Abstract
In this note we will generalize the results deduced in Figalli and Glaudo (Arch Ration Mech Anal 237(1):201–258, 2020) and Deng et al. (Sharp quantitative estimates of Struwe’s Decomposition. Preprint http://arxiv.org/abs/2103.15360, 2021) to fractional Sobolev spaces. In particular we will show that for \(s\in (0,1)\), \(n>2s\) and \(\nu \in \mathbb {N}\) there exists constants \(\delta = \delta (n,s,\nu )>0\) and \(C=C(n,s,\nu )>0\) such that for any function \(u\in \dot{H}^s(\mathbb {R}^n)\) satisfying,
where \(\tilde{U}_{1}, \tilde{U}_{2},\ldots \tilde{U}_{\nu }\) is a \(\delta \)-interacting family of Talenti bubbles, there exists a family of Talenti bubbles \(U_{1}, U_{2},\ldots U_{\nu }\) such that
for \(\Gamma =\left\| \Delta u+u|u|^{p-1}\right\| _{H^{-s}}\) and \(p=2^*-1=\frac{n+2s}{n-2s}.\)
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1 Introduction
1.1 Background
Recall that the Sobolev inequality for exponent 2 states that there exists a positive constant \(S>0\) such that
for \(n\ge 3\), \(u\in C^{\infty }_c(\mathbb {R}^n)\) and \(2^{*}=\frac{2n}{n-2}.\) By density, one can extend this inequality to functions in the homogeneous Sobolev space \(\dot{H}^1(\mathbb {R}^n)\). Talenti [20] and Aubin [1] independently computed the optimal \(S>0\) and showed that the extremizers of (1.1) are functions of the form
where \(c_n>0\) is a dimenion dependent constant. Furthermore, it is well known that the Euler-Lagrange equation associated with the inequality (1.1) is
In particular, Caffarelli, Gidas, and Spruck in [2] showed that the Talenti bubbles are the only positive solutions of the equation
where \(p=2^{*}-1.\) Geometrically, Eq. (1.4) arises in the context of the Yamabe problem which given a Riemannian manifold \((M,g_0)\), in dimension \(n\ge 3\), asks for a metric g conformal to \(g_0\) with prescribed scalar curvature equal to some function R. In the case when \((M,g_0)\) is the round sphere and R is a positive constant then the result of Caffarelli, Gidas, and Spruck can be seen as a rigidity result in the sense that the only possible choice of conformal functions are the Talenti bubbles defined in (1.2). For more details regarding the geometric perspective see the discussion in Section 1.1 in [4].
Following the rigidity result discussed above, it is natural to consider the case of almost rigidity. Here one asks if being an approximate solution of (1.3) implies that u is close to a Talenti bubble. We first observe that this statement is not always true. For instance, consider two bubbles \(U_1 = U[Re_1,1]\) and \(U_2=U[-Re_1,1]\), where \(R\gg 1\) and \(e_1=(1,0,\ldots ,0)\in \mathbb {R}^n\). Then \(u=U_1+U_2\) will be an approximate solution of (1.3) in some well-defined sense, however, it is clear that u is not close to either of the two bubbles. Thus, it is possible that if u approximately solves, (1.3) then it might be close to a sum of Talenti bubbles. However, Struwe in [19] showed that the above case, also known as bubbling, is the only possible worst case. More, precisely, he showed the following
Theorem 1.1
(Struwe). Let \(n \ge 3\) and \(\nu \ge 1\) be positive integers. Let \(\left( u_{k}\right) _{k \in \mathbb {N}} \subseteq \dot{H}^{1}\left( \mathbb {R}^{n}\right) \) be a sequence of nonnegative functions such that
with \(S=S(n)\) being the sharp constant for the Sobolev inequality defined in (1.1). Assume that
Then there exist a sequence \(\left( z_{1}^{(k)}, \ldots , z_{\nu }^{(k)}\right) _{k \in \mathbb {N}}\) of \(\nu \) -tuples of points in \(\mathbb {R}^{n}\) and a sequence \(\left( \lambda _{1}^{(k)}, \ldots , \lambda _{\nu }^{(k)}\right) _{k \in \mathbb {N}}\) of \(\nu \) -tuples of positive real numbers such that
Morally, the above statement says that if the energy of a function \(u_k\) is less than or equal to the energy of \(\nu \) bubbles as in (1.5) and if \(u_k\) almost solves (1.3) in the sense that the deficit (1.7) is small then \(u_k\) is close to \(\nu \) bubbles in \(\dot{H}^1\) sense. For instance, when \(\nu =1\) the energy constraint in (1.5) forbids us from writing u as the sum of two bubbles since that would imply that u has energy equal to \(2S^n\). Thus, Struwe’s result gives us a qualitative answer to the almost rigidity problem posed earlier.
Building upon this work, Ciraolo, Figalli, and Maggi in [4] proved the first sharp quantitative stability result around one bubble, i.e. the case when \(\nu =1\). Their result implied that the distance between the bubbles (1.7) should be linearly controlled by the deficit as in (1.6). Thus one might be inclined to conjecture that the same should hold in the case of more than one bubble or when \(\nu \ge 2.\)
Figalli and Glaudo in [12] investigated this problem and gave a positive result for any dimension \(n\in \mathbb {N}\) when the number of bubbles \(\nu = 1\) and for dimension \(3\le n \le 5\) when the number of bubbles \(\nu \ge 2.\) More precisely they proved the following
Theorem 1.2
(Figalli–Glaudo). Let the dimension \(3\le n\le 5\) when \(\nu \ge 2\) or \(n\in \mathbb {N}\) when \(\nu =1\). Then there exists a small constant \(\delta (n,\nu ) > 0\) and constant \(C(n,\nu )>0\) such that the following statement holds. Let \(u\in \dot{H}^1(\mathbb {R}^n)\) such that
where \(\left( \tilde{U}_{i}\right) _{1 \le i \le \nu }\) is any family of \(\delta \)-interacting Talenti bubbles. Then there exists a family of Talenti bubbles \((U_i)_{i=1}^{\nu }\) such that
where \(p=2^{*}-1=\frac{n+2}{n-2}.\)
Here a \(\delta -\) interacting family is a family of bubbles \((U_i[z_i, \lambda _i])_{i=1}^{\nu }\) where \(\nu \ge 1\) and \(\delta > 0\) if
Note that the definition \(\delta \)-interaction between bubbles follows naturally by estimating the \(\dot{H}^1\) interaction between any two bubbles. This is explained in Remark 3.2 in [12] which we recall here for the reader’s convenience. If \(U_1=U\left[ z_1, \lambda _1\right] \) and \(U_2=U\left[ z_2, \lambda _2\right] \) are two bubbles, then from the interaction estimate in Proposition B.2 in [12] and the fact that\(-\Delta U_i=U_i^{2^*-1}\) for \(i=1,2\) we have
In particular, if \(U_1\) and \(U_2\) belong to a \(\delta \)-interacting family then their \(\dot{H}^1\)-scalar product is bounded by \(\delta ^{\frac{n-2}{2}}\).
In the proof of the above theorem, the authors first approximate the function u by a linear combination \(\sigma = \sum _{i=1}^\nu \alpha _i U_i[z_i,\lambda _i]\) where \(\alpha _i\) are real-valued coefficients. They then show that each coefficient \(\alpha _i\) is close to 1. To quantify this notion we say that the family \((\alpha _i, U_i[z_i, \lambda _i])_{i=1}^{\nu }\) is \(\delta \)-interacting if (1.10) holds and
The authors [12] also constructed counter-examples for the higher dimensional case when \(n\ge 6\) proving that the estimate (1.9) does not hold. The optimal estimates in dimension \(n\ge 6\) were recently established by Deng, Sun, and Wei in [8]. They proved the following
Theorem 1.3
(Deng-Sun-Wei). Let the dimension \(n \ge 6 \) and the number of bubbles \(\nu \ge 2\). Then there exists \(\delta =\delta (n, \nu )>0\) and a large constant \(C=C(n,\nu )>0\) such that the following holds. Let \(u \in \dot{H}^{1}\left( \mathbb {R}^{n}\right) \) be a function such that
where \(\left( \tilde{U}_{i}\right) _{1 \le i \le \nu }\) is a \(\delta \)-interacting family of Talenti bubbles. Then there exists a family of Talenti bubbles \((U_i)_{i=1}^{\nu }\) such that
for \(\Gamma =\left\| \Delta u+u|u|^{p-1}\right\| _{H^{-1}}\) and \(p=2^{*}-1=\frac{n+2}{n-2}.\)
Given these results, it is natural to wonder if they generalize to some reasonable setting. One possible way is to observe that the Sobolev inequality (1.1) can be generalized to hold for functions in fractional spaces. This is better known as the Hardy-Littlewood-Sobolev (HLS) inequality, which states that there exists a positive constant \(S>0\) such that for all \(u\in C^{\infty }_{c}(\mathbb {R}^n)\), we have
where \(s\in (0,1)\) and \(n>2s\) and \(2^* = \frac{2n}{n-2\,s}.\)Footnote 1 Observe that by a density argument, the HLS inequality also holds for all functions \(u\in \dot{H}^s(\mathbb {R}^n),\) where we define the fractional homogeneous Sobolev space \(\dot{H}^s(\mathbb {R}^n)=\dot{W}^{s,2}(\mathbb {R}^n)\) as the closure of the space of test functions \(C_c^{\infty }(\mathbb {R}^n)\) with respect to the norm
equipped with the natural inner product for \(u,v\in \dot{H}^s\)
where \(C_{n,s}>0\) is a constant depending on n and s. Furthermore Lieb in [16] found the optimal constant and established that the extremizers of (1.14) are functions of the form
for some \(\lambda >0\) and \(z\in \mathbb {R}^n\), where we choose the constant \(c_{n,s}\) such that the bubble \(U(x)=U[z,\lambda ](x)\) satisfies
Similar to the Sobolev inequality, the Euler Lagrange equation associated with (1.14) is as follows
where \(p=2^*-1=\frac{n+2s}{n-2s}.\) Chen, Li, and Ou in [3] showed that the only positive solutions to 1.16 are the bubbles described in (1.15). Following this rigidity result Palatucci and Pisante in [18] proved an analog of Struwe’s result on any bounded subset of \(\mathbb {R}^n\). However, as noted in [9] the same proof works carries over to \(\mathbb {R}^n\). We state the result in the multi-bubble case (i.e. when \(\nu \ge 1\)). See also Lemma 2.1 in [9] for the single bubble case.
Theorem 1.4
(Palatucci–Pisante, Nitti–König). Let \(n \in \mathbb {N}, 0<s<n / 2\), \(\nu \ge 1\) be positive integers, and \(\left( u_{k}\right) _{k \in \mathbb {N}} \subseteq \dot{H}^{s}\left( \mathbb {R}^{n}\right) \) be a sequence of functions such that
with \(S=S(n,s)\) being the sharp constant for the HLS inequality defined in (1.14). Assume that
Then there exist a sequence \(\left( z_{1}^{(k)}, \ldots , z_{\nu }^{(k)}\right) _{k \in \mathbb {N}}\) of \(\nu \) -tuples of points in \(\mathbb {R}^{n}\) and a sequence \(\left( \lambda _{1}^{(k)}, \ldots , \lambda _{\nu }^{(k)}\right) _{k \in \mathbb {N}}\) of \(\nu \) -tuples of positive real numbers such that
Thus following the above qualitative result, we establish the following sharp quantitative almost rigidity for the critical points of the HLS inequality
Theorem 1.5
Let the dimension \(n >2s\) where \(s\in (0,1)\) and the number of bubbles \(\nu \ge 1\). Then there exists \(\delta =\delta (n,\nu , s)>0\) and a large constant \(C=C(n,\nu , s)>0\) such that the following holds. Let \(u \in \dot{H}^{s}\left( \mathbb {R}^{n}\right) \) be a function such that
where \(\left( \tilde{U}_{i}\right) _{1 \le i \le \nu }\) is a \(\delta \)-interacting family of Talenti bubbles. Then there exists a family of Talenti bubbles \(\left( {U}_{i}\right) _{1 \le i \le \nu }\) such that
for \(\Gamma =\left\| \Delta u+u|u|^{p-1}\right\| _{H^{-s}}\) and \(p=2^*-1=\frac{n+2s}{n-2s}.\)
Note that analogous to the local case, i.e. when \(s=1\), the definition of \(\delta \)-interaction between bubbles carries over naturally since given any two bubbles \(U_1=U\left[ z_1, \lambda _1\right] , U_2=U\left[ z_2, \lambda _2\right] \) we can estimate their interaction using Proposition B.2 in [12] and the fact that\((-\Delta )^s U_i=U_i^p\) for \(i=1,2\) we have
In particular, if \(U_1\) and \(U_2\) belong to a \(\delta \)-interacting family then their \(\dot{H}^s\)-scalar product is bounded by \(\delta ^{\frac{n-2s}{2}}\). Furthermore, the conditions on the scaling and the translation parameters in Theorem 1.1 of Palatucci and Pisante in [18] imply that bubbling observed in the non-local setting is caused by the same mechanism as in the local case, i.e. either when the distance between the centers of the bubbles or their relative concentration scales (such as \(\lambda _i/\lambda _j\) in 1.32) tend to infinity.
The above theorem, besides being a natural generalization, also provides a quantitative rate of convergence to equilibrium for the non-local fast diffusion equation
with
In particular, Nitti and König in [9] established the following result
Theorem 1.6
(Nitti-König). Let \(n \in \mathbb {N}\) and \(s \in (0, \min \{1, n / 2\})\). Let \(u_0 \in C^2\left( \mathbb {R}^n\right) \) such that \(u_0 \ge 0\) and \(|x|^{2\,s-n} u_0^m\left( x /|x|^2\right) \) can be extended to a positive \(C^2\) function in the origin. Then, there exist an extinction time \(\bar{T}=T\left( u_0\right) \in (0, \infty )\) such that the solution u of (1.22) satisfies \(u(t, x)>0\) for \(t \in (0, \bar{T})\) and \(u(\bar{T}, \cdot ) \equiv 0\). Moreover, there exist \(z \in \mathbb {R}^n\) and \(\lambda >0\), such that
for some \(C_*>0\) depending on the initial datum \(u_0\) and all \(\kappa <\kappa _{n, s}\), where
with \(p=\frac{n+2\,s}{n-2\,s}\), \(\gamma _{n, s}=\frac{4\,s}{n+2\,s+2}\) and
with \(\frac{p}{p-1}=\frac{1}{1-m}=\frac{n+2 s}{4 s}\).
Using Theorem 1.5 one can also establish Theorem 1.6 by following the arguments of the proof of Theorem 5.1 in [12]. However, such a result would not provide explicit bounds on the rate parameter \(\kappa \). The remarkable point of the Theorem 1.6 is that the authors provide an explicit bound on the parameter \(\kappa .\)
1.2 Proof sketch and challenges
The proof of the Theorem 1.5 follows by adapting the arguments in [12] and [8] however we need to overcome several difficulties introduced by the non-locality of the fractional laplacian.
To understand the technical challenges involved, we sketch the proof of Theorem 1.5. First, consider the case when \(n\in (2s,6s)\). Then the linear estimate in (1.21) follows by adapting the arguments in [12]. The starting point in their argument is to approximate the function u by a linear combination of Talenti Bubbles. This can be achieved for instance by solving the following minimization problem
where \(U_i = U[z_i, \lambda _i]\) and \(\sigma = \sum _{i=1}^{\nu } \alpha _i U_i\) is the sum of Talenti Bubbles closest to the function u in the \(\dot{H}^{s}(\mathbb {R}^n)\) norm. Denote the error between u and this approximation by \(\rho \), i.e.
Now, the idea is to estimate the \(H^s\) norm of \(\rho \). Thus
where in the third equality we made use of the orthogonality between \(\rho \) and \(\sigma \). This follows from differentiating in the coefficients \(\alpha _i\). On expanding \(u=\sigma +\rho \) we can further estimate the second term in the final inequality as follows
The argument for controlling the second and third terms follows in the same way as in [12] and this is where one uses the fact that \(n<6s.\) However, it is not so clear how to estimate the first and the last term. For the first term, ideally, we would like to show that
where \(\tilde{c}<1\) is a positive constant. In [12], the authors make use of a spectral argument that essentially says that the third eigenvalue of a linearized operator associated with (1.16) is strictly larger than p. Such a result was not proved in the non-local setting and thus, our first contribution is to prove this fact rigorously in Sect. 2.2.
By linearizing around a single bubble we deduce that
however, recall that our goal is to estimate \(\int _{\mathbb {R}^n} |\sigma |^{p-1} \rho ^2\) instead of \(\int _{\mathbb {R}^n} |U|^{p-1} \rho ^2.\) Thus one would like to localize \(\sigma \) by a bump function \(\Phi _i\) such that \(\sigma \Phi _i \approx \alpha _i U_i.\) This allows us to show that
where \(\Lambda >p.\) Now observe that in the local setting, estimating the first term on the right-hand side of the above inequality would be elementary since we could simply write the following identity
However, this clearly fails in the non-local setting. To get around this issue we observe that
where \(\mathcal {C}(\rho ,\Phi _i)\) is error term introduced by the fractional laplacian. Thus to estimate the product of the fractional gradient we must control this error term, which is fortunately controlled by the commutator estimates from Theorem A.8 in [13]. These estimates state that the remainder term \(\mathcal {C}(\rho , \Phi _i)\) satisfies
provided that \(s_{1}, s_{2} \in [0, s], s=s_{1}+s_{2}\) and \(p_{1}, p_{2} \in (1,+\infty )\) satisfy
Setting \(s_1=s, s_2=0, p_1=n/s\) and \(p_2=2^* = \frac{2n}{n-2\,s}\) we get
Thus it remains to control the fractional gradient of the bump function which we establish in Lemma 2.1.
Next, we consider the case when \(n\ge 6s\). We follow the proof strategy of [8]. The argument proceeds in the following steps.
Step (i). First we obtain a family of bubbles \((U_i)_{i=1}^{\nu }\) as a result of solving the following minimization problem
and denote the error \(\rho = u - \sum _{i=1}^{\nu }U_i= u - \sigma .\) Since \((-\Delta )^s U_i = U_i^p\) for \(i=1,\ldots ,\nu \) we have
Thus the error \(\rho \) satisfies
where
Clearly (1.20) implies that \(\left\Vert \rho \right\Vert _{\dot{H}^s}\le \delta \) and furthermore the family of bubbles \((U_i)_{i=1}^{\nu }\) are also \(\delta '-\)interacting where \(\delta '\) tends to 0 as \(\delta \) tends to 0. In the first step we decompose \(\rho = \rho _0 + \rho _1\) where we will prove the existence of the first approximation \(\rho _0\), which solves the following system
where \((c_a^i)\) is a family of scalars and \(Z^a_i\) are the rescaled derivative of \(U[z_i,\lambda _i]\) defined as follows
for \(1\le i\le \nu \) and \(1\le a\le n.\)
Step (ii). The next step is to establish point-wise estimates on \(\rho _0.\) In order to do this we argue as in the proof of Proposition 3.3 in [8] to show that
which along with an a priori estimate and a fixed point argument allows us to show that
where \(y_i=\lambda _i(x-z_i)\) and
This will also give us a control on the energy of \(\rho _0\), in particular arguing as in the proof of Proposition 3.9 in [8] one can show that
Here we define the interaction term Q by first defining the interaction between two bubbles \(U_i\) and \(U_j\)
and then setting
Since the bubbles are \(\delta \)-interacting, \(Q<\delta .\)
Step (iii). Next, we can estimate the second term \(\rho _1\). First observe that \(\rho _1\) satisfies
Using the above equation and the decomposition
we can estimate \(\rho _2\) and the absolute value of the scalar coefficients \(\beta ^i\) and \(\beta ^i_a\). This allows us to estimate the energy of \(\rho _1.\) In particular using similar arguments as in the proofs of Proposition 3.10, 3.11, and 3.12 in [8] one can show that
Step (iv). In the final step, combining the energy estimates for \(\rho _0\) and \(\rho _1\) from Step (ii) and (iii) one can finally arrive at the desired estimate (1.21) using the same argument as in Lemma 2.1 and Lemma 2.3 in [8]. This will conclude the proof of Theorem 1.5 in the case when the dimension \(n\ge 6s.\)
Even though the proof sketch outlined above seems to carry over in a straightforward manner we decided to include this case for a few reasons. First, we were curious to understand how the fractional parameter s would influence the estimates in our main Theorem. Second, we tried to explain the intuition behind some technical estimates which we hope might be useful for readers not very familiar with the finite-dimensional reduction method (see [6] for an introduction to this technique). Finally, we encountered an interesting technical issue in Step (ii) of the proof. The authors of [8] make use of the following pointwise differential inequality in the case when \(s=1\).
Proposition 1.7
The functions \(\tilde{W}\) and \(\tilde{V}\) satisfy
where \(\alpha _{n,s}>0\) is a constant depending on n and s.
For precise definitions of \(\tilde{W}\) and \(\tilde{V}\) see (3.30). After some straightforward reductions of \(\tilde{W}\) and \(\tilde{V}\), it suffices to establish
for some constant \(\alpha _{n,s}>0.\) By direct computations the above inequality is true in the local setting \(s=1\) however it is not clear whether it generalizes to the fractional case as well. We show that this is indeed true.
Such pointwise differential inequality seems to be new and for instance, does not follow from the well-known pointwise differential inequality
where \(\varphi \in C^1(\mathbb {R})\) is convex and \(f\in \mathcal {S}(\mathbb {R}^n)\). We prove (1.35) inequality directly by using an integral representation of the fractional derivative of \(\frac{1}{(1+|x|^2)^s}\) and by counting the number of zeros of a certain Hypergeometric function.
2 Case when dimension \(n<6s \)
The goal of this section is to prove Theorem 1.5 in the case when the dimension satisfies \(2s<n<6s\). The proof is similar to the proof of Theorem 3.3 in [12], however, we need to make some modifications to establish the spectral and interaction integral term estimate. We first prove Theorem 1.5 assuming that the desired spectral estimate holds.
Proof
Consider the following minimization problem
where \(U_i = U[z_i, \lambda _i]\) and \(\sigma = \sum _{i=1}^{\nu } \alpha _i U_i\) is the sum of Talenti Bubbles closest to the function u in the \(\dot{H}^{s}(\mathbb {R}^n)\) norm. Thus if we set
we immediately have that \(\left\Vert \rho \right\Vert _{\dot{H}^s}\le \delta \) and since the family \((\tilde{U}_i)_{i=1}^{\nu }\) is \(\delta \)-interacting we also deduce that the family \((\alpha _i, U_i)_{i=1}^{\nu }\) is \(\delta '-\)interacting where \(\delta '\rightarrow 0\) as \(\delta \rightarrow 0.\)
Furthermore, for each bubble \(U_i\) with \(1\le i \le \nu \), we have the following orthogonality conditions as before
for any \(1\le j\le n.\) Using Lemma 2.3, we deduce that \(U, \partial _{\lambda } U\) and \(\partial _{z_j} U\) are eigenfunctions for the operator \(\frac{(-\Delta )^s}{U^{p-1}}\) and thus for each \(1\le i,j\le n\), we get
Next using integration by parts and (2.1), we get
In order to estimate the second term, consider the following two estimates
where \(C_{n,s}>0\) is some constant depending on n and s. Thus using triangle inequality and the orthogonality condition (2.4), we get
For the first term by Lemma 2.3, we get
provided \(\delta '\) is small, where \(\tilde{c} = \tilde{c}(n,\nu )<1\) is a positive constant. Using Hölder and Sobolev inequality for the remainder terms with the constraint \(n<6s\) yields
For the last estimate using an integral estimate similar to Proposition B.2 in Appendix B of [12], we get
where
Furthermore, for the interaction term using the same argument as in Proposition 3.11 in [12], we have that for any \(\varepsilon > 0\), there exists a small enough \(\delta '>0\) such that
and
Thus using (2.8), (2.9), (2.10), and (2.11) into (2.7), we get
Therefore if we choose \(\varepsilon \) such that the quantity \( (\tilde{c}+ C\varepsilon ) <1\) then we can absorb the term \((\tilde{c}+ C\varepsilon ) \left\Vert \rho \right\Vert _{\dot{H}^s}^2\) on the left hand side of the inequality to obtain
Assuming w.l.o.g. that the quantity \(\left\Vert \rho \right\Vert _{\dot{H}^s}\ll 1\), we can now deduce that
Finally, if we decompose u in the following manner
then using (2.12) and (2.13), we get
Thus \(U_1, U_2, \ldots , U_{\nu }\) is the desired family of Talenti bubbles. \(\square \)
We now turn to prove the spectral estimate (2.8). For this, we begin by constructing bump functions that will localize the linear combination of bubbles.
2.1 Construction of bump functions
The construction of bump functions allows us to localize the sum of bubbles \(\sigma \) such that in a suitable region we can assume \(\sigma \Phi _i \approx \alpha _i U_i\) for some bubble \(U_i\) and associated bump function \(\Phi _i.\) As a preliminary step we start by constructing cut-off functions with suitable properties.
Lemma 1.8
(Construction of Cut-off Function). Let \(n > 2s.\) Given a point \(\overline{x}\in \mathbb {R}^n\) and two radii \(0<r< R\), there exists a Lipschitz cut-off function \(\varphi =\varphi _{\overline{x}, r, R}:\mathbb {R}^n \rightarrow [0,1]\) such that the following holds
-
(i)
\(\varphi \equiv 1\) on \(B(\overline{x}, r).\)
-
(ii)
\(\varphi \equiv 0\) outside \(B(\overline{x}, R).\)
-
(iii)
\(\int _{\mathbb {R}^n} |(-\Delta )^{s/2} \varphi |^{n/s} \lesssim \log (R/r)^{1-n/s}.\)
Proof
For simplicity, we set \(\overline{x}=0\). Then we define the function \(\varphi :\mathbb {R}^n\rightarrow [0,1]\) as follows
where \(F(x)=F(|x|)=\frac{\log (R/|x|)}{\log (R/r)}.\) The function \(\varphi \) clearly satisfies the first and second conditions. Next, we estimate the fractional gradient. Recall the formula for the fractional derivative
When \(|x|>2R\) then since the integral is non-zero when \(|y|<R\) we have \(|x-y|>\frac{1}{2}|x|\) and therefore
Thus
On the other hand when \(|x|<2R\) then we have three possible cases. When \(|x|<r\) then we consider three cases \(r<|y|<2r\) and \(2r<|y|<R\) and \(|y|>R.\) Since \(|x|<r\) and \(r<|y|<2r\) implies that \(|x-y|<3r\) we have
Denote
Estimating each term
where we used the fact the \(\varphi \) is Lipschitz for estimating \(\textrm{I}\) and \(K\approx \log (R/r)\). This implies that
When \(r<|x|<R\) then we can use the expression in Table 1 of [14] to deduce that
Following a similar argument as in the case when \(|x|<r\) we can estimate in the regime when \(R<|x|<2R\) to get
Thus combining the above estimates we get
\(\square \)
The construction of cut-off functions allows us to deduce the following lemma whose proof is identical to the proof of Lemma 3.9 in [12], except for the \(L^{n/s}\) estimate which is a consequence of property (iii) in Lemma 2.1.
Lemma 1.9
(Construction of Bump Function). Given dimension \(n>2s\), number of bubbles \(\nu \ge 1\) and parameter \(\hat{\varepsilon }>0\) there exists a \(\delta = \delta (n, \nu , \hat{\varepsilon },s) > 0\) such that for a \(\delta \)-interacting family of Talenti bubbles \((U_i)_{i=1}^{\nu }\) where \(U_i = U[z_i, \lambda _i]\) there exists a family of Lipschitz bump functions \(\Phi _i:\mathbb {R}^n \rightarrow [0,1]\) such that the following hold,
-
(i)
Most of the mass of the function \(U_i^{p+1}\) is in the region \(\{\Phi _i = 1\}\) or more precisely,
$$\begin{aligned} \int _{\left\{ \Phi _{i}=1\right\} } U_{i}^{p+1} \ge (1-\hat{\varepsilon }) S^{n/s}. \end{aligned}$$(2.15) -
(ii)
The function \(\Phi _i\) is much larger than any other bubble in the region \(\{\Phi _i > 0\}\) or more precisely,
$$\begin{aligned} \hat{\varepsilon } U_{i}>U_{j} \end{aligned}$$(2.16)for each index \(j\ne i.\)
-
(iii)
The \(L^{n/s}\) norm of the the function \((-\Delta )^{s/2} \Phi _{i}\) is small, or more precisely,
$$\begin{aligned} \left\| (-\Delta )^{s/2} \Phi _{i}\right\| _{L^{n/s}} \le \hat{\varepsilon }. \end{aligned}$$(2.17) -
(iv)
Finally, for all \(j\ne i\) such that \(\lambda _j \le \lambda _i\), we have
$$\begin{aligned} \frac{\sup _{\left\{ \Phi _{i}>0\right\} } U_{j}}{\inf _{\left\{ \Phi _{i}>0\right\} } U_{j}} \le 1+\hat{\varepsilon }. \end{aligned}$$(2.18)
2.2 Spectral properties of the linearized operator
Consider the linearized equation,
where \(\varphi \in \dot{H}^s\) and \(U(x)=U[z,\lambda ](x).\) By exploiting the positivity of the second variation of \(\delta (u) = \left\Vert u\right\Vert _{\dot{H}^s}^2 - S^2 \Vert u\Vert _{L^{{2}^*}}^2\) around the bubble U and using Theorem 1.1 in [11], we can deduce the following result
Lemma 1.10
The operator \(\mathcal {L}=\frac{(-\Delta )^s}{U^{p-1}}\) has a discrete spectrum with increasing eigenvalues \(\{\lambda _{i}\}_{i=1}^{\infty }\) such that,
-
(i)
The first eigenvalue \(\alpha _{1} = 1\) with eigenspace \(H_1 = {\text {span}}(U).\)
-
(ii)
The second eigenvalue \(\alpha _{2} = p\) with eigenspace \(H_2 ={\text {span}}(\partial _{z_1}U,\partial _{z_2}U,\ldots , \partial _{z_n}U,\partial _{\lambda }U)\).
Proof
Since the embedding \(\dot{H}^s \hookrightarrow L^{2^*}_{U^{p-1}}\) is compact, the spectrum of the operator \(\mathcal {L}\) is discrete. This can be proved by following the same strategy as in the proof of Proposition A.1 in [12] along with a fractional Rellich-Kondrakov Theorem which is stated as Theorem 7.1 in [10]. Furthermore, as the following identities hold
for \(1\le j\le n\), it is clear that 1 and p are eigenvalues with eigenfunctions U and the partial derivatives \(\partial _{z_j}U\) and \(\partial _{\lambda } U\) respectively. For the first part, since the function \(U>0\), we deduce that the first eigenvalue is \(\alpha _1=1\) which is simple and therefore \(H_1={\text {span}}(U).\) For the second part, recall the min–max characterization of the second eigenvalue
We first show that \(\lambda _2\le p\). For this consider the second variation of the quantity \(\delta (u) = \left\Vert u\right\Vert _{\dot{H}^s}^2 - S^2 \Vert u\Vert _{L^{{2}^*}}^2\) around the bubble U. Since U is an extremizer for the HLS inequality we know that
Furthermore using \(\int _{\mathbb {R}^n} U^{2^*} = S^{n/s}\) for any \(\varphi \in \dot{H}^s\), we get
Thus, when \(\langle \varphi ,U\rangle _{L^2_{U^{p-1}}}=\int _{\mathbb {R}^n} \varphi U^p = 0\) we can drop the second term to get
This implies that \(\lambda _2\le p\) and the equality is attained when \(\varphi = \partial _{\lambda } U\) or \(\varphi = \partial _{z_j} U\) for \(1\le j\le n.\) Thus the second eigenvalue \(\lambda _2 = p.\) Finally, we need to argue that \(H_2 ={\text {span}}(\partial _{z_1}U,\partial _{z_2}U,\ldots , \partial _{z_n}U,\partial _{\lambda }U)\). For this we make use of Theorem 1.1 in [11] which states the following
Theorem 1.11
Let \(n>2s\) and \(s\in (0,1)\). Then the solution
of the equation, \((-\Delta )^s U = U^p\) is nondegenerate in the sense that all bounded solutions of equation \((-\Delta )^s \varphi = pU^{p-1}\varphi \) are linear combinations of the functions
Thus, if we argue that the solutions to the linearized equation \((-\Delta )^s \varphi = p U^{p-1}\varphi \) are bounded, then we can apply the above theorem to deduce that \(H_2 ={\text {span}}(\partial _{z_1}U,\partial _{z_2}U,\ldots , \partial _{z_n}U,\partial _{\lambda }U)\). For this, we use a bootstrap argument. Let \(\varphi \in \dot{H}^s\) satisfy the equation \((-\Delta )^s \varphi = pU^{p-1}\varphi \). Then since \(pU^{p-1}\varphi \in \dot{H}^s\), we get that \((-\Delta )^s \varphi \in \dot{H}^s\), which implies that \(\varphi \in \dot{H}^{3s}.\) Indeed using the definition of \(\dot{H}^{3s}\) norm, we have
However, now since \(\varphi \in \dot{H}^{3s}\) we can repeat the same argument to get \(\varphi \in \dot{H}^{5s}.\) Proceeding in this manner we deduce that \(\varphi \in \dot{H}^{(2k+1)s}\) for any \(k\in \mathbb {N}.\) Thus for large enough \(k\in \mathbb {N}\), we have \(2(2k+1)s > n\) and therefore by Sobolev embedding for fractional spaces (see for instance Theorem 4.47 in [7]) we get that \(\varphi \in L^{\infty }(\mathbb {R}^n)\). This allows us to use Theorem 2.4 and thus we deduce that \(H_2 ={\text {span}}(\partial _{z_1}U,\partial _{z_2}U,\ldots , \partial _{z_n}U,\partial _{\lambda }U)\). \(\square \)
As done in [12], by localizing the linear combination of bubbles using bump functions and the spectral properties derived in Lemma 2.3, we can show the following inequality,
Lemma 1.12
Let \(n>2s\) and \(\nu \ge 1.\) Then there exists a \(\delta > 0\) such that if \((\alpha _i, U_i)_{i=1}^{\nu }\) is a family of \(\delta \)-interacting family of bubbles and \(\rho \in \dot{H}^s(\mathbb {R}^n)\) is a function satisfying the orthogonality conditions (2.1), (2.2) and (2.3) where we denote \(U_i = U[z_i,\lambda _i]\) then there exists a constant \(\tilde{c}<1\) such that following holds
where \(\sigma = \sum _{i=1}^{\nu } \alpha _i U_i.\)
Proof
Let \(\varepsilon > 0\) be such that there exists a \(\delta >0\) and a family of bump functions \((\Phi _i)_{i=1}^{\nu }\) as in Lemma 2.2. Then using (2.16), we get
where o(1) denotes a quantity that tends to 0 as \(\delta \rightarrow 0.\) We can estimate the second term using Hölder and Sobolev inequality
To estimate the first term, we first show that
where \(\Lambda > p\) is the third eigenvalue of the operator \(\frac{(-\Delta )^s}{U_i^p}.\) To prove this estimate we show that \(\rho \Phi _i\) almost satisfies the orthogonality conditions (2.1), (2.2) and (2.3). Let \(f:\mathbb {R}^n\rightarrow \mathbb {R}\) be a function equal to \(U_i, \partial _{\lambda } U_i\) or \(\partial _{z_j} U_i\) up to scaling and satisfying the identity \(\int _{\mathbb {R}^n} f U_{i}^{p-1} =1.\) Then using (2.1), (2.2) and (2.3), we get
Using Lemma 2.3, we can now conclude (2.21). We can further estimate the first term in (2.21) by using Theorem A.8 in [13], which states that the remainder term \(\mathcal {C}(\rho , \Phi _i)=(-\Delta )^{s/2}(\rho \Phi _i)-\rho (-\Delta )^{s/2}\Phi _i -\Phi _i(-\Delta )^{s/2}\rho \) satisfies the following estimate
provided that \(s_{1}, s_{2} \in [0, s], s=s_{1}+s_{2}\) and \(p_{1}, p_{2} \in (1,+\infty )\) satisfy
Setting \(s_1=s, s_2=0, p_1=n/s\) and \(p_2=2^* = \frac{2n}{n-2\,s}\) using (2.22) and (2.17), we get
We can estimate the term \(\rho (-\Delta )^{s/2}\Phi _i\) using Hölder and Sobolev inequality along with (2.17) as follows
Thus
Since the bump functions have disjoint support by construction, we get
Therefore using (2.20), (2.21), (2.24) and (2.25), we get
which implies the desired estimate. \(\square \)
2.3 Interaction integral estimate
In this section, we will prove (2.11) and (2.12).
Lemma 1.13
Let \(2\,s< n < 6\,s\) and \(\nu \ge 1.\) For any \(\varepsilon >0\) there exits \(\delta >0\) such that the following holds. Let \((\alpha _i, U_i)_{i=1}^{\nu }\) be a \(\delta \)-interacting family, \(u = \sum _{i=1}^{\nu } \alpha _i U_i + \rho \) and \(\rho \) satisfies (2.1), (2.2), (2.3) with \(\left\Vert \rho \right\Vert _{\dot{H}^s} \le 1.\) Then for all \(i=1,2,\ldots , \nu \) we have
and for each \(j\ne i\)
Proof
We begin by assuming that the bubbles are ordered by \(\lambda _i\) in descending order. Thus \(U_1\) is the most concentrated bubble and so on. The proof of this Lemma then proceeds by induction on the index i. Assume that the claim holds for all indices \(j<i\) where \(1\le i\le \nu \) and let \(U_i\) be the corresponding bubble and let \(V= \sum _{j=1,j\ne i }^{\nu } \alpha _j U_j.\)
For \(\varepsilon >0\) (in particular \(\varepsilon =o(1)\)) let \(\Phi _i\) be the bump function associated to \(U_i\) as in Lemma 2.2. Then consider the following decomposition
The term \((\alpha _i-\alpha _i^p)U_i^p\) allows us to estimate \(|\alpha _i-1|\) while the term \(U_i^{p-1}V\) will help us to establish the integral estimate. Furthermore, we want to establish a control that is linear in \(\left\Vert {(-\Delta )^s u-u|u|^{p-1}}\right\Vert _{H^{-s}}\), therefore, we introduce the laplacian term on the right-hand side. Notice that on the region \(\{\Phi _i > 0\}\) we can use (2.16) to get
Thus combining the above estimates we get
Testing the above estimate with \(f\Phi _i\) where \(f = U_i\) or \(f=\partial _{\lambda } U_i\) and using orthogonality conditions (2.1), (2.2) and (2.3) we get
To estimate the first two terms we make use of the Kato-Ponce inequality (1.23). Thus for the first term, we have
where we used made use of the orthogonality conditions (2.1) and (2.2) for the last term and \(\mathcal {C}(f,\Phi _i) = (-\Delta )^{s/2}(f\Phi _i)- f (-\Delta )^{s/2}\Phi _i -\Phi _i (-\Delta )^{s/2}f.\) Then using (1.23), Hölder and Sobolev inequalities we get
Combining the above estimates yields
For the second term
as \(\left\Vert (-\Delta )^{s/2}(f\Phi _i)\right\Vert _{L^2} \lesssim 1\) by (1.23). The other terms can be estimated in the same way as in [12]. Thus
Thus combining the above estimates we finally get
Now if we split \(V = V_1 + V_2\) where \(V_1 =\sum _{j<i} \alpha _jU_j\) and \(V_2 = \sum _{j>i}\alpha _jU_j\) then we know by our induction hypothesis that our claim holds for all indices \(j<i.\) Thus
Furthermore using (2.18) we have
If \(\alpha _i = 1\) then we have nothing to prove, otherwise define \(\theta = \frac{p\alpha _i^{p-1}V_2(0)}{\alpha _i-\alpha _i^{p}}\) and thus using the previous estimate and (2.29) we can re-write (2.28) as
Using (2.15) we can expand the integral on the left-side as follows
where the last two terms can be estimated using (2.15) and \(|f|\lesssim U_i\)
Thus we have
To prove (2.26) we need to show that left-side of the (2.31) cannot be too small for \(f=U_i\) or \(f=\partial _{\lambda } U_i.\) It is enough to check that
because otherwise, we would have that
which can be made arbitrarily small. To check (2.32) observe that
while,
Thus combining (2.31) and (2.32) we get
for either \(f=U_i\) or \(f=\partial _{\lambda } U_i\). Choosing f for which the above integral is maximized we get
This proves (2.26). To prove (2.27) we use (2.26) and (2.28) with \(f=U_i\) to get
which in particular implies that for all indices \(j\ne i\), we have
Using the integral estimate similar to Proposition B.2 in Appendix B of [12] we deduce that the (2.27) also holds for all indices \(j>i\) and thus we are done. \(\square \)
3 Case when dimension \(n\ge 6s \)
The goal of this section is to prove Theorem 1.5 in the case when the dimension satisfies \(n\ge 6s\). We follow the steps outlined in Sect. 1.2.
3.1 Existence of the first approximation \(\rho _0\)
In this section, our goal is to find a function \(\rho _0\) and a set of scalars \(\{c^i_a\}\) such that the following system is satisfied
Here \(U_i = U[z_i,\lambda _i]\) are a family of \(\delta -\) interacting bubbles, \(\sigma = \sum _{i=1}^{\nu } U_i\), \(Z_i^{a} \) are derivatives as defined in (1.30), \(h = \sigma ^{p}-\sum _{j=1}^{\nu } U_{j}^{p}\). From the finite-dimensional reduction argument we first strive to solve the equation with the linear operator
Since the proof follows from a fixed point argument, we need to define the appropriate normed space. Denote for \(i\ne j\) and \(i,j\in I\) where \(I=\{1,\ldots , \nu \}\)
The quantity \(R_{ij}\) gives us an idea of how concentrated or how far the two bubbles are with respect to each other. We further identify two regimes of interest in the subsequent definition.
Definition 1.14
If \(R_{ij} = \sqrt{\lambda _{i} \lambda _{j}}\left| z_{i}-z_{j}\right| \) then we call the bubbles \(U_i\) and \(U_j\) a bubble cluster. Otherwise, we call them a bubble tower.
We now define two norms that capture the behavior of the interaction term h.
Definition 1.15
Define the norm \(\left\Vert {\cdot }\right\Vert _{*}\) as
and the norm \(\left\Vert {\cdot }\right\Vert _{**}\) as
where the functions V and W are defined using \(y_i = \lambda _i(x-z_i)\)
Using the norm \(\Vert \cdot \Vert _{**}\) we can obtain control on the interaction term h for small enough \(\delta \). This is the content of the next lemma.
Lemma 1.16
There exists a small constant \(\delta _0 =\delta _{0}(n)\) and large constant \(C=C(n)\) such that if \(\delta <\delta _0\) then
Proof
We first prove the desired estimate when \(\nu =2\). Then since
we can conclude the proof in the case when \(\nu >2.\) Thus for the remainder of the proof assume that \(\nu = 2.\) As the bubbles are weakly interacting, the bubbles \(U_1\) and \(U_2\) either form a bubble tower or a bubble cluster. We will analyze each case separately.
Case 1. (Bubble Tower) Assume w.l.o.g. that \(\lambda _1 > \lambda _2\), in other words \(U_1\) is more concentrated than \(U_2.\) Then \(R_{12} = \sqrt{\frac{\lambda _1}{\lambda _2}} \gg 1.\) We will estimate the function \(h =(U_1+U_2)^p - U_1^p-U_2^p\) in different regimes.
Core region of \(U_1\): We define the core region of \(U_1\) as the following set
Working in the \(z_1\) centered co-ordinates \(y_1 = \lambda _1(x-z_1)\), we first have that
Furthermore denoting \(\lambda _2 = \lambda _1 R_{12}^{-2}\) and \(\xi _2 = \lambda _1(z_2-z_1)\) we have the following identity
Thus we can express \(U_2\) as follows
Note that by definition of \(R_{12}\) in (3.3), we have
Now we are in position to estimate h in the core region of \(U_1.\) Observe that for \(x\in {\text {Core}}(U_1)\) we have
and thus if \(|y_1|\le \frac{R_{12}}{2}\) we have that \(U_2\lesssim U_1\). Therefore
Outside the Core region of \(U_1\): In this case we consider \(R_{12}/3 \le |y_1| \le 2 R_{12}^2\) and thus we get \(U_1 \approx \lambda _1^{(n-2\,s)/2} |y_1|^{2\,s-n}\). This is because
where for the second estimate we used the fact that \(\frac{1}{3} \le \frac{R_{12}}{3} \le |y_1|\) implies \(9|y_1|^2 \ge 1.\) On the other hand, \(U_2 \approx \lambda _1^{(n-2s)/2} R_{12}^{2s-n}.\) This is because
where for the second estimate we used the fact that \(|\xi _2|^2\le R_{12}^4\) and \(|y_1|^2\le 4R_{12}^4\) to get
Thus we get
Core region of \(U_2\): We define the core region of \(U_2\) as
Note that in the \(z_2\) centered co-ordinates \(y_2 = \lambda _2(x-z_2)\) the points \(x\in \mathbb {R}^n\) satisfying \(|y_2|< R_{12}\) are precisely the ones forming \({\text {Core}}(U_2).\) Like before, in these new co-ordinates we can re-write \(U_2(x) = \lambda _2^{(n-2\,s)/2} U(y_2)\) and \(U_1\) as
where \(\xi _1 = \lambda _2(z_1-z_2)\) such that
Since \(y_2 - \xi _1 = R_{12}^{-2}y_1\), then in the region \(1\le |y_2-\xi _1| \le R_{12}/2\) we have \(R_{12}^2 \le |y_1| \le R_{12}^3/2.\) This is indeed the core region of \(U_2\) as
and thus \(U_1\lesssim U_2\) which implies that
Outside the Core region \(U_2\): In this region \(|y_2-\xi _2| \ge R_{12}/3 \) and, therefore
Thus if we put together estimates (3.10), (3.11), (3.13) and (3.14) we get
which in particular implies that \(h(x)\le V(x) C(n,s)\) for some positive dimension and s dependent constant \(C(n,s)>0\). Thus, if the bubbles form a tower then Lemma 3.3 holds. The proof when the bubbles form a cluster also follows from the same argument as in the proof of Proposition 3.3 in [8] with the minor modifications in the exponents involving the parameter s. \(\square \)
Next, we look at a system similar to (3.2) and deduce an a priori estimate on its solution.
Lemma 1.17
There exists a positive \(\delta _{0}\) and a constant C, independent of \(\delta \), such that for all \(\delta \leqslant \delta _{0}\), if \(\left\{ U_{i}\right\} _{1 \le i \le \nu }\) is a \(\delta \) -interacting bubble family and \(\varphi \) solves the equation
then
where the norms \(\Vert \varphi \Vert _{*} = \sup _{x\in \mathbb {R}^n} |\varphi (x)|W^{-1}(x)\) and \(\Vert h\Vert _{**} = \sup _{x\in \mathbb {R}^n}|h(x)|V^{-1}(x)\) with weight functions
Proof
We will prove this result following the same contradiction-based argument outlined in [8]. Thus, assume that the estimate (3.17) does not hold. Then there exists sequences \((U^{(k)}_i)_{i=1}^{\nu }\) of \(\frac{1}{k}-\)interacting family of bubbles, \(h = h_k\) with \(\Vert h_k\Vert _{**}\rightarrow 0\) as \(k\rightarrow \infty \) and \(\varphi = \varphi _k\) such that \(\Vert \varphi _k\Vert _{*} = 1\) and satisfies
Here \(Z_i^{a^{(k)}}\) are the partial derivatives of \(U_i^{(k)}\) defined as in (1.30). We will often drop the superscript for convenience and set \(U_i = U_i^{(k)}\) and \(\sigma = \sigma ^{(k)} =\sum _{i=1}^{\nu } U_i^{(k)}.\) Next denote, \(z_{ij}^{(k)} = \lambda _i^{(k)}(z_j^{(k)}-z_i^{(k)})\) for \(i\ne j.\) For each such sequence we can assume that either \(\lim _{k\rightarrow \infty } z_{ij}^{(k)} = \infty \) or \(\lim _{k\rightarrow \infty } z_{ij}^{(k)}\) exists and \(\lim _{k\rightarrow \infty } |z_{ij}^{(k)}|<\infty \). Thus define the following two index sets for \(i\in I=\{1,2,\ldots ,\nu \}\)
Given a bubble \(U_i\) we can further segregate it from bubble \(U_j\), for \(j\ne i\), based on whether they form a bubble cluster or bubble tower with respect to each other
Setting \(y_i^{(k)} = \lambda _i^{(k)}(x-z_i^{(k)})\) and \(L_i = \max _{j\in I_{i,1}}\{\lim _{k\rightarrow \infty } |z_{ij}^{(k)}|\}+L\) we define
where \(i\in I\), \(j\ne i\), \(L>0\) is a large constant and \(\varepsilon >0\) is a small constant to be determined later. Observe that for large k, our choice of \(L_i\) ensures that \(\{|y_i^{(k)}-z_{ij}^{(k)}|=\varepsilon \}\subset \{|y_i^{(k)}|\le L_i\}\) for \(j\in T_i^+\cup C_i^+.\)
Dropping superscripts we re-write the weight functions W and V as defined earlier in the following manner
where for each \(i\in I\) and \(R=\frac{1}{2}\min _{j\ne i} R_{ij}^{(k)}\rightarrow \infty \) as \(k\rightarrow \infty \), we have
Since \(\Vert \varphi _k\Vert _{*}=1\), by definition we have that \(|\varphi _k|(x) \le W(x)\). In particular, there exists a subsequence such that \(|\varphi _k|(x_k)=W(x_k).\) We will now analyze several cases based on where this subsequence \(\{x_k\}_{k \in \mathbb {N}}\) lives.
Case 1. Suppose \(\{x_k\}_{k \in \mathbb {N}}\in \Omega _i\) for some \(i\in I.\) W.l.o.g. \(i=1\) and set
Then since \(\varphi _k\) solves the system (3.16), the rescaled function \(\tilde{\varphi }_k\) and \(\tilde{h}_k\) satisfy
where \(U = U[0,1](y_1)\). Set, \(\tilde{z}_j = z_{1j}^{(k)}\), \(\bar{z}_j = \lim _{k\rightarrow \infty } \tilde{z}_{1j}^{(k)}\) and define
If we choose M to be large enough then 
for k large enough. To prove estimate (3.17) in Case 1 we need to make use of the following proposition.
Proposition 1.18
In each compact subset \(K_{M}\), it holds that, as \(k \rightarrow \infty \)
uniformly. Moreover, we have
Postponing the proof of Proposition 3.5 we strive to prove (3.17). First note that by elliptic regularity theory and (3.26) we get that \(\tilde{\varphi }_k\) converges upto a subsequence in \(K_M\). When \(M\rightarrow \infty \), using a diagonal argument we get that \(\tilde{\varphi }_k\rightarrow \tilde{\varphi }\) weakly such that \(\tilde{\varphi }\) satisfies
The orthogonality condition implies \(\tilde{\varphi }\) is orthogonal to the kernel of the linearized operator \((-\Delta )^s - pU^{p-1}\) and thus \(\tilde{\varphi } =0.\) On the other hand since \(|Y_k|= |\lambda _1(x_k-z_1)|\le L_1\) upto a subsequence we have that \(Y_k\rightarrow Y_{\infty }\) and thus \(\tilde{\varphi }(Y_{\infty }) = 1\) which is clearly a contradiction.
Case 2. Suppose \(\{x_k\}_{k \in \mathbb {N}}\subset \Omega {\setminus } \bigcup _{i\in I}\Omega _i.\) Then there exists an index \(i\in I\) such that there exists a subsequence \(\{x_k\}_{k \in \mathbb {N}}\) satisfying
We assume w.l.o.g. that \(i=1\). For \(\mu \in (0,1/2)\) define the function
which approximates the function \(\min \{a,b\}.\) Using F we define new weight functions \(\tilde{W}\) and \(\tilde{V}\) as follows
Then
where \(J_1 = T_{1}^{+} \cup C_{1}^{+}\). We need the following three propositions to conclude estimate (3.17) for Case 2.
Proposition 1.19
The functions \(\tilde{W}\) and \(\tilde{V}\) satisfy
where \(\alpha _{n,s}>0\) is a constant depending on n and s.
Proposition 1.20
For two bubbles \(U_{i}\) and \(U_{j}\), suppose k large enough, in the region \(\left\{ \left| y_{i}\right| \ge L_i,\left| y_{j}\right| \ge L_j\right\} \) it holds that
with \(\varepsilon _1>0\) depending on L, n, s and \(\varepsilon .\)
Proposition 1.21
For bubble \(U_{i}\), let \(J_{i}=T_{i}^{+} \cup C_{i}^{+}.\) In the region \(A_{i}\), we have
when k is large enough.
From Proposition 3.6 we have \((-\Delta )^s \tilde{W}\ge \alpha _{n,s} \tilde{V}\). Furthermore in \(A_1\) it is clear that \(U_1 \gg \sum _{j\in T_1^{-}\cup C_1^{-}} U_j\) and therefore
where \(C=C(n,s)>0\) is constant depending on n and s. This estimate combined with (3.33) and (3.34) implies
in the region \(A_1.\) Thus choosing large L and small \(\varepsilon \), we get
From Case 1, for large k, we have
where \(C=C(n,\nu ,s)>0\) is a large constant. This is because if (3.37) were not true then for any \(\gamma \in \mathbb {N}\) there exists \(k=k(\gamma )\) and \(x_k\in A\) such that
W.l.o.g. assume that \(|\varphi _k|(x_k)W^{-1}(x_k)=1\), then \(\Vert h_k\Vert _{**}\le \frac{1}{\gamma }\rightarrow 0\) as \(\gamma \rightarrow +\infty \) and \(k=k(\gamma )\rightarrow +\infty .\) However this cannot happen as shown in Case 1. Next, for \(i\in T_{1}^{-}\) with \(|y_i|=\frac{\lambda _i}{\lambda _1}|y_1- \tilde{z}_i|\le L_i,\) for large k, we have
Similarly if \(i\in C_1^{-}\) then \(|y_i| =\frac{\lambda _2}{\lambda _1}|y_1- \tilde{z}_i| \ge \frac{\lambda _i}{2\lambda _1}|\tilde{z}_i| = \frac{\lambda _i|z_1-z_i|}{2}\), which in turn implies
Combining estimates (3.38) and (3.39) we get the following estimate in the region \(A_1\)
For large k, \(\partial A_1 \subset \cup _{i\in I}\partial \Omega _i\) and therefore from (3.37) and (3.40) we conclude that
which along with (3.36) implies that \(\pm C\Vert h_k\Vert _{**}\tilde{W}\) is an upper/lower barrier for \(\varphi _k\) and therefore
This contradicts the fact that \(|\varphi _k|(x_k) =W(x_k).\)
Case 3. Finally consider the case \(\{x_k\}_{k \in \mathbb {N}}\subset \Omega ^c = \bigcup _{i\in I}\{|y_i|> L_i\}.\) Working with similar approximations of W and V as defined in (3.30) and arguing as in the proof of Proposition 3.6 we get that \((-\Delta )^s \tilde{W}\ge \alpha _{n,s} \tilde{V}\). Next using Proposition 3.7, we get
Consequently in the region \(\Omega ^c\), we get
From the previous two cases we know that for large k, we have
in the region \(\Omega \) and thus the above estimate also holds on the boundary \(\partial \Omega = \partial \Omega ^c.\) Thus \(\pm C \Vert h_k\Vert _{**} \tilde{W}(x)\) is an upper/lower barrier for the function \(\varphi _k\). This implies that
which in turn implies
This contradicts the fact \(|\varphi _k|(x_k) =W(x_k).\) To complete the proof we prove Proposition 3.6 since the proof of the other propositions can be found in [8] by modifying the exponents of the weights by the parameter s.
Proof of Proposition 3.6
Using the concavity of F and the integral representation of the fractional laplacian we deduce that
where \(a_j = R^{2\,s-n}\langle y_j\rangle ^{-2\,s}\), \(b_j=R^{-4\,s}\langle y_j\rangle ^{4\,s-n}\) and \(y_j=\lambda _j(x-z_j).\) Thus to obtain a lower bound for \((-\Delta )^s\tilde{W}\) we first show that
We prove the first estimate since the argument for the second inequality is the same. Furthermore, by scaling we can consider the case when \(\lambda _j=1\) and \(z_j=0\). Thus we need to show that
Using the hypergeometric function as in Table 1 in [14], we get
for constant \(c_{n,s}>0\) depending on n and s. By the integral representation as in (15.6.1) in [17], we have
as \(n>4s.\) Next observe that the left-hand side has no roots. This is because if we use transformation (15.8.1) in [17] we get
and therefore we can compute the number of zeros using (15.3.1) in [17] as suggested in [15] to get
where \(S = {\text {Sign}}(\Gamma (-s)\Gamma (2\,s)\Gamma (n/2+s)\Gamma (n/2-2\,s))=-1\). Then since \(n>4s\), we get \(N(n,s) = 0.\) Thus the function \({}_{2}F_{1}\left( n/2+s,2s,n/2;-|x|^2\right) \) must be strictly positive and in particular there exists a constant \(\alpha _{n,s}>0\) such that
which gives us the desired estimate. Finally using the homogeneity of the function F we can conclude the proof of Proposition 3.6. \(\square \)
The next result gives an estimate for the coefficients \(c^i_a\) of the system (3.2).
Lemma 1.22
Let \(\sigma \) be a sum of \(\delta \)-interacting bubbles and let \(\varphi , h\) and \(c^j_b\) for \(j=1,\ldots ,\nu \) and \(b=1,2,\ldots , n+1\) satisfy the system (3.2). Then
where Q is the interaction term as defined in (1.32).
Proof
For simplicity we set \(j=1\) and \(\nu =2.\) Multiplying \(Z^b_1\) to (3.2) and integrating we get
Using Lemma 3.16 we get
Using Lemma 3.18 we can also estimate the first term on the RHS as follows
Moving to the LHS, since \(\varphi \) satisfies
for \(b=1,2,\ldots , n+1\) and \(|Z_1^b|\lesssim U_1\) we have
where we used Lemma 3.3 to control the interaction term \(\sigma ^p-\sum _{i=1}^\nu U_i^p\) and Lemma 3.17 to control the integral term \(\int _{\mathbb {R}^n} V W.\) Thus we get that \(\{c^1_b\}_{b=1}^{n+1}\) satisfies the following system
and since \(q_{12}< Q < \delta \) the above system is solvable such that the estimate (3.42) also holds. To prove the estimate for \(\nu >2\) bubbles, one just needs to observe that for each \(j\in I\)
and thus repeating the same argument as above one ends up with the following system
which is a solvable system for \(\delta \) small enough. \(\square \)
Next, we prove that the system (3.2) has a unique solution under some smallness conditions.
Lemma 1.23
There exists a constant \(\delta _0 > 0\) and \(C>0\) independent of \(\delta \) such that for \(\delta _0 \le \delta \) and any h such that \(\Vert h\Vert _{**}<\infty \) the system (3.2) has a unique solution \(\varphi \equiv L_{\delta }(h)\) such that
Proof
We imitate the proof of Proposition 4.1 in [5]. For this consider the space of functions
endowed with the natural inner product
In the weak form solving the system (3.2) is equivalent to finding a function \(\varphi \in H\) that for all \(\psi \in H\) satisfies
which in operator form can be written as
where \(\tilde{h}\) depends linearly on h and \(T_\delta \) is a compact operator on H. Then Fredholm’s alternative implies that there exists a unique solution \(\varphi \in H\) to the above equation provided the only solution to
is \(\varphi \equiv 0\) in H. In other words we want to show that the following equation has a trivial solution in H
We proceed by contradiction. Suppose there exists a non-trivial solution \(\varphi \equiv \varphi _{\delta }\). W.l.o.g. assume that \(\Vert \varphi _{\delta }\Vert _{*} =1.\) However from Lemma 3.4 and Lemma 3.9 we get that \(\Vert \varphi _{\delta }\Vert _{*}\rightarrow 0\) as \(Q \rightarrow 0\) since
which is a contradiction. Thus for each h, the system (3.2) admits a unique solution in H. Furthermore the estimates (3.43) also follow from Lemma 3.4 and Lemma 3.9. \(\square \)
With this Lemma in hand, we can now prove the main result of this section.
Proposition 1.24
Suppose \(\delta \) is small enough. There exists \(\rho _0\) and a family of scalar \(\{c^i_a\}\) which solves (3.1) with
Proof
Set
and denote \(L_{\delta }(h)\) to be the solution to the system (3.2). Then
Thus solving (3.1) is the same as solving
where \(L_{\delta }\) is defined in Lemma 3.10. We show that A is a contraction map in a suitable normed space and show the existence of a solution \(\rho _0\) using the fixed point theorem. First using Lemma 3.3 we have that
for some large constant \(C_2 = C_2(n,s)>0.\) To control \(N_1\) observe that
which implies that
for some large constant \(C_1= C_1(n,s)>0\) that also satisfies the estimate
Define the space
We show that the operator A is a contraction on the space \((E, \Vert \cdot \Vert _{*}).\) First we show that \(A(E)\subset E.\) For this take \(\varphi \in E.\) Then
for \(R\gg 1\) when \(\delta \) is small. Next, we show that A is a contraction map. Observe that for \(\varphi _1, \varphi _2\in E\) we have
Since when \(n\ge 6\,s\), \(|N'(t)|\le C |t|^{p-1}\) which implies
where we choose small enough \(\delta \) such that \(R\gg 1.\) Then we get
This shows that the equation \(\varphi = A(\varphi )\) has a unique solution. Finally from Lemma 3.10 we get
for a large constant \(C>0.\) \(\square \)
3.2 Energy estimates of the first approximation \(\rho _0\)
In this section, our goal is to establish \(L^2\) estimate for \((-\Delta )^{s/2} \rho _0\) where \(\rho _0\) is the solution of the system (3.1) as in Proposition 3.11.
Using Lemma 3.17, Lemma 3.20, and Proposition 3.11 we obtain the following estimate.
Proposition 1.25
Suppose \(\delta \) is small enough. Then, for \(n\ge 6s\)
Proof
Testing the equation in (3.1) with \(\rho _0\) we get
where we used the inequality \(|(\sigma + \rho _0)-\sigma ^p-p\sigma ^{p-1}\rho _0|\lesssim |\rho _0|^p.\) Using \(\sigma ^{p-1} \lesssim U_1^{p-1} + \cdots +U_\nu ^{p-1}\), \(|\rho _0(x)|\lesssim W(x)\) from Proposition 3.11 and Lemma 3.19 we get
For the second term using the Sobolev inequality we have
Finally for the interaction term recall that Lemma 3.3 implies that \(|h| \lesssim V(x).\) Using this estimate along with \(|\rho _0| \lesssim W(x)\) we get
Using Lemma 3.17 we get that the RHS is bounded by (up to a constant) \(R^{-n-2\,s} \approx Q^{p}\) when \(n\ge 6\,s\). When \(n=6s\) we can obtain a more precise estimate using Lemma 3.20 since \(p=2\)
This concludes the proof of Proposition 3.12. \(\square \)
3.3 Energy estimate of the second approximation \(\rho _1\)
In our attempt to estimate the energy of the error term \(\rho \) as defined from the minimization process in (1.25), we define \(\rho _1 = \rho -\rho _0.\) Then since \(\rho \) satisfies (1.27) and \(\rho _0\) satisfies (3.1) the second approximation \(\rho _1\) satisfies
Here recall that \(f= (-\Delta )^s u -u|u|^{p-1}.\) We further decompose \(\rho _1\) as
where \(\rho _2\) satisfies the following orthogonality conditions
for \(i=1,\ldots ,\nu \) and \(a=1,\ldots , n+1.\) Thus in order to estimate the \(L^2\) norm of \((-\Delta )^{s/2} \rho _1\) we estimate the \(L^2\) norm of \((-\Delta )^{s/2} \rho _2.\)
Lemma 1.26
Suppose \(\delta \) is small enough. Then
Proof
Testing (3.47) with \(\rho _2\) and using (3.49) we get
The second term can be trivially estimated as follows
To estimate the first term, we make use of the following inequality
and therefore we get
Using the decomposition of \(\rho _1\) in (3.48) we get
where \(\mathcal {B} = \sum _i|\beta ^i| + \sum _{i,a} |\beta ^{i}_a|.\) For the first term, we have
where we made use of the Sobolev inequality and the spectral inequality of the same form as (2.8) with constant \(\tilde{c}<1.\) For the other terms, using Sobolev inequality we have
and
Thus combining the above estimates we get
For small enough \(\delta \) using Proposition 3.12 we can make \(\left\Vert (-\Delta )^{s/2} \rho _0\right\Vert _{L^2}\ll 1.\) Furthermore for small enough \(\delta \) we can also make \(\mathcal {B}<1\) and \(\left\Vert (-\Delta )^{s/2} \rho _2\right\Vert _{L^2}<1\) and thus using (3.52) and (3.51) we get (3.50). \(\square \)
Estimate (3.50) for \(\rho _2\) suggests that the next natural step is to control the absolute sum of coefficients \(\mathcal {B}\) as defined in (3.48).
Lemma 1.27
If \(\delta \) is small, then
Proof
We start by multiplying the bubble \(U_k\) to equation (3.47) and integrating by parts. Thus we get
Using the inequality
we get the following estimate
Estimating the three terms separately we get
where o(1) denotes a quantity that tends to 0 when \(\delta \rightarrow 0.\) Next, using integral estimate similar to Proposition B.2 in Appendix B of [12] and \(|Z_j^a|\lesssim U_j\) we get that
when \(j\ne k\) otherwise the above integral is equal to 0. Furthermore, the coefficients \(c^i_a\) can be estimated using Lemma 3.3 and Lemma 3.9 to get
Thus we get
Writing \(\int _{\mathbb {R}^n} (-\Delta )^{s/2} \rho _1(-\Delta )^{s/2} U_k+p\int _{\mathbb {R}^n} U_k^p\rho _1 = (p-1)\int _{\mathbb {R}^n} U_k^p\rho _1\) and using the decomposition of \(\rho _1\) in (3.48) along with the orthogonality condition for \(\rho _2\) in (3.49) we get
where we made use of the integral estimate similar to Proposition B.2 in Appendix B of [12] in the last step. On the other hand the orthogonality condition \(\langle \rho _1,Z_k^b\rangle _{\dot{H}^s} =0\), (3.48), (3.49) and Lemma 3.16 implies
Thus combining (3.56) and (3.55) we get
which along with (3.54) gives us the desired bound. \(\square \)
Lemma 1.28
Let \(\delta \) be small enough. Then,
Proof
From (3.48), (3.50) and (3.53) we get that
\(\square \)
3.4 Conclusion
Assuming Step (i), (ii), and (iii) we conclude the proof of Theorem 1.5 when \(n\ge 6s\).
Proof
The proof proceeds in four steps.
Step 1. Recall the equation satisfied by the error \(\rho \)
where,
Multiplying \(Z_k^{n+1}\), integrating by parts and using the orthogonality condition for \(\rho \) we get
Using \(|I_2|\lesssim |\rho |^p\) for the second term and \(|Z^{n+1}_k|\lesssim U_k\) for the third term we get
Next, we further estimate the first two terms in the above estimate.
Step 2. For \(\delta \) small from Lemma 3.13 in [8] we deduce that
where Q is defined as in (1.32) and o(Q) denotes a quantity that satisfies \(o(Q)/Q\rightarrow 0\) as \(Q\rightarrow 0.\)
Step 3. Thus from (3.58), (3.59) we get
Thus using Lemma 2.1 from [8] we get that
where \(i\ne k.\)
Step 4. Arguing as in Lemma 2.3 in [8] we deduce that \(Q\lesssim \Vert f\Vert _{{H}^{-s}}.\) Thus using the fact that, \(\rho = \rho _0 + \rho _1\) and estimates (3.45) and (3.57) we get that
which concludes the proof of Theorem 1.5. \(\square \)
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Notes
Note that we abuse notation by denoting S, \(2^*\) and \(p=2^{*}-1\) to be the fractional analogs of the best constant and the critical exponents related to the Sobolev inequality. For the remainder of this paper, we will only use the fractional version of these constants.
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Acknowledgements
The author would like to thank Prof. Alessio Figalli, Prof. Mateusz Kwaśniciki, Federico Glaudo, and the referees for their valuable comments and suggestions.
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Appendix
Appendix
Here we recall some integral estimates from Appendix A and B in [8]. The only difference is that the exponents have been modified by the parameter s. Thus one can follow the same proof as in Appendix A and B of [8] except minor modifications.
Lemma 1.29
For the \(Z_{i}^{a}\) defined in (1.30), there exist some constants \(\gamma ^{a}=\gamma ^{a}(n,s)>0\) such that
If \(i \ne j\) and \(1 \le a, b \le n+1\), we have
where for \(i\ne j\), \(q_{ij}=\left( \frac{\lambda _i}{\lambda _j} + \frac{\lambda _j}{\lambda _i} + \lambda _i\lambda _j|z_i-z_j|^2\right) ^{-(n-2\,s)/2}.\)
Lemma 1.30
Suppose \(n\ge 6\,s\) and \(R\gg 1\), we have
Lemma 1.31
Suppose \(n\ge 6\,s\) and \(R \gg 1\), we have
Lemma 1.32
Suppose \(n\ge 6\,s\) and \(R \gg 1\), then
Lemma 1.33
Suppose \(n=6\,s\) and \(R\gg 1\), then
where \(y_i = \lambda _i(x-z_i)\) is the \(z_i-\)centered co-ordinate for \(i=1,2.\)
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Aryan, S. Stability of Hardy Littlewood Sobolev inequality under bubbling. Calc. Var. 62, 223 (2023). https://doi.org/10.1007/s00526-023-02560-0
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DOI: https://doi.org/10.1007/s00526-023-02560-0