Asymptotic decay of solutions for sublinear fractional Choquard equations

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation $$(-\Delta)^s u + \mu u = (I_{\alpha}*F(u))f(u) \quad \hbox{on $\mathbb{R}^N$}$$ where $s \in (0,1)$, $N\geq 2$, $\alpha \in (0,N)$, $\mu>0$, $I_{\alpha}$ denotes the Riesz potential and $F(t) = \int_0^t f(\tau) d \tau$ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than $\sim\frac{1}{|x|^{N+2s}}$. The result is new even for homogeneous functions $f(u)=|u|^{r-2}u$, $r\in [\frac{N+\alpha}{N},2)$, and it complements the decays obtained in the linear and superlinear cases in [D'Avenia, Siciliano, Squassina (2015)] and [Cingolani, Gallo, Tanaka (2022)]. Differently from the local case $s=1$ in [Moroz, Van Schaftingen (2013)], new phenomena arise connected to a new"$s$-sublinear"threshold that we detect on the growth of $f$. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers.


Introduction
The present paper is devoted to the study of the following doubly nonlocal equation where s ∈ (0, 1), N ≥ 2, α ∈ (0, N ), µ > 0, |x| N−α is the Riesz potential and (−∆) s denotes the fractional Laplacian.The nonlinearity F (t) = ´t 0 f (τ )dτ is assumed to be quite general, in the spirit of the papers by Berestycki and Lions [7] and Moroz and Van Schaftingen [45], but the result is new even for the power case.In particular, we aim to study the asymptotic behaviour at infinity of the solutions: qualitative properties of this type have been already discussed when f is linear or superlinear in [17] by the author, Cingolani and Tanaka, that is why we will restrict to the case of f sublinear (in the origin).
Physically, this doubly nonlocal model has different applications, in particular in the study of exotic stars: minimization properties related to (1.1) play indeed a fundamental role in the mathematical description of the dynamics of pseudo-relativistic boson stars [26] and their gravitational collapse [29], as well as the evolution of attractive fermionic systems, such as white dwarf stars [34].In fact, the study of the ground states to (1.1) gives information on the size of the critical initial conditions for the solutions of the corresponding pseudo-relativistic equation: in particular, when s = 1  2 , N = 3, α = 2 and f is a power, we obtain √ −∆u + µu = 1 4πr|x| * |u| r |u| r−2 u in R 3 related to the so called massless boson stars equation [35], where the pseudo-relativistic operator √ −∆ + m 2 − m collapses to the square root of the Laplacian; we refer to [30] and references therein for a soft introduction.Other applications can be found in relativistic physics and in quantum chemistry [4,22] and in the study of graphene [41].Mathematically, when s = 1 and f is a power, that is Cingolani, Clapp and Secchi in [13,Proposition A.2] obtained an exponential decay of positive solutions whenever r ≥ 2, which means that the effect of the classical Laplacian prevails.Afterwards Moroz and Van Schaftingen in [44] (see also [46] and [12,19]) extended the previous analysis in the case of ground state solutions to all the possible values of r in the range [ N +α N , N +α N −2 ], in particular by finding a polynomial decay when f is sublinear (i.e., the Choquard term effect prevails).They prove the following result [44,Theorem 4].
Notice that, when µ = 1, the frequency µ influences both the limiting constants and -when r ≥ 2 -the speed of the exponential decays.We refer also to [20,Section 8.2] for some results on convolution equations with non-variational structure.
The case of the fractional Choquard equation s ∈ (0, 1) with homogeneous f , that is has been studied by D'Avenia, Siciliano and Squassina in [21] (see also [8,42,57] for other related results).In this paper the authors gain existence of ground states, multiplicity and qualitative properties of solutions: in particular they obtain asymptotic decay of solutions whenever the source is linear or superlinear, that is when r ≥ 2 (see also [6] for the p-fractional Laplacian counterpart); in this case the rate is polynomial, as one can expect dealing with the fractional Laplacian.More specifically, it does not depend on α, and they prove the following theorem.In this paper, we aim to study the fractional Choquard case s ∈ (0, 1), α ∈ (0, N ), in presence of general, sublinear nonlinearities.We point out that the arguments in [44] cannot be directly adapted to the fractional framework: for instance, we see that the explicit computation of the fractional Laplacian of some comparison function is not possible, and the choice of the comparison functions itself is hindered by some growth condition typical of the fractional framework; moreover, it is not obvious that all the weak solutions are pointwise solutions, and neither one can deduce that the concave power of a pointwise solution is indeed a solution (of a different equation) itself.
We start by presenting the case of homogeneous powers f , which has an interest on its own.Since in the superlinear case the rate of convergence is of the type ∼ 1 |x| N+2s , in the sublinear case we generally expect a slower decay.Actually this is what we find, as the following theorem states.
Theorem 1.3 Let u ∈ H s (R N ), strictly positive, radially symmetric and decreasing, be a weak solution of (1.4).Let r ∈ [ N +α N , 2) and set Moreover, in the case r ∈ [ N +α N , N +α+4s N +2s ) (i.e.β < N + 2s), we have the sharp decay (1.7) We notice that, if µ = 1, the constant in (1.7) is coherent with (1.3).We refer to Remark 2.4 for some comments and generalizations on the assumptions.This result in particular applies to ground states solutions (see Definition 7.3).
Corollary 1.4 Let u be a positive ground state of (1.4).Then the conclusions of Theorem 1.3 hold.
We highlight that the found decay of the ground states might give information, when r < 2, also on the twice Gateaux differentiability of the corresponding functional and on the nondegenaracy of the ground state solution itself, see [44] (see also [46,Section 3.3.5]).Moreover this information on the decay may be exploited to study fractional Choquard equations with potentials V = V (x) approaching, as |x| → +∞, some V ∞ > 0 from above or oscillating, in the spirit of [43].It might be further used, for example, in the semiclassical analysis of concentration phenomena, see e.g.[14].
In both the estimates from above and below in Theorem 1.3 we rely on some comparison principle and the use of some auxiliary function whose fractional Laplacian is related to the Gauss hypergeometric function.For the estimate from above we succeed in working with the weak formulation of the problem; on the other hand, in order to deal with the estimate from below, we find the necessity of working with u 2−r , where 2 − r ∈ (0, 1): this concave power of the solution may fail to lie in H s (R N ), and thus we cannot treat the problem with its weak formulation.The pointwise formulation seems to arise some problems as well, since the fractional Laplacian of u 2−r needs some restrictive assumption on α, s, N and r in order to be well defined.This is why we work with a viscosity formulation of the problem, obtaining a Córdoba-Córdoba type inequality for concave functions (see Lemma 6.1).We remark that the estimate from above may be treated with the viscosity formulation as well.
The paper is organized as follows.In Section 2 we make some comments on the found results and present some generalizations, in particular for the case of a general nonlinearity f = f (t) in (1.1).In Section 3.1 we introduce definitions and notations, collecting some existence and comparison results in Appendix A.1.In Section 3.2 we introduce some suitable auxiliary function (see Appendix A.2 for some related asymptotic property) and establish some asymptotic behaviour on suitable comparison functions; other preliminary estimates are studied in Section 4. Then in Section 5 we deal with the estimate from above, by working with the weak formulation, while in Section 6 we study the asymptotic behaviour from below, by exploiting a viscosity formulation and proving a fractional Chain Rule, suitable for concave functions.Finally in Section 7 we conclude the proofs of the main results.

Comments and generalizations
Joining the results in Theorem 1.2 and Theorem 1.3 we obtain the following picture of the asymptotic decay of fractional Choquard equations.
Corollary 2.1 Let u be a positive ground state of (1.4), with r ∈ By the previous Corollary we see that the exponent r * s,α := , separates the cases where the fractional Laplacian influences more the rate of convergence (which does not depend on α), from the cases where the asymptotic behaviour is dictated by the Choquard term (which does not depend on s).This phenomenon seems to highlight a difference between the fractional and the local case, where the separating exponent is r = 2 (see Theorem 1.1): indeed, when r ∈ r * 1,α , 2 , the arbitrary big (as r → 2) polynomial behaviour keeps being slower than the exponential decay induced by the classical Laplacian; this is not the case when compared with the polynomial decay induced by the fractional Laplacian, and this is why this new phenomenon appears in this range.Thus r * s,α can ben seen as a kind of s-subquadratic threshold for the growth of F ; set instead it can be seen as a s-sublinear threshold for the growth of f .Notice that It might be interesting to investigate other possible phenomena on fractional Choquard equations when r is above and below this exponent r * s,α , or also possible phenomena in (r * 1,α , 2) for the local Choquard equation.We refer also to the recent paper [32,Theorem 1.4] where asymptotic decay results are studied in a different framework (still involving the fractional Laplacian and the Riesz potential); here a threshold different from the classical case s = 1 is detected as well.
When µ = 0 and ρ(x) ≤ 1 |x| γ with γ > N , this fractional sublinear equation (r ∈ (0, 2)) has been studied in [47] (see also [33,Theorem 4.4] where they extend the result to γ > 2s): here the authors find an estimate from above of the asymptotic decay of the solutions, which is strictly slower than ∼ 1 |x| N .Notice that, in our case, ρ = I α * u r decays at most as ∼ 1 |x| N−α (see [33,Lemma 4.6] and [46, page 801]) and we discuss the strict positivity of µ.See also [21,38] for more results on the zero mass case.
We pass now to more general nonlinearities, and study (1.1).For the whole paper we assume the following conditions on f in order to give sense to appearing integrals: or equivalently there exists C > 0 such that for every t ∈ R, In particular, (f2) implies or equivalently that there exists C > 0 such that for every t ∈ R, These conditions have been introduced in [45] for the local case s = 1, extending [7] where the seminal case of local nonlinearities is treated.These critical exponents have then been adapted to the fractional case s ∈ (0, 1) in [21], while the general case (f1)-(f2) has been introduced in [16].This set of assumptions covers different types of nonlinearities, such as pure powers, both odd f (u) = |u| r−1 u or even f (u) = |u| r , combination of powers f (u) = u r ± u q (standing for cooperation or competition), asymptotically linear (saturable) nonlinearities u r+1 1+u r (which appear in nonlinear optics [24]) and many others.Notice that these assumptions include the case of critical nonlinearities, both in the origin and at infinity.
In the papers [15][16][17][18] (see also [30]) the authors study existence and multiplicity of normalized solutions and of Pohozaev minima for (1.1), as well as qualitative properties of solutions, such as regularity, positivity, radial symmetry and Pohozaev identities.In particular in [17] they extend Theorem 1.2 to the case of general nonlinearities, by proving the polynomial asymptotic behaviour of solutions whenever f is linear or superlinear in the origin.That is, by assuming lim sup t→0 |f (t)| |t| < +∞ they gain that every positive weak solution u satisfies (1.5).In this paper, we further investigate the asymptotic behaviour of the solutions of the fractional Choquard equation (1.1) when f is sublinear in the origin.Thus we consider the following additional assumptions: i.e., for some C > 0 and δ ∈ (0, 1) we have i.e., for some C > 0 and δ ∈ (0, 1) we have (2.10) A sufficient condition for (f3) is clearly given by lim sup which means that C can be taken arbitrary small in (2.9) (up to taking δ sufficiently small); in particular it includes logarithmic nonlinearities f (t) = t log(t 2 ), where r can be chosen arbitrary close to 2. A sufficient condition for (f4) is instead given (for example) by a local Ambrosetti-Rabinowitz condition (f (t)t ≥ rF (t) > 0 for t ∈ (0, δ)).The restriction in (f3) and (f4) to right neighborhoods of zero is due to the fact we deal with positive solutions.
We eventually come up with the following generalization of Theorem 1.3.
If both conditions in (i) and (ii) hold, together with C = C (i.e., f is a power near the origin) and r ∈ [ N +α N , N +α+4s N +2s ), then we have the sharp decay where C N,α > 0 is given in (1.3).

Remark 2.4
We highlight that the conclusions of Theorem 2.3 (as well as of Theorem 1.3) hold in more general cases.
• The case ) is included, and as we expect the decay is of order ∼ 1 |x| N+2s .It is sufficient to apply the argument of Remark 4.6 (since f (t) ≥ Ct for t small and positive), and the results in Proposition 5.1 (after having chosen a whatever r ∈ [r * α,s , 2)).
• The conclusions hold also without assuming radial symmetry and monotonicity of u, but by assuming a priori that lim sup N , is radially symmetric and decreasing, this is the case with ω = N q (see Remark 4.1); in particular, if q = 1, we have ω = N .Notice that u is automatically radially symmetric and decreasing when Theorem 1] (see also [55,Theorem 1.3]).
• In light of the previous remark, we highlight that the estimate from above actually holds true also for nonnegative solutions u ≥ 0; see Proposition 5.1; moreover, it can be further extended to |u| in the case of changing sign solutions, by applying a Kato's inequality [2, Theorem 3.2].
• The conclusions hold also for solutions u ∈ L 1 (R N ) ∩ C(R N ) in the viscosity sense, without assuming f Hölder continuous (which is needed in (ii) only to pass from weak to viscosity solutions): see Section 6.
• When (f4) holds, we actually have We highlight that the energy term ´RN I α * F (u) F (u) is always positive (see e.g.[18]).
• We find some estimates on the asymptotic constants, which are coherent, when r ∈ [ N +α N , r * s,α ), with the one found in Theorem 1.1 and Theorem 2.3: see Propositions 5.1 and 6.3.We notice that (1.4) is obtained by (1.1) formally choosing f (t) = √ r|t| r−2 t.In the paper -up to well posedness and regularity -we do not use that F is the primitive of f : in particular, we do not apply (f3) and (f4) to F .Thus we can arbitrary move constants from f to F in our arguments to adjust -for example -the value of C, and this allows to gain the result for every µ > 0.
Our results apply in particular to Pohozaev minima of the equation (see Definition 7.3), whenever some symmetric assumption is assumed on f , that is (f5) f is odd or even, with constant sign on (0, +∞) and locally Hölder continuous.
We refer to [15] for discussions on the assumption (f5).We notice that, since every Pohozaev minimum has strict constant sign [15], it is not restrictive to assume a priori the sign of u.
We finally want to highlight that our results may be adapted to the local case s = 1, extending Theorem 1.1 to general nonlinearities, studied in [45].We leave the details to the reader, observing that in this case the rate of decaying is simply given by β = N −α 2−r , since, as already observed, the solutions of the homogeneous linear (associated) equation decay exponentially.
Theorem 2.6 Let s = 1, and assume (f1)-(f2) (where the upper critical exponent is substituted by N +α N −2 ).Let u ∈ H 1 (R N ), strictly positive, radially symmetric and decreasing, be a solution of in particular, u may be a ground state.Let r ∈ [ N +α N , 2).
If both conditions (i) and (ii) hold, together with C = C, then (2.12) holds.

Definitions and notations
Let s ∈ (0, 1) and α ∈ (0, N ), where N ≥ 2. We will denote by C k,σ (R N ) the space of the functions in C k (R N ) with σ-Hölderian k-derivatives, and more briefly we will write The same notations apply to the local case C γ loc (R N ).Moreover we write • p = • L p (R N ) for the classical L p norm in the entire space, p ∈ [1, +∞], and we will use also the following notation Let the fractional Laplacian be defined via Fourier transform [23] while, when u is regular enough, we can write [23, Proposition 3.3] where C N,s := π N/2 |Γ(−s)| > 0 and the integral is in the principal value sense.A sufficient condition in order to have (−∆) s u well defined pointwise is given by [52, Proposition 2.4] (see also [31,Proposition 2.15] and [18, Proposition 2.1]).
We introduce, for any Ω ⊂ R N and s ∈ (0, 1), We recall that [23, Theorem 5.4 and 6.7], when Ω is for example an open set with C 0,1 bounded boundary, we have In the case Ω = R N we also have the following relation [23,Proposition 3.6] which leads to the following formulation via Fourier transform this definition extends also to every s > 0 [27].
We further recall the Riesz potential where 2 ) > 0: by the Hardy-Littlewood-Sobolev inequality we have Remark 3.2 Arguing as in [17, Proposition 4.5] we see that We recall now the definitions of weak solution, and of viscosity solution (see for instance [50, page 136] or [11,Definition 2.1]).

Definition 3.3 (Weak solution)
Let Ω ⊆ R N and g : Ω → R be measurable.We say that u ∈ H s (Ω) is a weak subsolution [supersolution] of is well defined and holds for each nonnegative ϕ ∈ X s 0 (Ω).We say that u is a weak solution if it is both a weak subsolution and a weak supersolution, i.e. if it satisfies the equality in (3.14) for every ϕ ∈ X s 0 (Ω).Notice that, when Ω = R N , we have We say that u is a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution.
We observe that, generally, the function v appearing in the definition of viscosity solution might be discontinuous.More generally, this definition involves lower and upper semicontinuity of u (see for instance [9,Definition 2.2]).Furthermore, one can easily check that every (continuous) classical solution is a viscosity solution, that the sum of two subsolutions is still a subsolution (with source the sum of the sources), and that the notion of subsolution is conserved on subdomains Ω ′ ⊂ Ω.
We refer to [49,Remark 2.11] and [50, Theorem 1] for some discussions on the relation between classical, weak and viscosity solutions on bounded domains.
The above definitions apply, mutatis mutandis, to equation depending on u, i.e.where the right hand side is of the form h u (x); in this case (fixed u) the definition applies to g(x) := h u (x).
In particular this adapts to our nonlocal equation by substituting (3.14) with where we implicitly assume (f1)-(f2) to give sense to the integrals, and substitute (3.15) with in this last case, we need some assumptions on f and u to have I α * F (u) well defined pointwise, see Remark 3.2.
In Appendix A.1 we collect some standard lemmas on existence results and comparison principles, both for weak and viscosity solutions.

Fractional auxiliary functions
In order to implement some comparison argument, we search for a function which behaves like ∼ 1 |x| β , β > 0, and which lies in H s (R N ): in order to handle the presence of a pole in the origin when β ≥ N , we make the following choice, by considering, for any β > 0, notice that, when β = N + 2s, this function is related to the extremals of the fractional Sobolev inequality [40] and to the solutions of the zero mass critical fractional Choquard equation [38].Chosen h β in this way, we have [37, Table 1 page 168] where > 0 and 2 F 1 denotes the Gauss hypergeometric function (see also [25,Corollary 2], observed that ).Notice that we will be interested in β ∈ (0, N + 2s].
In Appendix A.2 we collect some results on Gauss hypergeometric functions and their asymptotic behaviour at infinity.We use now this auxiliary function to study some comparison function.

Indeed, we impose
which is satisfied if we impose (recall that σ > 0) . We notice that both the minimum and the maximum of w in the ball are finite and strictly positive, since w > 0 is continuous.Thus, summing up By joining (3.19) with the assumption on u, we obtain By the weak version of the Comparison Principle (Lemma A.3) we obtain and hence, by the assumption on w, which gives the claim passing to the limit |x| → +∞, since θ > β and N + 2s > β.
Assume now β = N + 2s, and choose θ = β = N + 2s.Now we have where C σ := γ λ C ′ N +2s,N,s + σC ′ N +2s,N,s + λσ; recall that C ′ N +2s,N,s < 0. We can choose proper σ ∈ R such that C σ < 0, and thus the first equation in (3.19) still hold.Since the sign of σ may be now different, we choose τ ≥ . We come up then with the same proof, obtaining lim sup Notice that the appearing constants depend on u, γ, λ, ρ, β, N, s.Proof.The proof goes as the previous Lemma, with the difference that at the end we apply the pointwise version of the Comparison Principle (Lemma A.4).

Some preliminary estimates
We start with some observations.Remark 4.1 Let u ∈ L q (R N ), for some q ∈ [1, +∞), be continuous and such that |u| is radially symmetric and decreasing.Then, for every where ω N −1 denotes the area of the N − 1 dimensional sphere.Thus We keep with some preliminary lemmas; see [44,Lemma 6.2] (and [28,Lemma C.3]) for the first.
) continuous be such that |u| is radially symmetric and decreasing.Let f satisfy (f1) and (f2,i), and let θ ∈ (N, N + α].Then there exists C = C(N, α) > 0 such that Thus F (u(x)) |x| θ is bounded on a ball B R (since F (u) is bounded), and it is bounded on the complement of this ball since by considering the growth condition (f2,i) of F in zero (when R ≫ 0, not depending on θ) and the restriction on θ.Thus sup x∈R N F (u(x)) |x| θ < +∞ and Lemma 4.2 applies with g(x) := F (u(x)), which concludes the proof.We further notice that for any θ ∈ (N, N + α] and any R ≫ 0 (not depending on θ, but depending on u).

Remark 4.4
In what follows, for the sake of exposition we will restrict our analysis to the space of radially symmetric and decreasing functions in L 1 (R N ), but we highlight that this assumption is needed only to get the a priori asymptotic decay of Remark 4.1.By the above proof, actually we see that we may ask only for some ω such that ω > N 2 N + α .
In particular ω = N , obtained in Remark 4.1, fits this condition.Alternatively, one may assume this a priori asymptotic decay on u (and adapt the restrictions on θ by θ ∈ (N, N +α N ω]).
By Remarks 4.6 and 4.1, we obtain that every strictly positive, continuous, radially symmetric and decreasing solution of (3.16) whenever f satisfies (f1)-( f2) and (f4), together with ´RN F (u) > 0: indeed in this case, by Lemma 4.3, we have Thus the goal is to improve the asymptotic decay (4.22) in the case of sublinear nonlinearities.
We highlight that, by Lemma 3.5, Corollary 4.5, and a bootstrap argument one can give a first qualitative (not rigorous) proof of the main result.We refer to [30,Remark 4.6.22]for details.

Estimate from above
First, we deal with the estimate from above.In this case we succeed in arguing in the weak sense with no additional assumption on f .In what follows we notice that, when r > N +α N , we are actually improving (4.22).Proposition 5.1 Assume (f1) and (f3).Let u ∈ H s (R N ) ∩ L 1 (R N ), continuous, nonnegative, radially symmetric and decreasing, be a weak solution of (3.16).Assume moreover Then, set β as in (1.6) we have, for some if β < N + 2s, the constant C u depends on u in the following way: where C N,α > 0 is given in (3.13).
Proof.We start noticing that, by the Young product inequality, we obtain In particular we choose b = 1 r−1 and thus a = 1 2−r > 0 (possible thanks to the sublinearity restriction on r); with this choice, by (2.9) and the fact that u(x) → 0 as |x| → +∞, we obtain for |x| ≥ R, where R = R(u) ≫ 0 is sufficiently large.By Corollary 4.5, for a whatever fixed θ ∈ (N, N + α] and any ε > 0 we obtain Notice that F (u) ≡ 0 (otherwise, by the equation, u ≡ 0 and the claim is trivial), thus we set notice that we use the fact that We have Joining the first equation with (5.24) we obtain Notice that, if we assume (2.11), then one can choose every C > 0, and thus in particular every µ > 0 is allowed (see Remark 7.2).Anyway, in the proof of Theorem 2.3, we will see how to drop the restriction on µ.
We observe that the previous estimate from above is still valid by considering viscosity solutions u ∈ L 1 (R N ) ∩ C(R N ), see Section 6.We leave the details to the reader.

Fractional concave Chain Rule and estimate from below
Next, we deal with the estimate from below.We need first some preliminary results, in order to deal with the fractional Laplacian of the concave power of a function: since it might happen that u θ / ∈ H s (R N ) when u ∈ H s (R N ) and θ ∈ (0, 1), the weak formulation seems not to be appropriate.Similarly, (−∆) s u θ might be not well defined pointwise, even if u is regular enough.Notice that knowing a priori that u is continuous, radially symmetric and decreasing seems of no use.The idea is thus to treat the problem via viscosity formulation.
The following lemma is a well known result in the case of convex and Lipschitz functions (see [10,Theorem 1.1], [31,Theorem 19.1]).We state it here in the case of concave (not globally Lipschitz) function, in the framework of viscosity solutions.Notice that we do not require u to be in L 2 (R N ).Lemma 6.1 (Córdoba-Córdoba chain rule inequality) Let ϕ : I → R be a concave function, I ⊆ R interval, such that ϕ ∈ C 1 (I).Let u : R N → I.
• Assume in addition ϕ invertible, increasing, with ϕ −1 ∈ C 2 increasing.If u is a continuous viscosity supersolution of (−∆) s u ≥ g in Ω for some function g and Ω ⊆ R N , then ϕ(u) is a viscosity supersolution of Proof.The first claim is a direct consequence of the Lipschitz continuity Secondly, by the concavity of ϕ, for each t, r ∈ I we have We move to the third part.Let x 0 ∈ U ⊂ Ω and φ ∈ C 2 (U ) be such that φ(x 0 ) = ϕ(u(x 0 )) and φ ≤ ϕ(u) in U , and set v := φχ U + ϕ(u)χ U c .Let now By the assumptions on ϕ −1 we have ψ ∈ C 2 (U ), ψ(x 0 ) = u(x 0 ) and ψ ≤ u in U .Thus (−∆) s w(x 0 ) ≥ g(x 0 ).
On the other hand, w = ψ ∈ C 2 on U and ϕ(w) = φ ∈ C 2 on U , hence both the functions are regular enough in a neighborhood of x 0 to state that both the fractional Laplacians are well defined (see Proposition 3.1).Thus we may apply the previous point and obtain Since w(x 0 ) = u(x 0 ), ϕ(w) = v and ϕ ′ is positive, we obtain, by joining the two previous inequalities which is the claim.This concludes the proof.
As a corrollary, we obtain the following result.
• We have • If (−∆) s u is well defined pointwise, then for every x ∈ R N such that (−∆) s u θ (x) is well defined.
• If u is a viscosity supersolution of (−∆) s u ≥ g in Ω for some function g and Ω ⊆ R N , then u θ is a viscosity supersolution of We are now ready the prove the estimate from below.Proposition 6.3 Assume (f1)-(f2,i) and the sublinear condition (f4).Let u ∈ L 1 (R N ) ∩ C(R N ), strictly positive, radially symmetric and decreasing, be a viscosity solution of (3.16).Assume ´RN F (u) > 0.Then, and C N,α > 0 is given in (3.13).Moreover, set β as in (1.6), we have, for some The result in particular applies to pointwise solutions.
Observe that, by Lemma 3.6, we have v(x) → 0 as |x| → +∞.Since (u r−2 − v)(x) → 0 as |x| → +∞, by the viscosity version of the Comparison Principle (Lemma A.4) we obtain By Lemma 3.6 we gain Combining the previous inequalities and sending ε → 0 + , we have the first claim.We conclude by adapting Remark 4.6 to the viscosity case (notice that u ∈ By the results in [15,17], we gain sufficient conditions in order to state that a weak solution is a pointwise solution. Corollary 6.4 Assume (f1)-(f2,i) and the sublinear condition (f4).Let u ∈ H s (R N ) ∩ L 1 (R N ) ∩ C(R N ), strictly positive, radially symmetric and decreasing, be a weak solution of (3.16).Assume moreover that f ∈ C 0,σ loc (R) for some σ ∈ (0, 1] and ´RN F (u) > 0. Then u is a classical solution and the conclusions of Proposition 6.3 hold.
Notice that, by the sublinearity in zero, σ can lie only in (0, r−1].We conjecture anyway that the conclusion of Corollary 6.4 holds in more general cases, by assuming f merely continuous.

Proofs of the main theorems
We can sum up some of the results of the previous sections in the following.
We can now conclude the proof of the main theorem.
Proof of Theorem 2.3.First, we show how to remove the restriction on µ in Proposition 5.1.Indeed, for any κ > 0 we can write I α * F (u) f (u) ≡ I α * F κ (u) f κ (u), where f κ := 1 κ f and F κ := κF .We can thus rewrite (f3) as Since in Proposition 5.1 we did not use that F is the primitive of f (in particular, we did not apply (f3) to F ), fixed a whatever µ > 0 we can choose κ such that that is a large κ given by κ > r−1 µ r−1 C, and obtain We notice, as we expect, that as µ → 0 then κ → +∞ and C u,κ → +∞, while C ′ u defined in Proposition 6.3 is invariant under κ-transformations.
We show now the sharp decay; indeed, we search for a κ such that C u,κ = C ′ u .By a straightforward analysis of g(κ)  Here C i , i = 1, 2, 3, are some strictly positive constants.
Notice that a = N 2 + s, b = β 2 + s, c = N 2 satisfy the assumptions of the previous Lemma, whenever s ∈ (0, 1) and β ∈ (0, N + 2s].Thus, exploiting the representation of (−∆) s h β given in (A.28) and the results on Gauss hypergeometric functions, we come up with the following estimates.

Remark 2 . 2
We notice that, fixed a positive solution u, by setting ρ := I α * u r equation (1.4) can be rewritten as

Lemma 3 . 5 (
Comparison for weak equation) Let u ∈ C(R N ) be a weak solution of

Lemma 3 . 6 (
Comparison for pointwise equation) Let u ∈ C(R N ) be a pointwise solution of(3.18).Then the conclusions of Lemma 3.5 holds.

2 u
and that I α * F (u) and ´RN F (u) are finite and well defined.By Remark 4.1 we have |u(x)| ≤ C |x| N → 0.