) must blow up as \(\varepsilon \rightarrow 0\).
The following is the main result of this section.
Proposition 3.1
(Non-existence of a minimizer for S(a)) Suppose that \(a \in C({\overline{\Omega }})\) 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 \(s \in (0,1)\) not completely straightforward is its use of the product rule for ordinary derivatives. Instead, we shall use the identity
where
with \(C_{N,s}\) as in (1.6). While the relation (3.1) can be verified by a simple computation (see e.g. [26, Lemma 20.2]), it leads to more complicated terms than those arising in Druet’s proof. To be more precise, the term \(\int _\Omega u^2 |\nabla \varphi |^2\) from [20] is replaced by the term \({\mathcal {I}}(\varphi )\) defined in (3.6), which is more involved to evaluate for the right choice of \(\varphi \).
Proof of Proposition 3.1
For the sake of finding a contradiction, we suppose that there exists u which achieves S(a), normalized so that
Then u satisfies 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 \(\varphi \in C^\infty ({\mathbb {R}}^N)\) and \(\varepsilon > 0\), and abbreviating \(p = \frac{2N}{N-2s}\),
We shall expand both sides of (3.4) in powers of \(\varepsilon \). 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
Hence,
where we write
Writing out \({(-\Delta )^{s}}\varphi \) 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 Eq. (3.3). After expanding the square \((1 + \varepsilon \varphi )^2\), the terms of orders 1 and \(\varepsilon \) on both sides of (3.4) cancel precisely. For the coefficients of \(\varepsilon ^2\), we thus recover the inequality
We now make a suitable choice of \(\varphi \), which turns (3.8) into the desired contradiction. As in [20], we choose
where \({\mathcal {S}}: {\mathbb {R}}^N \rightarrow {\mathbb {S}}^N\) is the (inverse) stereographic projection, i.e. [34, Sect. 4.4]
Moreover, we may assume (up to scaling and translating \(\Omega \) if necessary) that the balancing condition
is satisfied. Since [20] is rather brief on this point, we include some details in Lemma 3.2 below for the convenience of the reader.
By definition, we have \(\sum _{i = 1}^{N+1} \varphi _i^2 = 1\). Testing (3.8) with \(\varphi _i\) and summing over i thus yields, by (3.10),
To obtain a contradiction and finish the proof, we now show that \(\sum _{i = 1}^{N+1} {\mathcal {I}}(\varphi _i) < S(p-2)\). By definition of \(\varphi _i\), we have
To evaluate this integral further, we pass to \({\mathbb {S}}^N\). Set \(J_{{\mathcal {S}}^{-1}}(\eta ):= \det D {\mathcal {S}}^{-1} (\eta )\) and define
so that \(\int _{{\mathbb {S}}^N} U^p \mathop {}\!\textrm{d}\eta = 1\). Since the distance transforms as
changing variables in (3.12) gives
By applying first Cauchy–Schwarz and then Hölder’s inequality, we estimate
where the last inequality is strict. Indeed, U vanishes near the south pole of \({\mathbb {S}}^N\), hence there cannot be equality in Hölder’s inequality applied with the functions \(U^2\) and 1. Moreover, above we abbreviated
(note that this number is independent of \(\eta \in {\mathbb {S}}^N\)). By transforming back to \({\mathbb {R}}^N\) and evaluating a Beta function integral, the explicit value of \(\delta _{N,s}\) 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
It can be easily shown by induction over N that
for every \(N \in {\mathbb {N}}\), and hence
This is the desired contradiction to (3.11). \(\square \)
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 with \(\int _\Omega h = 1\).
Lemma 3.2
Let \(\Omega \subset {\mathbb {R}}^N\) be an open bounded set and \(0 \le h \in L^1(\Omega )\) with \(\Vert h\Vert _1 = 1\). Then there is \((y, t) \in {\mathbb {R}}^N \times (0, \infty )\) such that
Proof
Define \(H: {\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}^{N+1}\) by
We claim that
whenever \(|y|^2 + s^2\) is large enough. Thus, for large enough radii \(R >0\), the map H sends \(B(0,R) \subset {\mathbb {R}}^{N+1}\) into itself. By the Brouwer fixed point theorem, H has a fixed point (y, s). Then the pair \((y,\frac{s + \sqrt{s^2 + 4}}{2})\) satisfies the property stated in the lemma.
To prove (3.15), it is more natural to set \(t:= \frac{s + \sqrt{s^2 + 4}}{2} > 0\), so that \(s = t - t^{-1}\). By writing out \(|H(y,s)|^2\), (3.15) is equivalent to
whenever \(|y|^2 + (t - t^{-1})^2\) is large enough.
First, it is easy to see that \(y \cdot F(y,t)\), F(y, t) and G(y, t) are bounded in absolute value uniformly in \((y,t) \in {\mathbb {R}}^N \times (0,\infty )\). Moreover, there is \(C > 0\) such that
Therefore, \((t - t^{-1}) G(y,t) \rightarrow -\infty \) as \(t \rightarrow 0\) or \(t \rightarrow \infty \). Thus (3.16) holds whenever \((t - t^{-1})^2\) is large enough.
Thus, in what follows, we assume that \(t \in [1/C, C]\) and prove that (3.16) holds if |y| is large enough. For convenience, fix some sequence (y, t) with \(|y| \rightarrow \infty \) and \(t \rightarrow t_0 \in (0,\infty )\). Then \(|F(y,t)| \rightarrow 0\) and \(G(y,t) \rightarrow -1\). Moreover, since \(\Omega \) is bounded, \(\frac{|x-y|}{|y|} \rightarrow 1\) uniformly in \(x \in \Omega \) and hence
Altogether, the quantity on the left side of (3.16) thus tends to \(-2 t_0 - 2 t_0^{-1} + 1 \le -3 < 0\), which concludes the proof of (3.16). \(\square \)
3.2 Profile decomposition
The following proposition gives an asymptotic decomposition of a general sequence of normalized (almost) minimizers of \(S(a+\varepsilon V)\).
Proposition 3.3
(Profile decomposition) Let \(a \in C({\overline{\Omega }})\) be critical and let \(V \in C({\overline{\Omega }})\) be such that \({\mathcal {N}}_a(V) \ne \emptyset \). Suppose that \((u_\varepsilon ) \subset {{\widetilde{H}}^s(\Omega )}\) is a sequence such that
where \(U_{0,1}\) is given by (1.13). Then, there are sequences \((x_\varepsilon ) \subset \Omega \), \((\lambda _\varepsilon ) \subset (0, \infty )\), \((w_\varepsilon ) \subset T_{x_\varepsilon , \lambda _\varepsilon }^\bot \), and \((\alpha _\varepsilon ) \subset {\mathbb {R}}\) such that, up to extraction of a subsequence,
Moreover, as \(\varepsilon \rightarrow 0\), we have
In what follows, we shall always work with a sequence \(u_\varepsilon \) that satisfies the assumptions of Proposition 3.3. For readability, we shall often drop the index \(\varepsilon \) from \(\alpha _\varepsilon \), \(x_\varepsilon \), \(\lambda _\varepsilon \) and \(w_\varepsilon \), and write \(d:= d_\varepsilon := d(x_\varepsilon )\). Moreover, we adopt the convention that we always assume the strict inequality
In Theorems 1.4 and 1.6, we assume \({\mathcal {N}}_a(V) \ne \emptyset \), so assumption (3.19) is certainly justified in view of Corollary 2.3. For Theorem 1.5, where we assume \({\mathcal {N}}_a(V) = \emptyset \), we discuss the role of assumption (3.19) in the proof of that theorem in Sect. 6.
Proof
Step 1. We derive a preliminary decomposition in terms of the Sobolev optimizers \(U_{z, \lambda }\) and without orthogonality condition on the remainder, see (3.20) below.
The assumptions imply that the sequence \((u_\varepsilon )\) is bounded in \({{\widetilde{H}}^s(\Omega )}\), hence, up to a subsequence, we may assume \(u_\varepsilon \rightharpoonup u_0\) for some \(u_0 \in {{\widetilde{H}}^s(\Omega )}\). By the argument given in [25, Proof of Proposition 3.1, Step 1], the fact that \({\mathcal {S}}_{a + \varepsilon V}[u_\varepsilon ] \rightarrow S(a) = S\) implies that \(u_0\) is a minimizer for S(a), unless \(u_0 \equiv 0\). Since such a minimizer does not exist by Proposition 3.1, we conclude that, in fact, \(u_\varepsilon \rightharpoonup 0\) in \({{\widetilde{H}}^s(\Omega )}\).
By Rellich’s theorem, \(u_\varepsilon \rightarrow 0\) strongly in \(L^2(\Omega )\), in particular \(\int _\Omega (a + \varepsilon V) u_\varepsilon ^2 = o(1)\). The assumption (1.17) thus implies that \((u_\varepsilon )\) is a minimizing sequence for the Sobolev quotient \(\int _{{\mathbb {R}}^N} |{(-\Delta )^{s/2}}u|^2 \mathop {}\!\textrm{d}y / \Vert u\Vert ^2_\frac{2N}{N-2\,s}\). Therefore, the assumptions of [35, Theorem 1.3] are satisfied, and we may conclude by that theorem that there are sequences \((z_\varepsilon ) \subset {\mathbb {R}}^N\), \((\mu _\varepsilon ) \subset (0, \infty )\), \((\sigma _\varepsilon )\) such that
in \({\dot{H}}^s({\mathbb {R}}^N)\), for some \(\beta \in {\mathbb {R}}\). By the normalization condition from (1.17), we have \(\beta \in \{ \pm 1\}\). Now, a change of variables \(y = z_\varepsilon + \mu _\varepsilon ^{-1} x\) implies
where still \(\sigma _\varepsilon \rightarrow 0\) in \({\dot{H}}^s({\mathbb {R}}^N)\), since the \({\dot{H}}^s({\mathbb {R}}^N)\)-norm is invariant under this change of variable.
Moreover, since \(\Omega \) is smooth, the fact that
implies \(\mu _\varepsilon \text {dist}(z_\varepsilon , {\mathbb {R}}^N {\setminus } \Omega ) \rightarrow \infty \).
Step 2. We make the necessary modifications to derive (3.18) from (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 \(s \in (0,1)\). It states the following. Suppose that \(u \in {{\widetilde{H}}^s(\Omega )}\) with \(\Vert u\Vert _{{\widetilde{H}}^s(\Omega )}= A_{N,s}\) satisfies
for some \(\eta > 0\). Then, if \(\eta \) is small enough, the minimization problem
has a unique solution.
By the decomposition from Step 1 and Lemma A.2, we have
as \(\varepsilon \rightarrow 0\), so that (3.21) is satisfied by \(u_\varepsilon \) for all \(\varepsilon \) small enough, with a constant \(\eta _\varepsilon \) tending to zero. We thus obtain the desired decomposition
by taking \((x_\varepsilon , \lambda _\varepsilon , \alpha _\varepsilon )\) to be the solution to (3.22) and \(w_\varepsilon := \alpha _\varepsilon ^{-1} u_\varepsilon - PU_{x_\varepsilon , \lambda _\varepsilon }\). To verify the claimed asymptotic behavior of the parameters, note that since \(\eta _\varepsilon \rightarrow 0\), by definition of the minimization problem (3.22), we have \(\Vert {(-\Delta )^{s/2}}w_\varepsilon \Vert _2 < \eta _\varepsilon \rightarrow 0\) and \(\lambda _\varepsilon d(x_\varepsilon ) > (4 \eta _\varepsilon )^{-1} \rightarrow \infty \). Since \(\Omega \) is bounded, the convergence \(x_\varepsilon \rightarrow x_0 \in {\overline{\Omega }}\) is ensured by passing to a suitable subsequence. Finally, using (1.17), we have
which implies \(\alpha _\varepsilon = \pm 1 + o(1)\). \(\square \)
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 [38, Eq. (D.1)] and Esposito [22, Lemma 2.1], respectively, whose proofs inspired those given below.
Proposition 3.4
(Coercivity inequality) For all \(x \in {\mathbb {R}}^n\) and \(\lambda > 0\), we have
for all \(v \in T_{x, \lambda }^\bot \).
As a corollary, we can include the lower order term \(\int _\Omega a v^2\), at least when \(d(x) \lambda \) 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 refine our error bounds in Sects. 3.4 and 5.2.
Proposition 3.5
(Coercivity inequality with potential a) Let \((x_n) \subset \Omega \) and \((\lambda _n) \subset (0, \infty )\) be sequences such that \({{\,\textrm{dist}\,}}(x_n, \partial \Omega ) \lambda _n \rightarrow \infty \). Then there is \(\rho > 0\) such that for all n large enough,
Proof
Abbreviate \(U_n:= U_{x_n, \lambda _n}\) and \(T_n:= T_{x_n, \lambda _n}\). We follow the proof of [22] and define
Then \(C_n\) is bounded from below, uniformly in n. We first claim that \(C_n\) is achieved whenever \(C_n <1\). Indeed, fix n and let \(v_k\) be a minimizing sequence. Up to a subsequence, \(v_k \rightharpoonup v_\infty \) in \({{\widetilde{H}}^s(\Omega )}\) and consequently \(\Vert {(-\Delta )^{s/2}}v_\infty \Vert \le 1\) and \(\int _\Omega a v_k^2 - c_{N,s} (p-1) \int _\Omega U_n^{p-2} v_k^2 \mathop {}\!\textrm{d}y \rightarrow \int _\Omega a v_\infty ^2 - c_{N,s} (p-1) \int _\Omega U_n^{p-2} v_\infty ^2 \mathop {}\!\textrm{d}y\), by compact embedding \({{\widetilde{H}}^s(\Omega )}\hookrightarrow L^2(\Omega )\). Thus
On the other hand, the left hand side of the above inequality must itself be non-negative, for otherwise \({\tilde{v}}:= v_\infty /\Vert {(-\Delta )^{s/2}}v_\infty \Vert \) (notice that \(C_n < 1\) enforces \(v_\infty \not \equiv 0\)) yields a contradiction to the definition of \(C_n\) as an infimum. Thus the above inequality must be in fact an equality, whence \(\Vert {(-\Delta )^{s/2}}v_\infty \Vert = 1\). We have thus proved that \(C_n\) is achieved if \(C_n < 1\).
Now, assume for contradiction, up to passing to a subsequence, that \(\lim _{n \rightarrow \infty } C_n =: L \le 0\). By the first part of the proof, let \(v_n\) be a minimizer satisfying
for all \(w \in T_n^\bot \). Up to passing to a subsequence, we may assume \(v_n \rightharpoonup v \in {{\widetilde{H}}^s(\Omega )}\). We claim that
Assuming (3.26) for the moment, we obtain a contradiction as follows. Testing (3.26) against \(v \in {{\widetilde{H}}^s(\Omega )}\) gives
On the other hand, by coercivity of \({(-\Delta )^{s}}+ a\), the left hand side must be nonnegative and hence \(v \equiv 0\). By compact embedding, we deduce \(v_n \rightarrow 0\) strongly in \(L^2(\Omega )\) and thus
This is the desired contradiction to \(\lim _{n \rightarrow \infty } C_n \le 0\).
At last, we prove (3.26). Let \(\varphi \in {{\widetilde{H}}^s(\Omega )}\) be given and write \(\varphi = u_n + w_n\), with \(u_n \in T_n\) and \(w_n \in T_n^\bot \). By (3.25) and using \( \int _{{\mathbb {R}}^N} {(-\Delta )^{s/2}}v_n {(-\Delta )^{s/2}}u_n = 0\), we have
On the one hand,
because \(\varphi ^\frac{p}{p-1} \in L^{p-1} = (L^\frac{p-1}{p-2}(\Omega ))'\) and \(U_n^\frac{(p-2)p}{p-1} \rightharpoonup 0\) weakly in \(L^\frac{p-1}{p-2}(\Omega )\). Thus, by weak convergence, the expression in (3.27) tends to
as desired. In view of (3.28), the proof of (3.26) is thus complete if we can show \(\Vert {(-\Delta )^{s/2}}u_n \Vert \rightarrow 0\). This is again a consequence of weak convergence to zero of the \(U_n\). Indeed, by Lemmas B.1 and B.2, we have
and, similarly,
Here, we used again that \(U_n^{p-1} \rightharpoonup 0\) in \(L^\frac{p}{p-1}\) and \(U_n^\frac{(p-2)p}{p-1} \rightharpoonup 0\) in \(L^\frac{p-1}{p-2}\) weakly, and \(\int _\Omega U^\frac{(p-2)p}{p-1} \varphi ^\frac{p}{p-1} \mathop {}\!\textrm{d}y = o(1)\) by weak convergence.
From the convergence to zero of these scalar products, one can conclude \(u_n \rightarrow 0\) by using the fact that the \(PU_n\), \(\partial _{\lambda }PU_n\), \(\partial _{x_i}PU_n\) are ‘asymptotically orthogonal’ by the bounds of Lemma B.2. For a detailed argument, we refer to Lemma 5.2 below, see also [25, Lemma 6.1]. \(\square \)
Let us now prepare the proof of Proposition 3.4. We recall that \({{\mathcal {S}}}: {\mathbb {R}}^N \rightarrow {{\mathbb {S}}^N}{\setminus } \{-e_{N+1}\}\) (where \(-e_{N+1}= (0,\ldots ,0,1) \in {\mathbb {R}}^{N+1}\) is the south pole) denotes the inverse stereographic projection defined in (3.9), with Jacobian \(J_{{\mathcal {S}}}(x):= \det D {{\mathcal {S}}}(x) = \left( \frac{2}{1+|x|^2}\right) ^N\).
Given a function v on \({\mathbb {R}}^N\), we may define a function u on \({{\mathbb {S}}^N}\) by setting
The inverse of this map is of course given by
The exponent in the determinant factor is chosen such that \(\Vert v\Vert _{L^p({\mathbb {R}}^N)} = \Vert u\Vert _{L^p({{\mathbb {S}}^N})}\).
For a basis \((Y_{l,m})\) of \(L^2({{\mathbb {S}}^N})\) consisting of \(L^2\)-normalized spherical harmonics, write \(u \in L^2({{\mathbb {S}}^N})\) as \(u = \sum _{l,m} u_{l,m} Y_{l,m}\) with coefficients \(u_{l,m} \in {\mathbb {R}}\). With
the Paneitz operator \({\mathcal {P}}_{2s}\) is defined by
for every \(u \in L^2({{\mathbb {S}}^N})\) such that \(\sum _{l,m} \lambda _l u_{l,m}^2 < \infty \).
It is well-known (see [6]) that, for every \(v \in C^\infty _0({\mathbb {R}}^N)\), we have,
where \(u = v_{{\mathcal {S}}}\). Thus, we have
Since \(C^\infty _0({\mathbb {R}}^N)\) is dense in the space \({\mathcal {D}}^{s,2}({\mathbb {R}}^N):= \{ v \in L^\frac{2N}{N-2\,s}({\mathbb {R}}^N) \,: \, {(-\Delta )^{s/2}}v \in L^2({\mathbb {R}}^N) \}\) (see, e.g., [8]), the equality
extends to all \(v \in {\mathcal {D}}^{s,2}({\mathbb {R}}^N)\). In particular, it holds for \(v \in {{\widetilde{H}}^s(\Omega )}\).
Proof of Proposition 3.4
We first prove (3.23) for \((x,\lambda ) = (0,1)\). Let \(v \in T_{0,1}^\perp \) and denote \(u = v_{{\mathcal {S}}}\). We claim that the orthogonality conditions on v imply that \(u_{l,m} = 0\) for \(l = 0,1\). Indeed, e.g. from \(v \perp \partial _{\lambda }PU_{0,1}\) and recalling \(J_{{\mathcal {S}}}(y) = (\frac{2}{1+|y|^2})^N = 2^N U_{0,1}^\frac{2N}{N-2s}\), we compute
Analogous calculations show that \(v \perp PU_{0,1}\) implies \(\int _{{\mathbb {S}}^N}u = 0\) and that \(v \perp \partial _{x_i}PU_{0,1}\) implies \(\int _{{\mathbb {S}}^N}u \omega _i = 0\) for \(i = 1,\ldots ,N\). Since the functions 1 and \(\omega _i\) (\(i = 1,\ldots ,N+1\)) form a basis of the space of spherical harmonics of angular momenta \(l = 0\) and \(l=1\) respectively, we have proved our claim.
Since the eigenvalues \(\lambda _l\) of \({{\mathcal {P}}_{2s}}\) are increasing in l, changing back variables to \({\mathbb {R}}^N\), we deduce from (3.31) that
By an explicit computation using the numerical values of \(\lambda _2\) given by (3.29) and \(c_{N,s}\) given in Lemma B.5, this is equivalent to
which is the desired inequality.
The case of general \((x, \lambda ) \in \Omega \times (0, \infty )\) can be deduced from this by scaling. Indeed, for \(v \in T_{x, \lambda }^\bot \), set \(v_{x, \lambda }(y):= v(x - \lambda ^{-1}y)\). Then \(v_{x, \lambda }\in T_{0,1}^\bot \) with respect to the set \(\lambda (x - \Omega )\), so that by the above \(v_{x, \lambda }\) satisfies
Changing back variables now yields (3.32). \(\square \)
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 \(\Omega \) and gives an optimal quantitative bound on w.
Proposition 3.6
As \(\varepsilon \rightarrow 0\),
and
In particular, \(x_0 \in \Omega \).
The proposition will readily follow from the following expansion of \({\mathcal {S}}_{a + \varepsilon V}[u_\varepsilon ]\) with respect to the decomposition \(u_\varepsilon = \alpha (PU_{x, \lambda }+ w)\) obtained in the previous section.
Lemma 3.7
As \(\varepsilon \rightarrow 0\), 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 \(\phi _0(x) \gtrsim d^{-N+2s}\). Using the estimate
we obtain, by taking \(\delta \) small enough,
Since all three terms on the right side are nonnegative, the proposition follows. \(\square \)
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 \(PU_{x, \lambda }= U_{x, \lambda }- \lambda ^{-\frac{N-2\,s}{2}} H_0(x,\cdot ) + f_{x, \lambda }\) with \(\Vert f_{x, \lambda }\Vert _\infty \lesssim \lambda ^\frac{-N+4-2\,s}{2} d^{-N -2 +2\,s}\), by Lemma A.2. Thus,
Next, we have
and thus
Finally, using the fact that \(H_0(x,y) = \phi _0(x) + {\mathcal {O}}( \Vert \nabla _y H_0(x,\cdot )\Vert _\infty |x-y|) = \phi _0(x) + {\mathcal {O}}( d^{-N+2\,s-1} |x-y|)\) by Lemma A.1, we have
Since \(H_0(x,y) \lesssim d^{-N+2s}\) by Lemma A.1, the last term is
Similarly,
Finally,
(where one needs to distinguish the cases where \(1-2s\) is positive, negative or zero because the \(\mathop {}\!\textrm{d}r\)-integral is divergent if \(1-2s \ge 0\)).
Collecting all the previous estimates, we have proved
The potential term splits as
and we can estimate
as well as
In summary, we have, for the numerator of \({\mathcal {S}}_{a+ \varepsilon V}[u_\varepsilon ]\),
Step 2: Expansion of the denominator. By Taylor’s formula,
Note that, strictly speaking, we use this formula if \(p \ge 3\). If \(2< p \le 3\), the same is true without the remainder term \(PU^{p-3} |w|^3\), which does not affect the rest of the proof. To evaluate the main term, we write \(PU_{x, \lambda }= U_{x, \lambda }- \varphi _{x, \lambda }\) with \(\varphi _{x, \lambda }:= \lambda ^{-1/2} H_0(x, \cdot ) + f_{x, \lambda }\) (see Lemma A.2). Then,
where we used that, by Lemmas A.2 and B.1, \(\int _\Omega U_{x, \lambda }^{p-2} \varphi _{x, \lambda }^2 \mathop {}\!\textrm{d}y \le \Vert U_{x, \lambda }\Vert _{p-2}^{p-2} \Vert \varphi _{x, \lambda }\Vert ^2_\infty \lesssim (d\lambda )^{-2N+4\,s} = o((d\lambda )^{-N+2\,s})\) and \(\Vert \varphi _{x, \lambda }\Vert _p^p \lesssim (d\lambda )^{-N} = o((d\lambda )^{-N+2\,s})\).
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 \(\Vert \varphi _{x, \lambda }\Vert _p \lesssim (d \lambda )^{-\frac{N-2\,s}{2}}\) by Lemma A.2, we get
Using additionally that \(\Vert \varphi _{x, \lambda }\Vert _\infty \lesssim d^{-N+2\,s} \lambda ^{-\frac{N-2\,s}{2}}\) by Lemma A.2, by the same computation as in [23, Lemma A.1] we get \(\Vert U_{x, \lambda }^{p-2} \varphi _{x, \lambda }\Vert _{\frac{p}{p-1}} \lesssim (d\lambda )^{-N+2\,s}\) and therefore
In summary, we have, for the denominator of \({\mathcal {S}}_{a+ \varepsilon V}[u_\varepsilon ]\),
Step 3: Expansion of the quotient. Using Taylor’s formula, we find, for the denominator,
Multiplying this with the expansion for the denominator found above, we obtain
Expressing the various constants using Lemma B.5, we find
This yields the expansion claimed in the lemma. \(\square \)
4 Proof of Theorem 1.2
At this point, we have collected sufficiently precise information on the behavior of a general almost minimizing sequence to prove Theorem 1.2.
The main difficulty of the argument consists in constructing, for a critical potential a, a point \(x_0 \in \Omega \) at which \(\phi _a(x_0) = 0\). To do so, we carry out some additional analysis for a sequence \(u_\varepsilon \) which we assume to consist of true minimizers of \(S(a - \varepsilon )\), not only almost minimizers as in the rest of this paper. We make this additional assumption essentially for convenience and brevity of the argument, see the remark below Lemma 4.1.
Indeed, since a is critical, we have \(S(a - \varepsilon ) < S\) for every \(\varepsilon > 0\). By the results of [42], which adapts the classical lemma of Lieb contained in [10] to the fractional case, this strict inequality implies existence of a minimizer \(u_\varepsilon \) of \(S(a - \varepsilon )\). Normalizing \(\int _\Omega u_\varepsilon ^\frac{2N}{N-2\,s} \mathop {}\!\textrm{d}y = A_{N,s}\) as in (1.17), \(u_\varepsilon \) satisfies the equation
By using equation (4.1), we can conveniently extract the leading term of the remainder term \(w_\varepsilon \). We do this in the following lemma, which is the key step in the proof of Theorem 1.2.
Lemma 4.1
Let \(u_\varepsilon \) be minimizers of \(S(a - \varepsilon )\) which satisfy (4.1). Then we have
If \(8s/3 < N\), Lemma 4.1 is in fact implied by the more refined analysis carried out in Sect. 5 below, which does not use the Eq. (4.1). If \(2s < N \le 8s/3\), we speculate than one can prove Lemma 4.1 for almost minimizers not satisfying (4.1) by arguing like in [25, Sect. 5], but we do not pursue this explicitly here.
Proof of Lemma 4.1
Clearly, the analysis carried out in Sect. 3 so far applies to the sequence \((u_\varepsilon )\). Thus, up to passing to a subsequence, we may assume that \(u_\varepsilon = \alpha _\varepsilon (PU_{x_\varepsilon , \lambda _\varepsilon } + w_\varepsilon )\) with \(\alpha _\varepsilon \rightarrow 1\), \(x_\varepsilon \rightarrow x_0 \in \Omega \) and \(\Vert {(-\Delta )^{s/2}}w_\varepsilon \Vert _2 \lesssim \lambda ^{-\frac{N-2s}{2}}\) as \(\varepsilon \rightarrow 0\).
Thus the sequence \({\tilde{w}}_\varepsilon := \lambda _\varepsilon ^{\frac{N-2s}{2}} w_\varepsilon \) is bounded in \({{\widetilde{H}}^s(\Omega )}\) and converges weakly in \({{\widetilde{H}}^s(\Omega )}\), up to a subsequence, to some \({\tilde{w}}_0 \in {{\widetilde{H}}^s(\Omega )}\). Inserting the expansion \(u = \alpha ( PU_{x, \lambda }+ w)\) in (4.1), the equation fulfilled by \({\tilde{w}}_\varepsilon \) reads
By Lemma A.2, we can write
with h as in Lemma B.3. By the bounds on h and \(f_{x, \lambda }\) from Lemmas A.2 and B.3, this yields
Letting \(\varepsilon \rightarrow 0\) in (4.3), we obtain
for every \(\varphi \in C^\infty _c(\Omega \setminus \{x_0\})\). Now it is straightforward to show that \(C^\infty _c(\Omega {\setminus } \{x_0\})\) is dense in \({{\widetilde{H}}^s(\Omega )}\), by using a cutoff function argument together with identity (3.1). Thus, by approximation, (4.5) even holds for every \(\varphi \in {{\widetilde{H}}^s(\Omega )}\). In other words, \({\tilde{w}}_0\) weakly solves the equation
By uniqueness of solutions, we conclude \({\tilde{w}}_0 = H_0(x_0, \cdot ) - H_a(x_0, \cdot )\).
We will now use this information to prove the desired expansion (4.2) of the energy \(S(a-\varepsilon ) = {\mathcal {S}}_{a -\varepsilon } [PU_{x, \lambda }+ w]\). Indeed, using the already established bound \(\Vert {(-\Delta )^{s/2}}w\Vert \lesssim \lambda ^{-\frac{N-2s}{2}}\), the numerator is
By integrating the equation for w against w and recalling \(\frac{S(a-\varepsilon )}{A_{N,s}^{2s/N}} = c_{N,s} + o(1)\), we easily find the asymptotic identity (compare [22, Eq. (8)] for \(s= 1\))
Inserting this in (4.6), together with the expansion of \(\Vert {(-\Delta )^{s/2}}PU_{x, \lambda }\Vert ^2\) given in (3.35), the numerator of \({\mathcal {S}}_{a -\varepsilon } [PU_{x, \lambda }+ w]\) becomes
The numerator of \({\mathcal {S}}_{a -\varepsilon } [PU_{x, \lambda }+ w]\), by the computations in the proof of Lemma 3.7, is given by
Multiplying out (4.7) and (4.8), the terms in \(\int _\Omega U^{p-2} w^2 \mathop {}\!\textrm{d}y\) cancel precisely and we obtain
Now we are ready to return to our findings about \({\tilde{w}}_0\). Indeed, by (4.4), and observing that \(G_0(x, \cdot )\) is an admissible test function in (4.5), we get
By inserting this into (4.9) and observing that \(\gamma _{N,s} = c_{N,s} a_{N,s}\) by the numerical values given in Lemma B.5, the proof is complete. \(\square \)
Now we have all the ingredients to give a quick proof of our first main result.
Proof of Theorem 1.2
As explained after the statement of the theorem, it only remains to prove the implication \((ii) \Rightarrow (i)\). Suppose thus \(S(a) < S\) and let \(c > 0\) be the smallest number such that \({\bar{a}}:= a +c\) satisfies \(S({\bar{a}}) = S\). For \(\varepsilon > 0\), let \(u_\varepsilon \) be the sequence of minimizers \(S({\bar{a}} - \varepsilon )\), normalized to satisfy (4.1). By Lemma 4.1, we have
Letting \(\varepsilon \rightarrow 0\), this shows \(\phi _{{\bar{a}}}(x_0) \le 0\). By the resolvent identity, we have for every \(x \in \Omega \)
and hence \(\phi _a(x_0)\) is strictly monotone in a. Thus \(\phi _a(x_0) < \phi _{a+c}(x_0) = 0\), and the proof is complete. \(\square \)
5 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 sufficiently good bounds on the new error term. In Sect. 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
(We keep omitting the subscript \(\varepsilon \).) A refined expansion of \({\mathcal {S}}_{a + \varepsilon V}[u_\varepsilon ]\) then yields an error term in r which can be controlled using the coercivity inequality of Proposition 3.5. The refined expansion is derived in Sect. 5.2 below.
On the other hand, since t is an element of the \((N+2)\)-dimensional space \(T_{x, \lambda }\), it can be bounded by essentially explicit computations. This is achieved in Sect. 5.1.
Remark 5.1
The present Sect. 5 thus constitutes the analogon of [25, Sect. 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 [25]. Indeed, the argument in [25] relies on an intermediate step involving a spectral cutoff construction, through which the apriori bound \(\Vert \nabla q\Vert =o(\lambda ^{1/2}) = o(\lambda ^{-\frac{N-2s}{2}})\) is obtained.
On the contrary, we are able to conduct the following analysis with only the weaker bound \(\Vert \nabla q\Vert = {\mathcal {O}}(\lambda ^{-\frac{N-2s}{2}})\) 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 fulfilled when \(N=3\), \(s=1\), this simplified proof of course also works in the particular situation of [25].
5.1 A precise description of t
For \(\lambda \) large enough, the functions \(PU_{x, \lambda }\), \(\partial _{\lambda }PU_{x, \lambda }\) and \(\partial _{x_i}PU_{x, \lambda }\), \(i = 1,\ldots ,N\) are linearly independent. There are therefore uniquely determined coefficients \(\beta , \gamma , \delta _i\), \(i = 1,\ldots ,N\), such that
Here the choice of the different powers of \(\lambda \) multiplying the coefficients is justified by the following result.
Lemma 5.2
As \(\varepsilon \rightarrow 0\), we have \(\beta , \gamma , \delta _i = {\mathcal {O}}(1)\).
As a corollary, we obtain estimates on t in various norms.
Lemma 5.3
As \(\varepsilon \rightarrow 0\),
Proof
Recall that \(PU_{x, \lambda }= U_{x, \lambda }- \lambda ^{-\frac{N-2s}{2}} H_0(x,\cdot ) + f_{x, \lambda }\). Then all bounds follow in a straightforward way from (5.1) together with Lemma 5.2 and the standard bounds from Lemmas A.1, A.2, B.1 and B.2. \(\square \)
Proof of Lemma 5.2
Step 1. We introduce the normalized basis functions
and prove that
Since \(\lambda ^{-\frac{N-2\,s}{2}} (H_0(x,\cdot ) - H_a(x,\cdot )) + t + r = w \in T_{x, \lambda }^\bot \), and \(r \in T_{x, \lambda }^\bot \), we have
Thus,
where we used that by Lemma B.2, \(\Vert {(-\Delta )^{s/2}}PU_{x, \lambda }\Vert ^{-1} \lesssim 1\). The bound for \(a_2\) follows similarly. To obtain the claimed improved bound for \(a_j\), \(j = 3,\ldots ,N+2\), we write
Here, we wrote \(H_a(x,y) - H_0(x,y) = \phi _a(x) -\phi _0(x) + {\mathcal {O}}(|x-y|)\) and used that by oddness of \(\partial _{x_i}U\),
This concludes the proof of (5.3).
Step 2. We write
with
Our goal is to show that
Then, from (5.4), we conclude by using the estimates on the \(a_j\) from (5.3) and Lemma B.2.
To prove (5.4), we define the Gram matrix G by
By Lemma B.2 and the definition of the \({\tilde{\varphi }}_j\) it is easily checked that
Thus, for sufficiently large \(\lambda \), G is invertible with
By definition of G,
is an orthonormal basis of \(T_{x, \lambda }\). We can therefore write
Thus, \(b_l = \sum _k (G^{-1})_{l,k} a_k\) and (5.4) follows from (5.5). \(\square \)
Remark 5.4
By treating the terms in the above proof more carefully, it can be shown in fact that \(\lambda ^{N - 2\,s} \beta \), \(\lambda ^{N- 2\,s-1}\gamma \) and \(\lambda ^{N-2\,s+2} \delta _i\) have a limit as \(\lambda \rightarrow \infty \). Indeed, for instance, the leading orders of the expressions \(\int _\Omega U_{x, \lambda }^\frac{N+2\,s}{N-2\,s} (H_a(x, \cdot ) - H_0(x, \cdot )) \mathop {}\!\textrm{d}y\) and \(\Vert {(-\Delta )^{s/2}}PU_{x, \lambda }\Vert \) going into the leading behavior of \(\beta \) can be explicitly evaluated, see Lemma A.4 and the proof of Lemma B.2 respectively. We do not need the behavior of the coefficients \(\beta , \gamma , \delta _i\) to that precision in what follows, so we do not state them explicitly.
5.2 The new expansion of \({\mathcal {S}}_{a+\varepsilon V} [u]\)
Our goal is now to expand the value of the energy functional \({\mathcal {S}}_{a + \varepsilon V} [u_\varepsilon ]\) with respect to the refined decomposition introduced above, namely
In all that follows, we work under the important assumption that
so that \(\lambda ^{-3N + 6s} = o(\lambda ^{-2s})\). Assumption (5.7) has the consequence that, using the available bounds on t and r, we can expand the energy \({\mathcal {S}}_{a + \varepsilon V}[u]\) up to \(o(\lambda ^{-2s})\) errors in a way that does not depend on t. This is the content of the next lemma.
Lemma 5.5
As \(\varepsilon \rightarrow 0\), we have
Here,
and I[r] is as defined in (5.10) below.
We emphasize that the contribution of t enters only into the remainders \(o(\lambda ^{-2\,s}) + o(\varepsilon \lambda ^{-N+ 2\,s})+ o(\phi _a(x) \lambda ^{-N+2\,s})\). This is remarkable because t enters to orders \(\lambda ^{-N+2\,s} \gg \lambda ^{-2\,s}\) and \(\lambda ^{-2N+4\,s} \gg \lambda ^{-2\,s}\) (if \(N< 3\,s\)) into both the numerator and the denominator of \( {\mathcal {S}}_{a + \varepsilon V}[u_\varepsilon ]\), 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 \({{\widetilde{H}}^s(\Omega )}\) and perturbation by \(a + \varepsilon V\)) by definition 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 coefficients \(\lambda ^{-kN + 2ks}\) for \(3 \le k \le \tfrac{2N}{N-2\,s}\) would continue to cancel.
We prove Lemma 5.5 by separately expanding the numerator and the denominator of \({\mathcal {S}}_{a + \varepsilon V}[u_\varepsilon ]\). We abbreviate
and write \({\mathcal {E}}_\varepsilon [u,v]\) for the associated bilinear form.
Lemma 5.6
(Expanding the numerator) As \(\varepsilon \rightarrow 0\),
Proof
We write \(\alpha ^{-1} u_\varepsilon = \psi _{x, \lambda }+ t + r\) and therefore
The third term on the right side is
Now \(\int _{{\mathbb {R}}^N} {(-\Delta )^{s/2}}t {(-\Delta )^{s/2}}r \mathop {}\!\textrm{d}y = 0\) by orthogonality and therefore, by Lemma 5.3,
where the last equality is a consequence of assumption (5.7) and Young’s inequality. Finally, again by Lemma 5.3,
The second term on the right side of (5.9) is
To start with, using Lemma 5.3,
again by Young’s inequality. Moreover, using that \(r \in T_{x, \lambda }^\bot \), that \({(-\Delta )^{s}}H_a(x, \cdot ) = a G_a(x, \cdot )\) and \({(-\Delta )^{s}}H_0(x,\cdot ) = 0\), and integrating by parts,
Since we can write \(\lambda ^{-\frac{N-2s}{2}} G_a(x,y) - \psi _{x, \lambda }(y) = \lambda ^\frac{N-2s}{2}h(\lambda (x-y)) + f_{x, \lambda }\) with h as in Lemma B.3 and \(f_{x, \lambda }\) as in Lemma A.2, we get
by the bounds in those lemmas. Finally,
and \(\int _\Omega a t^2 \mathop {}\!\textrm{d}y \lesssim \Vert t\Vert _2^2 \lesssim \lambda ^{-3N+6\,s} = o(\lambda ^{-2\,s})\) by Lemma 5.3 and Assumption (5.7). \(\square \)
Lemma 5.7
(Expanding the denominator) As \(\varepsilon \rightarrow 0\),
Proof
Write \(\alpha ^{-1} u_\varepsilon = \psi _{x, \lambda }+t+ r\). We expand
By Lemma 5.3 together with assumption (5.7), the last term is \(o(\lambda ^{-2s})\). The third term is, by Lemma 5.3,
The second term is
The remaining term \(\int _\Omega \psi _{x, \lambda }^{p-2} r t\mathop {}\!\textrm{d}y\) needs to be expanded more carefully. Using \(\psi _{x, \lambda }= U_{x, \lambda }- \lambda ^{-\frac{N-2s}{2}} H_a(x, \cdot ) - f_{x, \lambda }\) with \(\Vert \lambda ^{-\frac{N-2s}{2}} H_a(x, \cdot ) + f_{x, \lambda }\Vert _\infty \lesssim \lambda ^{-\frac{N-2s}{2}}\), we write
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 \(\int _\Omega (\psi _{x, \lambda }+ r)^p\mathop {}\!\textrm{d}y\). We find
Using orthogonality of r, we get that \(\int _\Omega U_{x, \lambda }^{p-1} r \mathop {}\!\textrm{d}y = 0\) and hence
and \(\lambda ^{-\frac{N+2\,s}{2}} \Vert {(-\Delta )^{s/2}}r\Vert \lesssim \lambda ^{-N} = o(\lambda ^{-2\,s})\). Finally, we have
Collecting all the estimates gives the claim of the lemma. \(\square \)
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
where
and
Taylor expanding up to and including second order, we find
We now observe \({\mathcal {I}}[r] \lesssim \Vert {(-\Delta )^{s/2}}r\Vert ^2 + o(\phi _a(x) \lambda ^{-N+2\,s} + \lambda ^{-2\,s})\) since
Hence we can simplify the expression of the denominator to
Multiplying this with the expansion of the numerator from above, we find
We show in Lemma 5.8 below that the bracket involving the terms \(N_1\) and \(D_1\) involving s vanishes up to order \(o(\lambda ^{-2s})\), due to cancellations. Noting that \(D_0^{-2/p} N_0\) is nothing but \({\mathcal {S}}_{a+ \varepsilon V}[\psi ]\), the expansion claimed in Lemma 5.5 follows. \(\square \)
Lemma 5.8
Assume (5.7) and let \(N_0\), \(N_1\), \(D_0\), \(D_1\) be defined as in the proof of Lemma 5.5. Then
and
where we abbreviated \(B_{N,s}:= \int _{{\mathbb {R}}^N} U_{0,1}^{p-2} |\partial _{\lambda }U_{0,1}|^2 \mathop {}\!\textrm{d}y\).
In particular,
Proof
We start with expanding \(N_1 = 2{\mathcal {E}}_0[\psi , t] + \Vert {(-\Delta )^{s/2}}t\Vert ^2 \). From Lemma B.2 and the expansion (5.1) for t, one easily sees that
where we also used assumption (5.7). Next, recalling \(({(-\Delta )^{s}}+ a) \psi _{x, \lambda }= c_{N,s} U_{x, \lambda }^{p-1} - a (\lambda ^{\frac{N-2s}{2}} h(\lambda (x - \cdot ) + f_{x, \lambda })\) with h as in Lemma B.3, we easily obtain
(Observe that the leading order term with \(\gamma \) vanishes because \(\int _{{\mathbb {R}}^N} U_{0,1}^{p-1} \partial _{\lambda }U_{0,1} = 0\).) This proves the claimed expansion for \(N_1\). For \(D_1\), we have
Writing out \(\psi _{x, \lambda }= U_{x, \lambda }- \lambda ^{-\frac{N-2s}{2}} H_0(x, \cdot ) - f_{x, \lambda }\) and \(PU_{x, \lambda }= U_{x, \lambda }- \lambda ^{-\frac{N-2s}{2}} H_0(x, \cdot ) - f\), by the usual bounds together with assumption (5.7) we get
Similarly,
Observe that the leading order term with \(\gamma \) vanishes because \(\int _{{\mathbb {R}}^N} U_{0,1}^{p-1} \partial _{\lambda }U_{0,1} = 0\). Finally,
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. \(\square \)
Based on the refined expansion of \({\mathcal {S}}_{a+ \varepsilon V}[u_\varepsilon ]\) obtained in Lemma 5.5, we are now in a position to give the proofs of our main results.
We first 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 \(\rho > 0\) such that, as \(\varepsilon \rightarrow 0\),
Proof
Recalling the definition (5.10) of \({\mathcal {I}}[r]\) and observing that \(N_0/D_0 = c_{N,s}\), we find by Proposition 3.5 that
for some \(\rho > 0\). The remaining error term can be bounded as follows:
where we used Lemma A.4. By choosing \(\delta > 0\) small enough, we obtain the conclusion. \(\square \)
6 Proof of the main results
Combining Lemma 5.9 with Lemma 5.5 gives a lower bound on \({\mathcal {S}}_{a + \varepsilon V}[u_\varepsilon ]\). Using the almost-minimizing assumption (1.17) and the expansion from Theorem 2.1, this lower bound can be stated as follows:
where
for some \(\rho > 0\), and \({\mathcal {T}}_2(\phi _a(x), \lambda )\) as in (2.5).
Recall that \(\phi _a \ge 0\) by Corollary 2.2 and that \(\phi _a(x)\) is bounded because \(x_0 \in \Omega \). Since \({\mathcal {T}}_2\) is a sum of higher powers \((\phi _a(x) \lambda ^{-N+2\,s})^k\) with \(k \ge 2\), we have \({\mathcal {R}} \ge 0\) for \(\varepsilon \) small enough.
Lemma 6.1
As \(\varepsilon \rightarrow 0\), \(\phi _a(x) = o(1)\). In other words, \(x_0 \in {\mathcal {N}}_a\).
Proof
Since \(S - S(a + \varepsilon V) \ge 0\) and \(\Vert {(-\Delta )^{s/2}}r\Vert _2^2 \ge 0\), the bound (6.1) gives
Since \(\phi _a(x)\) is uniformly bounded, we can bound \({\mathcal {T}}_2(\phi _a(x), \lambda ) \lesssim \lambda ^{-2N + 4s}\), which concludes. \(\square \)
Lemma 6.2
If \({\mathcal {N}}_a(V) \ne \emptyset \), then \(x_0 \in {\mathcal {N}}_a(V)\).
In the proof of this lemma, we need the assumption (1.12), i.e. that \(a(x) < 0\) on \({\mathcal {N}}_a\).
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 \(S - S(a+ \varepsilon V)\) into (6.1), and using \({\mathcal {R}} \ge 0\), we obtain that
Here, the numbers \(C_1\) and \(C_2\) are given by
Using Lemma 6.1 and the assumption \(a < 0\) on \({\mathcal {N}}_a\), we have that \(C_2\) is strictly positive and remains bounded away from zero by assumption. Since \({\mathcal {N}}_a(V)\) is not empty, the same is clearly true for \(C_1\). Thus, by Young’s inequality,
for some \(c > 0\). This implies \(Q_V(x_0)< -c < 0\) as desired. \(\square \)
Now we are ready to prove our main results Theorems 1.4, 1.5, and 1.6.
Proof of Theorem 1.4
By using \({\mathcal {R}} \ge 0\) and minimizing the last term over \(\lambda \), like in the proof of Corollary 2.3, the bound (6.1) implies
where the last inequality follows from Lemma 6.2. This is equivalent to
Since the matching upper bound has already been proved in Corollary 2.3, the proof of the theorem is complete. \(\square \)
Proof of Theorem 1.5
Since \(x_0 \in {\mathcal {N}}_a\) by Lemma 6.1, by assumption we have \(Q_V(x_0) \ge 0\) and \(a(x_0) < 0\). Together with \({\mathcal {R}} \ge 0\), the bound (6.1) then implies
for some \(c > 0\). Since \(o(\varepsilon \lambda ^{-N+2\,s}) \ge - \frac{c}{2} \lambda ^{-2\,s} + o(\varepsilon ^\frac{2\,s}{4\,s - N})\) by Young, this implies \(S(a+\varepsilon V) \ge S + o(\varepsilon ^\frac{2s}{N-4s}\). Since the inequality
always holds (e.g. by Theorem 2.1), we obtain \(S(a+\varepsilon V) \ge S + o(\varepsilon ^\frac{2s}{N-4s})\) as desired.
Now assume that additionally \(Q_V(x_0) > 0\). With \({\mathcal {R}} \ge 0\), (6.1) implies, for \(\varepsilon > 0\) small enough and some \(C_1, C_2 > 0\)
which contradicts (6.2). Thus assumption (3.19), under which we have worked so far, cannot be satisfied, and we must have \(S(a + \varepsilon _0 V) = S\) for some \(\varepsilon _0 > 0\). Since \(S(a + \varepsilon V)\) is concave in \(\varepsilon \) (being the infimum of functions linear in \(\varepsilon \)) and since \(S(a) = S\), we must have \(S(a + \varepsilon V) = S\) for all \(\varepsilon \in [0, \varepsilon _0]\). \(\square \)
Proof of Theorem 1.6
We may first observe that the upper and lower bounds on \(S(a + \varepsilon V)\) already discussed in the proof of Theorem 1.4 imply
Now, by using additionally Lemma B.6, the estimate (6.1) becomes
where
in the notation of Lemma B.6, for \(A_\varepsilon := A_{N,s}^{-\frac{N-2\,s}{N}} (\alpha _{N,s} + c_{N,s} d_{N,s} b_{N,s} )(|a(x_0)|+o(1))\) and \(B_\varepsilon := A_{N,s}^{-\frac{N-2\,s}{N}} (|Q_V(x_0)| + o(1)) \), with \(\lambda _0(\varepsilon )\) given by (B.5). Now applying in (6.4) the upper bound on \(S(a +\varepsilon V)\) from Corollary 2.3 yields
The terms that make up \({\mathcal {R}}'\) being separately nonnegative, this implies \({\mathcal {R}} = o(\varepsilon ^\frac{2s}{4s-N})\) and \((\lambda ^{-1} - \lambda _0(\varepsilon )^{-1})^2 = o(\varepsilon ^\frac{2}{4\,s-N})\), that is,
and
Inserting the asymptotics of \(\lambda \) back into \({\mathcal {R}} = o\left( \varepsilon ^\frac{2\,s}{4\,s-N}\right) \) now gives
It remains to derive the claimed expansion for \(\alpha \). From Lemma 5.7, we deduce
Using the bound \(\Vert {(-\Delta )^{s/2}}s\Vert _2 \lesssim \lambda ^{-N+2s}\), together with (6.5), (6.6) and the expansion of \(\int _\Omega \psi _{x, \lambda }^\frac{2N}{N-2s}\) from Theorem 2.1, we obtain
This completes the proof of Theorem 1.6. \(\square \)
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Acknowledgements
We acknowledge the kind hospitality of the Universität Ulm during the winter school “Gradient Flows and Variational Methods in PDEs” (2019), where parts of this project were discussed. N. De Nitti is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). He has been partially supported by the Alexander von Humboldt Foundation and by the TRR-154 project of the Deutsche Forschungsgemeinschaft (DFG). T. König acknowledges partial support through ANR BLADE-JC ANR-18-CE40-002. We warmly thank the anonymous referee for his or her careful reading and pertinent remarks, which helped improve the exposition of the paper.
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Appendices
Appendix A: Green’s function
1.1 The Green’s function \(G_0\) and the projections \(PU_{x, \lambda }\)
We begin by studying the case \(a = 0\). The next lemma collects some important estimates on the regular part \(H_0(\cdot , \cdot )\) of the Green’s function and the Robin function \(\phi _0(x) = H_0(x,x)\), which will turn out very important for our analysis. Similar estimates for \(s=1\) have been derived in [38, Sect. 2 and Appendix A].
We denote in the following \(d(x):= \text {dist}(x, \partial \Omega )\).
Lemma A.1
Let \(x \in \Omega \) and \(N > 2s\). Then \(y \mapsto H_0(x,y)\) is continuous on \(\Omega \) and we have, for all \(y \in \Omega \),
Moreover, the Robin function \(\phi _0\) satisfies the two-sided bound
Proof
\(H_0(x, \cdot )\) satisfies
Thus we can write
where \(P_\Omega ^y\) denotes harmonic measure for \({(-\Delta )^{s}}\), see [33, Theorem 7.2]. Since \(P_\Omega ^y\) is a probability measure, this implies
Similarly, since
we have
The lower bound \(d(x)^{2s-N} \lesssim \phi _0(x)\) is proved in [18, Lemma 7.6]. \(\square \)
The following important lemma shows the relation between the regular part \(H_0(x, \cdot )\) and the projections \(PU_{x, \lambda }\) introduced in (1.16). For the classical case \(s = 1\), this is [38, Proposition 1]. For fractional \(s \in (0,1)\), a slightly weaker version relying on the extension formulation of \((-\Delta )^s\) appears in [16, Lemma C.1].
Lemma A.2
Let \(x \in \Omega \) and \(N > 2s\).
-
(i)
We have
$$\begin{aligned} 0 \le PU_{x, \lambda }\le U_{x, \lambda }\end{aligned}$$(A.2)and the function \(\varphi _{x, \lambda }:= U_{x, \lambda }- PU_{x, \lambda }\) satisfies the estimates
$$\begin{aligned} \Vert \varphi _{x, \lambda }\Vert _{L^\infty ({\mathbb {R}}^N)} \lesssim d(x)^{-N + 2s} \lambda ^{\frac{-N+2s}{2}} \end{aligned}$$(A.3)and
$$\begin{aligned} \Vert \varphi _{x, \lambda }\Vert _{L^p({\mathbb {R}}^N)} \lesssim \left( d(x) \lambda \right) ^{-\frac{N-2s}{2}}. \end{aligned}$$(A.4) -
(ii)
Moreover, the expansion
$$\begin{aligned} PU_{x, \lambda }= U_{x, \lambda }- \lambda ^\frac{N-2s}{2} H_0(x, y) + f_{x, \lambda }, \end{aligned}$$(A.5)holds with
$$\begin{aligned} \Vert f_{x, \lambda }\Vert _{L^\infty (\Omega )}&\lesssim d(x)^{-N-2 + 2s} \lambda ^{-\frac{N + 4-2s}{2}}. \end{aligned}$$
Proof
Claim (i). Our proof follows mostly [38, Appendix A]. Since
the maximum principle (see e.g. [43, Proposition 2.17]) implies that \(PU_{x, \lambda }\ge 0\). Similarly, \(\varphi _{x, \lambda }= U_{x, \lambda }- PU_{x, \lambda }\) satisfies
and, thus, \(\varphi _{x, \lambda }\ge 0\) by the maximum principle. This completes the proof of (A.2).
By (A.6), we can moreover write
Thus, \(\Vert \varphi _{x, \lambda }\Vert _{L^\infty ({\mathbb {R}}^N)} = \Vert U_{x, \lambda }\Vert _{L^\infty ({\mathbb {R}}^N {\setminus } \Omega )} \lesssim \lambda ^{\frac{-N+2\,s}{2}} d(x)^{-N + 2\,s}\).
Next, let us prove the \(L^p\) estimate on \(\varphi _{x, \lambda }\). Since \(\varphi _{x, \lambda }\in H^s({\mathbb {R}}^N)\), by the Sobolev inequality we have
The second summand in (A.7) can be written as
by (A.3).
Similarly, the third summand in (A.7) is
where we also used the bound
Collecting these estimates and returning to (A.7), we obtain
This concludes the proof of (A.4).
Claim (ii). The function \(f_{x, \lambda }:= \varphi _{x, \lambda }- \lambda ^{-\frac{N-2\,s}{2}} H_0(x, \cdot )\) satisfies
As in the proof of Lemma A.1, we have
and, hence, since \(P_\Omega ^x\) is a probability measure, we have
by Lemma B.3 below. \(\square \)
1.2 Expanding the regular part \(H_b(x,y)\) near the diagonal
We now turn to the Green’s function \(G_b\), for a general potential \(b \in C^1(\Omega ) \cap C({\overline{\Omega }})\) such that \({(-\Delta )^{s}}+ b\) is coercive. By calling the potential b rather than a, we emphasize the fact that criticality of b is not needed for the following expansions. Moreover, in contrast to the previous subsection, we specialize to the condition \(2s< N < 4s\) again, which plays a role in the proof of Lemmas A.3 and A.4 below.
Lemma A.3
Let \(x \in \Omega \) and \(2s< N < 4s\).
-
(i)
If \(4s - N < 1\), then, as \(y \rightarrow x\),
$$\begin{aligned} H_b(x,y) = \phi _b(x) - d_{N,s} b(x) |x-y|^{4s - N} + o( |x-y|^{4s - N}). \end{aligned}$$ -
(ii)
If \(4s - N \ge 1\), then there is \(\xi _x \in {\mathbb {R}}^N\) such that
$$\begin{aligned} H_b(x,y) = \phi _b(x) + \xi _x \cdot (y-x) - d_{N,s} b(x) |x-y|^{4s - N} + o( |x-y|^{4s - N}). \end{aligned}$$
Here, the constant \(d_{N,s} > 0\) is given by (B.2). The asymptotics are uniform for x in compact subsets of \(\Omega \).
Proof
Fix \(x \in \Omega \) and let
with \(d_{N,s}\) as in (B.2). We use the facts that, in the distributional sense,
and, by Lemma B.5,
Thus \(\psi _x\) solves, in the distributional sense, the equation
with
Since \(b \in C^1(\Omega )\), we have
We will deduce the assertion of the lemma in each case from elliptic estimates on the Eq. (A.8) and appropriate bounds on \(F_x\).
Case \(- N + 2s + 1 < 0\). Since the second summand \(b(y) H_b(x,y)\) is in \(L^\infty \), we have \(F_x \in L^p(\Omega )\) for every \(p < \frac{N}{N-2s-1}\). For the following, fix some \(p \in (\frac{N}{2\,s}, \frac{N}{N-2\,s-1})\). (The assumption \(N < 4\,s\) guarantees that this interval is not empty.)
Define \(\tilde{\psi _x}:= (-\Delta )^{-s} F_x\), where \((-\Delta )^{-s}\) is convolution with the Riesz potential. Then by [39, Theorem 1.6.(iii)] we have \([\tilde{\psi _x}]_{C^\alpha ({\mathbb {R}}^N)} \lesssim \Vert F_x\Vert _{L^p({\mathbb {R}}^N)}\), where \(\alpha = 2s - \frac{N}{p}\). Moreover, \((-\Delta )^s (\psi _x - \tilde{\psi _x}) = 0\) on \(\Omega \). Since s-harmonic functions are smooth (see, e.g., [1, Sect. 2]), we conclude that \(\psi _x \in C^{2s - \frac{N}{p}}(B_{d/2}(x))\).
Since \(\psi _x(x) = 0\), we conclude that as \(y \rightarrow x\),
If we choose \(p \in \left( \frac{N}{N-2\,s}, \frac{N}{N-2\,s-1}\right) \), then \(2\,s - \frac{N}{p} > 4\,s - N\). As a consequence of \(N < 4\,s\), we have the inclusion \(\left( \frac{N}{N-2\,s}, \frac{N}{N-2\,s-1}\right) \subset \left( \frac{N}{2\,s}, \frac{N}{N-2\,s-1}\right) \). Together with the definition of \(\psi _x\), (A.9) then implies
which is the assertion of the lemma.
Case \(- N + 2s + 1 \ge 0\). In this case \(F_x \in L^\infty (\Omega )\). More precisely, we have
Notice that we always have \( - N + 2s +1 < 1\), since \(N > 2s\). As above, define \(\tilde{\psi _x} = (-\Delta )^{-s} F_x\). By [43], we find using \(N < 4s\) that in any of the above cases, \(\tilde{\psi _x} \in C^{1, \alpha }\) for all \(\alpha \in (0,1]\) with \(\alpha < 4\,s - N\). Using Hölder continuity of the gradient, we easily find
Choosing \(\alpha > 4s - N - 1\) and inserting the definition of \(\psi _x\), we find
which is the assertion of the lemma with \(\xi _x:= \nabla \psi _x(x)\). \(\square \)
Lemma A.4
Let \(k \in {\mathbb {N}}\) with \(k \le p = \frac{2N}{N-2\,s}\). If \(k > \frac{2\,s}{N-2\,s}\), then
If \(2 \le k \le \frac{2s}{N-2s}\), then
If \(k = 1, \)
The asymptotics are uniform for x in compacts of \(\Omega \).
Proof
Let us start with the easy case of \(k > \frac{2s}{N-2s}\). In that case, since \(H_a(x, \cdot )\) is uniformly bounded, we have
In any case, this is \(o(\lambda ^{-2s})\).
Now, assume \(1 \le k \le \frac{2s}{N-2s}\). Let us abbreviate \(d = d(x)\) and \(B_d = B_d(x)\) and show that the integral over \(\Omega {\setminus } B_d\) is \(o(\lambda ^{-2s})\). Indeed, since \(H_a(x, \cdot )\) is uniformly bounded,
To evaluate the remaining integral over \(B_d\), we use the formula
by Lemma A.3 (where \(\xi _x\) may be zero if we are in case (i) of that lemma). After multiplying out the right side, every term containing the factor \(\xi _x \cdot (y-x)\) only once vanishes by oddness.
Let now \(k \ge 2\). Since \(\phi _a(x)\) and a(x) are uniformly bounded and \(\Omega \) is bounded, it is clear that we can estimate
For the last step we used that \(4s - N \le 2 + 2s - N < 2\). Now
In any case, this is \(o(\lambda ^{-2s})\).
Finally, if \(k=1\), plugging in expansion (A.12), the term involving a(x) is not negligible anymore. Instead, it gives
which completes the proof. \(\square \)
Appendix B: Auxiliary computations
In this Appendix, we collect some technical results and computations used throughout the paper.
First, we compute the \(L^q\) norm of \(U_{x,\lambda }\) for various values of q.
Lemma B.1
(\(L^q\)-norm of \(U_{x,\lambda }\)) Let \(x \in \Omega \) and \(q \in [1, \infty ]\). As \(\lambda \rightarrow \infty \), we have, uniformly for x in compact subsets,
Moreover, for \(\partial _{\lambda }U_{x, \lambda }= \frac{N-2\,s}{2} \lambda ^{\frac{N-2\,s-2}{2}} \frac{1 - \lambda ^2 |x-y|^2}{(1+ \lambda ^2 |x-y|^2)^\frac{N-2\,s+2}{2}}\), we have \(|\partial _{\lambda }U_{x, \lambda }| = {\mathcal {O}}(\lambda ^{-1} U_{x, \lambda })\) pointwise and therefore
Finally, for \(\partial _{x_i}U_{x, \lambda }= (-N +2\,s) \lambda ^\frac{N-2\,s+2}{2} \frac{\lambda (x-y)}{(1 + \lambda ^2 |x-y|^2)^\frac{N-2\,s+2}{2}}\), we have
Lemma B.2
We have
Moreover,
We remark that the bounds of Lemma B.2 are consistent with the ones proved in [38, Appendix B].
Lemma B.3
We have
with
Moreover \(h(z) \sim |z|^{-N-2 + 2\,s}\) and \(|\nabla h(z)| \sim |z|^{-N + 2\,s - 3}\) as \(|z| \rightarrow \infty \). Consequently, \(h \in L^p({\mathbb {R}}^N)\) for every \(p \in [1, \frac{N}{N-2\,s})\) and \(\nabla h \in L^p({\mathbb {R}}^N)\) for every \(p \in [1, \frac{N}{N-2s+1})\), where the latter interval is possibly empty.
Lemma B.4
Let \(b \in C({\overline{\Omega }}) \cap C^1(\Omega )\). As \(\lambda \rightarrow \infty \), uniformly for x in compact subsets of \(\Omega \),
The numerical value of \(\alpha _{N,s} = \int _{{\mathbb {R}}^N} U_{0,1}(y) h(y) \mathop {}\!\textrm{d}y\) is given in Lemma B.5 below.
Proof
Abbreviate \(d=d(x)\) and \(B_d = B_d(x)\). We integrate separately over \(B_d\) and over \(\Omega \setminus B_d\).
For the outer integral, from Lemma B.3 we get that \(U_{0,1}(y) h(y) \sim |y|^{-2N + 4s -2}\). Thus,
For the inner integral, using that \(b \in C^1(\Omega )\), we write \(b(y) = b(x) + \nabla b(x) \cdot (y-x) + o(|x-y|)\) for \(y \in B_d\). Then (the integral over \(\nabla b(x) \cdot (y-x)\) cancels due to oddness)
To show that the last term is \(o(\lambda ^{-2s})\) as well, note that by Lemma B.3 we have \(U_{0,1}(z) h(z) |z| \lesssim |z|^{-2N + 4\,s -1}\). Thus,
This is \(o(\lambda ^{-2s})\) in all cases. \(\square \)
We compute explicitly the constants that appear in the asymptotic expansions throughout the paper.
Lemma B.5
(Constants) For \(N > 2s\) and \(p = \frac{2N}{N-2s}\), let \(U_{0,1}(y) = \left( \frac{1}{1+|y|^2} \right) ^{\frac{N-2s}{2}} \) and \(h(y)= \frac{1}{|y|^{N-2\,s}} - \frac{1}{(1 + |y|^2)^{\frac{N-2\,s}{2}}}\). Then, for every \(0 \le k < \frac{N}{N-2\,s}\), we have
We denote \(A_{N,s}:= a_{N,s}(0)\) and \(a_{N,s}:= a_{N,s}(1)\). Furthermore,
Moreover, the constant in \((-\Delta )^s u(x):= C_{N,s} P.V. \int _{{\mathbb {R}}^N} \frac{u(x) - u(y)}{|x-y|^{N+2\,s}} \mathop {}\!\textrm{d}y\) is given by
and the constant in \({(-\Delta )^{s}}U_{0,1} = c_{n,s} U_{0,1}^\frac{N+2\,s}{N-2\,s}\) is given by
The explicit value of the best fractional Sobolev constant in \(\Vert {(-\Delta )^{s/2}}u\Vert ^2 \ge S \Vert u\Vert _\frac{2N}{N-2\,s}^2\) is
The constant in \({(-\Delta )^{s}}|x|^{4s - N} = -d_{N,s}^{-1} |x|^{2s-N}\) is given by
The constant \(\gamma _{N,s}\) in \(({(-\Delta )^{s}}+ a) G_a(x, \cdot ) = \gamma _{N,s} \delta _x\) is given by
Proof
The values of \(a_{n,s}(k)\) and \(b_{N,s}\) are a consequence of the following computation. For \(\alpha , \beta >0\),
To compute \(\alpha _{N,s} \), we write
If \(N > 4s\), then the summands of I(r, N, s) are separately integrable, in which case (B.3) gives
To extend this formula to the case \(2s< N < 4s\) which concerns us, we remark that the right side of (B.4) defines a holomorphic function of s in the complex subdomain \({\mathcal {D}}_N:= \{ 0< \text { Re}(s) < N/2 \} {\setminus } \{N/4\} \subset {{{\mathbb {C}}}}\). On the other hand, by a cancellation I(r, N, s) remains integrable in \(r \in (0, \infty )\) for every \(s \in (0,1)\) and \(N \in (2s, 4s)\). Indeed,
By a standard argument, this implies that \(\int _0^\infty I(r,N,s) \mathop {}\!\textrm{d}r\) is holomorphic in \({\mathcal {D}}_N\) as a function of s. By the identity theorem for analytic functions, the formula (B.4) thus holds also for \(s \in (N/4, N/2)\), which is what we wanted to show.
Finally, the claimed value of S can be found, e.g., in [17, Theorem 1.1] and that of \(d_{N,s}\) in [33, Table 1, p. 168]. \(\square \)
Lemma B.6
Let \(2\,s< N <4\,s\) and let \(f_\varepsilon : (0, \infty ) \rightarrow {\mathbb {R}}\) be given by
with \(A_\varepsilon , B_\varepsilon > 0\) uniformly bounded away from 0 and \(\infty \). The unique global minimum of \(f_\varepsilon \) is given by
with corresponding minimal value
Moreover, there is a \(c_0 > 0\) such that, for all \(\varepsilon > 0\), we have
Proof
The values of \(\lambda _0\) and \(f_\varepsilon (\lambda _0)\) are obtained by standard computations. Thus we only prove (B.7). Let \(F(t):= t^{2s} - t^{N-2s}\) and denote by \(t_0:= (\frac{2\,s}{N-2\,s})^{-\frac{1}{4\,s-N}}\) the unique global minimum of F on \((0, \infty )\). Then, there exists \(c > 0\) such that
The assertion of the lemma now follows by rescaling. Indeed, it suffices to observe that
and to use the boundedness of \(A_\varepsilon \) and \(B_\varepsilon \). \(\square \)
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De Nitti, N., König, T. Critical functions and blow-up asymptotics for the fractional Brezis–Nirenberg problem in low dimension. Calc. Var. 62, 114 (2023). https://doi.org/10.1007/s00526-023-02446-1
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DOI: https://doi.org/10.1007/s00526-023-02446-1