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Research article

On the new sine-Gordon solitons of the generalized Korteweg-de Vries and modified Korteweg-de Vries models via beta operator

  • Received: 12 December 2024 Revised: 15 February 2025 Accepted: 21 February 2025 Published: 11 March 2025
  • MSC : 35A24, 35Q53

  • In this paper, we applied the sine-Gordon expansion method (SGEM) and the rational sine-Gordon expansion method (RSGEM) for obtaining some new analytical solutions of the (2+1)-dimensional generalized Korteweg-de Vries (gKdV) and modified Korteweg-de Vries (mKdV) equations with a beta operator. The sine-Gordon expansion method (SGEM) has recently been extended to a rational form, referred to as the rational sine-Gordon expansion method (RSGEM). By applying a specific transformation, the equations are reduced to a nonlinear ordinary differential equation (NODE), allowing for the derivation of analytical solutions in various forms, including complex, hyperbolic, rational, and exponential. All these solutions are expressed through periodic functions using SGEM and RSGEM. The physical significance of the parametric dependencies of these solutions is also examined. Additionally, several simulations, including three-diemensional (3D) visualizations and revolutionary wave behaviors, are presented, based on different parameter selections. Revolutionary surfaces, defined by height and radius as independent variables, are extracted to further illustrate the wave dynamics.

    Citation: Yaya Wang, Md Nurul Raihen, Esin Ilhan, Haci Mehmet Baskonus. On the new sine-Gordon solitons of the generalized Korteweg-de Vries and modified Korteweg-de Vries models via beta operator[J]. AIMS Mathematics, 2025, 10(3): 5456-5479. doi: 10.3934/math.2025252

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  • In this paper, we applied the sine-Gordon expansion method (SGEM) and the rational sine-Gordon expansion method (RSGEM) for obtaining some new analytical solutions of the (2+1)-dimensional generalized Korteweg-de Vries (gKdV) and modified Korteweg-de Vries (mKdV) equations with a beta operator. The sine-Gordon expansion method (SGEM) has recently been extended to a rational form, referred to as the rational sine-Gordon expansion method (RSGEM). By applying a specific transformation, the equations are reduced to a nonlinear ordinary differential equation (NODE), allowing for the derivation of analytical solutions in various forms, including complex, hyperbolic, rational, and exponential. All these solutions are expressed through periodic functions using SGEM and RSGEM. The physical significance of the parametric dependencies of these solutions is also examined. Additionally, several simulations, including three-diemensional (3D) visualizations and revolutionary wave behaviors, are presented, based on different parameter selections. Revolutionary surfaces, defined by height and radius as independent variables, are extracted to further illustrate the wave dynamics.



    In this paper we deal with the following fractional Choquard equation

    (Δ)su+μu=(IαF(u))f(u)inRN (1.1)

    where N2, μ>0, s(0,1), α(0,N), (Δ)s and Iα denote respectively the fractional Laplacian and the Riesz potential defined by

    (Δ)su(x):=CN,sRNu(x)u(y)|xy|N+2sdy,Iα(x):=AN,α1|x|Nα,

    where CN,s:=4sΓ(N+2s2)πN/2|Γ(s)| and AN,α:=Γ(Nα2)2απN/2Γ(α2) are two suitable positive constants and the integral is in the principal value sense. Finally F:RR, F=f is a nonlinearity satisfying general assumptions specified below.

    When dealing with double nonlocalities, important applications arise in the study of exotic stars: minimization properties related to (1.1) play indeed a fundamental role in the mathematical description of the gravitational collapse of boson stars [31,53] and white dwarf stars [37]. 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 [48]. Moreover, when s=12, N=3, α=2 and F(t)=1r|t|r, we obtain

    Δu+μu=(12πr|x||u|r)|u|r2uinR3

    related to the well-known massless boson stars equation [29,39,50], where the pseudorelativistic operator Δ+m collapses to the square root of the Laplacian. Other applications can be found in relativistic physics and in quantum chemistry [1,22,38] and in the study of graphene [56], where the nonlocal nonlinearity describes the short time interactions between particles.

    In the limiting local case s=1, when N=3, α=2 and F(t)=12|t|2, the equation has been introduced in 1954 by Pekar in [63] to describe the quantum theory of a polaron at rest. Successively, in 1976 it was arisen in the work [51] suggested by Choquard on the modeling of an electron trapped in its own hole, in a certain approximation to Hartree-Fock theory of one-component plasma (see also [32,33,69]). In 1996 the same equation was derived by Penrose in his discussion on the self-gravitational collapse of a quantum mechanical wave-function [59,64,65,66] (see also [70,71]) and in that context it is referred as Schrödinger-Newton system. Variational methods were also employed to derive existence and qualitative results of standing wave solutions for more generic values of α(0,N) and of power type nonlinearities F(t)=1r|t|r [60] (see also [14,20,49,58,61,62]). The case of general functions F, almost optimal in the sense of Berestycki-Lions [5], has been treated in [19,61].

    The fractional power of the Laplacian appearing in (1.1), when s(0,1), has been introduced instead by Laskin [47] as an extension of the classical local Laplacian in the study of nonlinear Schrödinger equations, replacing the path integral over Brownian motions with Lévy flights. This operator arises naturally in many contexts and concrete applications in various fields, such as optimization, finance, crystal dislocations, charge transport in biopolymers, flame propagation, minimal surfaces, water waves, geo-hydrology, anomalous diffusion, neural systems, phase transition and Bose-Einstein condensation (see [6,25,31,45,46,55] and references therein). Equations involving the fractional Laplacian together with local nonlinearities have been largely investigated, and some fundamental contributions can be found in [9,10,30]. In particular, existence and qualitative properties of the solutions for general classes of fractional NLS equations with local sources have been studied in [7,11,28,41,42].

    Mathematically, doubly nonlocal equations have been treated in [23,24] in the case of pure power nonlinearities (see also [13] for some orbital stability results and [12] for a Strichartz estimates approach), obtaining existence and qualitative properties of the solutions. Other results can be found in [4,57,67] for superlinear nonlinearities, in [36] for L2-supercritical Cauchy problems, in [35] for bounded domains and in [72] for concentration phenomena with strictly noncritical and monotone sources.

    In the present paper we address the study of (1.1) when f satisfies the following set of assumptions of Berestycki-Lions type [5]:

    (f1) fC(R,R);

    (f2) we have

    i)lim supt0|tf(t)||t|N+αN<+,ii)lim sup|t|+|tf(t)||t|N+αN2s<+;

    (f3) F(t)=t0f(τ)dτ satisfies

    i)limt0F(t)|t|N+αN=0,ii)lim|t|+F(t)|t|N+αN2s=0;

    (f4) there exists t0R, t00 such that F(t0)0.

    We observe that (f3) implies that we are in a noncritical setting: indeed the exponents N+αN and N+αN2s have been addressed in [60] as critical for Choquard-type equations when s=1, and then generalized to s(0,1) in [23]; we will assume the noncriticality in order to obtain the existence of a solution, while most of the qualitative results will be given in a possibly critical setting. This kind of general nonlinearities include some particular cases such as pure powers f(t)tr, cooperating powers f(t)tr+th, competing powers f(t)trth and saturable functions f(t)t31+t2 (which arise, for instance, in nonlinear optics [27]).

    We deal first with existence of a ground state for (1.1), obtaining the following result.

    Theorem 1.1. Assume (f1)–(f4). Then there exists a radially symmetric weak solution u of (3.1), which satisfies the Pohozaev identity:

    N2s2RN|(Δ)s/2u|2dx+N2μRNu2dx=N+α2RN(IαF(u))F(u)dx. (1.2)

    This solution is of Mountain Pass type and minimizes the energy among all the solutions satisfying (1.2).

    We refer to Section 3 for the precise meaning of weak solution, of Mountain Pass type and energy, according to a variational formulation of the problem.

    We point out some difficulties which arise in this framework. Indeed, the presence of the fractional power of the Laplacian does not allow to use the fact that every solution satisfies the Pohozaev identity to conclude that a Mountain Pass solution is actually a (Pohozaev) ground state, as in [44] (see Remark 3.5). On the other hand, the presence of the Choquard term, which scales differently from the L2-norm term, does not allow to implement the classical minimization argument by [5,21]. Finally, the nonhomogeneity of the nonlinearity f obstructs the minimization approach of [23,61]. Thus, we need a new approach to get existence of solutions, in the spirit of [16,17,18].

    Under (f1)–(f4) it is moreover possible to state the existence of a constant sign solution (see Proposition 3.6). This motivates the investigation of qualitative properties for general positive solutions; in this case we consider weaker or stronger assumptions in substitution to (f1)–(f3), depending on the result. In particular, we observe that (f1)–(f2) alone imply

    |tf(t)|C(|t|N+αN+|t|N+αN2s),

    and

    |F(t)|C(|t|N+αN+|t|N+αN2s),

    where we notice that the last inequality is weaker than (f3); some of the qualitative results are still valid when F has this possible critical growth. Consider finally the following stronger assumption in the origin:

    (f5) lim supt0|tf(t)||t|2<+,

    and observe that

    (f5)(f2,i)and(f3,i).

    The main qualitative results that we obtain are the following ones.

    Theorem 1.2. Assume (f1)–(f2). Let uHs(RN) be a weak positive solution of (1.1).Then uL1(RN)L(RN). The same conclusion holds for generally signed solutions by assuming also (f5).

    The condition in zero of the function f assumed in (f5) leads also to the following polynomial decay of the solutions.

    Theorem 1.3. Assume (f1)– (f2) and (f5). Let uHs(RN) be a positive weak solution of (1.1). Then there exists C,C>0 such that

    C1+|x|N+2su(x)C1+|x|N+2s,forxRN.

    The previous results generalize some of the ones in [23] to the case of general, not homogeneous, nonlinearities; in particular, we do not even assume f to satisfy Ambrosetti-Rabinowitz type conditions nor monotonicity conditions. We observe in addition that the information uL1(RN)L2(RN) is new even in the power-type setting: indeed in [23] the authors assume the nonlinearity to be not lower critical, while here we include the possibility of criticality. Moreover, we improve the results in [57,67] since we do not assume f to be superlinear, and we have no restriction on the parameter α. Finally, we extend some of the results in [61] to the fractional framework, and some of the results in [7] to Choquard nonlinearities.

    The paper is organized as follows. We start with some notations and recalls in Section 2. In Section 3 we obtain the existence of a ground state in a noncritical setting, and in addition the existence of a positive solution. Section 4 is dedicated to the study of the boundedness of positive solutions, while in Section 5 we investigate the asymptotic decay. Finally in the Appendix A we obtain the boundedness of general signed solutions under some more restrictive assumption.

    Let N2 and s(0,1). Recalled the definition of the fractional Laplacian [25]

    (Δ)su(x)=CN,sRNu(x)u(y)|xy|N+2sdy

    for every s(0,1), we set the fractional Sobolev space as

    Hs(RN)={uL2(RN)(Δ)s/2uL2(RN)}

    endowed with

    u2Hs=u22+(Δ)s/2u22.

    In particular, we consider the subspace of radially symmetric functions Hsr(RN), and recall the continuous embedding [[25], Theorem 3.5]

    Hs(RN)Lp(RN)

    for every p[2,2s], 2s=2NN2s critical Sobolev exponent, and the compact embedding [54]

    Hsr(RN)↪↪Lp(RN)

    for every p(2,2s). In addition we have the following embedding of the homogeneous space [[25], Theorem 6.5] for some S>0

    u2sS1/2(Δ)su2. (2.1)

    Moreover the following relation with the Gagliardo seminorm holds [[25], Proposition 3.6], for some C(N,s)>0

    (Δ)s/2u22=C(N,S)R2N|u(x)u(y)|2|xy|N+2sdxdy. (2.2)

    Thanks to this last formulation, we obtain that if uHs(RN) and h:RR is a Lipschitz function with h(0)=0, then h(u)Hs(RN). Indeed

    h(u)22=RN|h(u)h(0)|2dxRNh2|u0|2dx=h2u22

    and

    (Δ)s/2h(u)22C(N,S)R2Nh2|u(x)u(y)|2|xy|N+2sdxdy=h2(Δ)s/2u22.

    We further have the following relation with the Fourier transform [[25], Proposition 3.3]

    (Δ)su=F1(|ξ|2s(F(u));

    notice that this last expression is suitable for defining the fractional Sobolev space Ws,p(RN) also for s1 and p1, by [28]

    Ws,p(RN)={uLp(RN)F1(|ξ|s(F(u))Lp(RN)}.

    Finally, set α(0,N), we recall the following standard estimates for the Riesz potential [[52], Theorem 4.3].

    Proposition 2.1 (Hardy-Littlewood-Sobolev inequality). Let α(0,N), and let r,h(1,+) be such that 1r1h=αN. Then the map

    fLr(RN)IαfLh(RN)

    is continuous. In particular, if r,t(1,+) verify 1r+1t=N+αN, then there exists a constant C=C(N,α,r,t)>0 such that

    |RN(Iαg)hdx|Cgrht

    for all gLr(RN) and hLt(RN).

    In this section we search for solutions to the fractional Choquard equation

    (Δ)su+μu=(IαF(u))f(u)inRN (3.1)

    by variational methods on the subspace of radially symmetric functions Hsr(RN). We recall that F=f and we assume (f1)–(f2) in order to have well defined functionals. We set D:Hsr(RN)R as

    D(u):=RN(IαF(u))F(u)dx

    and define the C1-functional Jμ:Hsr(RN)R associated to (3.1) by

    Jμ(u):=12RN|(Δ)s/2u|2dx12D(u)+μ2u22.

    We notice that, by the Principle of Symmetric Criticality of Palais, the critical points of Jμ are weak solutions of (3.1). Moreover, inspired by the Pohozaev identity

    N2s2(Δ)s/2u22+N2μu22=N+α2D(u) (3.2)

    we define also the Pohozaev functional Pμ:Hsr(RN)R by

    Pμ(u):=N2s2(Δ)s/2u22N+α2D(u)+N2μu22.

    Furthermore we introduce the set of paths

    Γμ:={γC([0,1],Hsr(RN))γ(0)=0,Jμ(γ(1))<0}

    and the Mountain Pass (MP for short) value

    l(μ):=infγΓμmaxt[0,1]Jμ(γ(t)). (3.3)

    Finally we set

    p(μ):=inf{Jμ(u)uHsr(RN){0},Pμ(u)=0}

    the least energy of Jμ on the Pohozaev set.

    Remark 3.1. Since of key importance in the good definition of the functionals, as well as in bootstrap argument in the rest of the paper, we write here in which spaces lie the considered quantities. Let uHs(RN)L2(RN)L2s(RN). By (f2) we have

    f(u)L2Nα(RN)LNα2NN2s(RN)+L2N2sα+2sL2Nα+2s(RN)L2Nα(RN)+L2Nα+2s(RN),F(u)L2NN+α(RN)LNN+α2NN2s(RN)+L2N2sN+α(RN)L2NN+α(RN)L2NN+α(RN).

    Thus by the Hardy-Littlewood-Sobolev inequality we obtain

    IαF(u)L2NNα(RN)L2N2N2(α+2s)N2sα(RN)+L2N(N2s)N2αN+4sα(RN)L2NNα(RN)L2NNα(RN).

    Finally, by the Hölder inequality, we have

    (IαF(u))f(u)L2(RN)L2N2N22sα(RN)+L2N(N2s)N2+2αs(RN)L2NN+2s(RN)L2(RN)+L2NN+2s(RN).

    In particular we observe that (IαF(u))f(u) does not lie in L2(RN), generally. On the other hand, if φHs(RN)L2(RN)L2s(RN), we notice that the found summability of (IαF(u))f(u) is enough to have

    RN(IαF(u))f(u)φdx

    well defined.

    We present now an existence result for (3.1).

    Theorem 3.2. Assume (f1)(f4). Let μ>0 be fixed. Then there exists a Mountain Pass solution u of (3.1), that is

    Jμ(u)=l(μ)>0.

    Moreover, the found solution satisfies the Pohozaev identity

    Pμ(u)=0.

    Proof. We split the proof in some steps.

    Step 1. We first show that Jμ satisfies the Palais-Smale-Pohozaev condition at every level bR, that is each sequence un in Hsr(RN) satisfying

    Jμ(un)b, (3.4)
    Jμ(un)0 stronglyin(Hsr(RN)), (3.5)
    Pμ(un)0, (3.6)

    converges up to a subsequence. Indeed (3.4) and (3.6) imply

    α+2s2(Δ)s/2un22+α2μun22=(N+α)b+o(1).

    Thus we obtain that b0 and un is bounded in Hsr(RN).

    Step 2. After extracting a subsequence, denoted in the same way, we may assume that unu0 weakly in Hsr(RN). Taking into account the assumptions (f1)–(f3), we obtain

    RN(IαF(un))f(un)u0dxRN(IαF(u0))f(u0)u0dx

    and

    RN(IαF(un))f(un)undxRN(IαF(u0))f(u0)u0dx.

    Thus we derive that Jμ(un),un0 and Jμ(un),u00, and hence

    (Δ)s/2un22+μun22(Δ)s/2u022+μu022

    which implies unu0 strongly in Hsr(RN).

    Step 3. Denote by

    [Jμb]:={uHsr(RN)Jμ(u)b}

    the sublevel of Jμ and by

    Kb:={uHsr(RN)Jμ(u)=b,Jμ(u)=0,Pμ(u)=0}

    the set of critical points of Jμ satisfying the Pohozaev identity. Then, by Steps 1–2, Kb is compact. Arguing as in [[40], Proposition 4.5] (see also [[43], Proposition 3.1 and Corollary 4.3]), we obtain for any bR, ˉε>0 and any U open neighborhood of Kb, that there exist an ε(0,ˉε) and a continuous map η:[0,1]×Hsr(RN)Hsr(RN) such that

    (1o) η(0,u)=uuHsr(RN);

    (2o) η(t,u)=u(t,u)[0,1]×[Jμbˉε];

    (3o) Jμ(η(t,u))Jμ(u) (t,u)[0,1]×Hsr(RN);

    (4o) η(1,[Jμb+ε]U)[Jμbε];

    (5o) η(1,[Jμb+ε])[Jμbε]U;

    (6o) if Kb=, then η(1,[Jμb+ε])[Jμbε].

    Step 4. By exploiting (f4) and arguing as in [[61], Proposition 2.1], we obtain the existence of a function vHsr(RN) such that D(v)>0. Thus defined γ(t):=v(/t) for t>0 and γ(0):=0 we have J(γ(t))<0 for t large and J(γ(t))>0 for t small; this means, after a suitable rescaling, that l(μ) is finite and strictly positive. In particular we observe that 0Kl(μ).

    Step 5. By applying the deformation result at level b=l(μ)>0, the existence of a Mountain Pass solution u is then obtained classically. Moreover, uKl(μ) by construction, thus u0 and Pμ(u)=0.

    We prove now that the found solution is actually a ground state over the Pohozaev set.

    Proposition 3.3. The Mountain Pass level and the Pohozaev minimum level coincide, that is

    l(μ)=p(μ)>0.

    In particular, the solution found in Theorem 3.2 is a Pohozaev minimum.

    Proof. Let uHsr(RN){0} such that Pμ(u)=0; observe that D(u)>0. We define γ(t):=u(/t) for t0 and γ(0):=0 so that t(0,+)Jμ(γ(t)) is negative for large values of t, and it attains the maximum in t=1. After a suitable rescaling we have γΓμ and thus

    Jμ(u)=maxt[0,1]Jμ(γ(t))l(μ). (3.7)

    Passing to the infimum in Eq (3.7) we have p(μ)l(μ). Let now γΓμ. By definition we have Jμ(γ(1))<0, thus by

    Pμ(v)=NJμ(v)s(Δ)s/2v22α2D(v),vHsr(RN),

    we obtain Pμ(γ(1))<0. In addition, since D(u)=o(uHs2) as u0 and γ(t)0 as t0 in Hsr(RN), we have

    Pμ(γ(t))>0forsmallt>0.

    Thus there exists a t such that Pμ(γ(t))=0, and hence

    p(μ)Jμ(γ(t))maxt[0,1]Jμ(γ(t));

    passing to the infimum we come up with p(μ)l(μ), and hence the claim.

    Proof of Theorem 1.1. We obtain the result by matching Theorem 3.2 and Proposition 3.3.

    We pass to investigate more in details Pohozaev minima, showing that it is a general fact that they are solutions of the Eq (3.1).

    Proposition 3.4. Every Pohozaev minimum is a solution of (3.1), i.e.,

    Jμ(u)=p(μ)andPμ(u)=0

    imply

    Jμ(u)=0.

    As a consequence

    p(μ)=inf{Jμ(u)uHsr(RN){0},Pμ(u)=0,Jμ(u)=0}.

    Proof. Let u be such that Jμ(u)=p(μ) and Pμ(u)=0. In particular, considered γ(t)=u(/t), we have that Jμ(γ(t)) is negative for large values of t and its maximum value is p(μ) attained only in t=1.

    Assume by contradiction that u is not critical. Let I:=[1δ,1+δ] be such that γ(I)Kp(μ)=, and set ˉε:=p(μ)maxtIJμ(γ(t))>0. Let now U be a neighborhood of Kp(μ) verifying γ(I)U=: by the deformation lemma presented in the proof of Theorem 3.2 there exists an η:[0,1]×Hsr(RN)Hsr(RN) at level p(μ)R with properties (1o)-(6o). Define then ˜γ(t):=η(1,γ(t)) a deformed path.

    For tI we have Jμ(γ(t))<p(μ)ˉε, and thus by (2o) we gain

    Jμ(˜γ(t))=Jμ(γ(t))<p(μ)ˉε,fortI. (3.8)

    Let now tI: we have γ(t)U and Jμ(γ(t))p(μ)p(μ)+ε, thus by (4o) we obtain

    Jμ(˜γ(t))p(μ)ε. (3.9)

    Joining (3.8) and (3.9) we have

    maxt0Jμ(˜γ(t))<p(μ)=l(μ)

    which is an absurd, since after a suitable rescaling it results that ˜γΓμ, thanks to (3o).

    Remark 3.5. We point out that it is not known, even in the case of local nonlinearities [7], if

    p(μ)=inf{Jμ(u)uHsr(RN){0},Jμ(u)=0}.

    On the other hand, by assuming that every solution of (3.1) satisfies the Pohozaev identity (see e.g., [[67], Proposition 2] and [[23], Eq (6.1)]), the claim holds true.

    We show now that, under the same assumptions of Theorem 3.2, we can find a solution with constant sign.

    Proposition 3.6. Assume (f1)(f4) and that F0 on (0,+) (i.e., t0 in assumption (f4) can be chosen positive). Then there exists a positive radially symmetric solution of (3.1), which is minimum over all the positive functions on the Pohozaev set.

    Proof. Let us define

    g:=χ(0,+)f.

    We have that g still satisfies (f1)–(f4). Thus, by Theorem 3.2 there exists a solution u of

    (Δ)su+μu=(IαG(u))g(u)inRN

    where G(t):=t0g(τ)dτ. We show now that u is positive. We start observing the following: by (2.2) we have

    (Δ)s/2|u|22=C(N,s)R2N(|u(x)||u(y)|)2|xy|N+2sdxdy=C(N,s)R2N|u|2(x)+|u|2(y)2|u|(x)|u|(y)|xy|N+2sdxdyC(N,s)R2Nu2(x)+u2(y)2u(x)u(y)|xy|N+2sdxdy=C(N,s)R2N(u(x)u(y))2|xy|N+2sdxdy=(Δ)s/2u22,

    thus

    (Δ)s/2|u|2(Δ)s/2u2.

    In particular, written u=u+u, by the previous argument we have u=|u|u2Hsr(RN). Thus, chosen u as test function, we obtain

    RN(Δ)s/2u(Δ)s/2udx+μRNuudx=RN(IαG(u))g(u)udx.

    By definition of g and (2.2) we have

    CN,sRN×RN(u(x)u(y))(u(x)u(y))|xy|N+2sdxdyμRNu2dx=0. (3.10)

    Splitting the domain, we gain

    RN×RN(u(x)u(y))(u(x)u(y))|xy|N+2sdxdy={u(x)0}×{u(y)<0}(u+(x)+u(y))(u(y))|xy|N+2sdxdy{u(x)<0}×{u(y)0}(u(x)+u+(y))(u(x))|xy|N+2sdxdy{u(x)<0}×{u(y)<0}(u(x)u(y))2|xy|N+2sdxdy.

    Thus we obtain that the left-hand side of (3.10) is sum of non positive pieces, thus u0, that is u0. Hence g(u)=f(u) and G(u)=F(u), which imply that u is a (positive) solution of (3.1).

    In this section we prove some regularity results for (3.1). We split the proof of Theorem 1.2 in different steps.

    We start from the following lemma, that can be found in [[61], Lemma 3.3].

    Lemma 4.1 ([61]). Let N2 and α(0,N). Let λ[0,2] and q,r,h,k[1,+) be such that

    1+αN1h1k=λq+2λr.

    Let θ(0,2) satisfying

    min{q,r}(αN1h)<θ<max{q,r}(11h),
    min{q,r}(αN1k)<2θ<max{q,r}(11k).

    Let HLh(RN), KLk(RN) and uLq(RN)Lr(RN). Then

    RN(Iα(H|u|θ))K|u|2θdxCHhKkuλqu2λr

    for some C>0 (depending on θ).

    By a proper use of Lemma 4.1 we obtain now an estimate on the Choquard term depending on Hs-norm of the function.

    Lemma 4.2. Let N2, s(0,1) and α(0,N). Let moreover θ(αN,2αN) and H,KL2Nα(RN)+L2Nα+2s(RN). Then for every ε>0 there exists Cε,θ>0 such that

    RN(Iα(H|u|θ))K|u|2θdxε2(Δ)s/2u22+Cε,θu22

    for every uHs(RN).

    Proof. Observe that 2θ(αN,2αN) as well. We write

    H=H+HL2Nα(RN)+L2Nα+2s(RN),
    K=K+KL2Nα(RN)+L2Nα+2s(RN).

    We split RN(Iα(H|u|θ))K|u|2θdx in four pieces and choose

    q=r=2,h=k=2Nα,λ=2,
    q=2,r=2NN2s,h=2Nα,k=2Nα+2s,λ=1,
    q=2,r=2NN2s,h=2Nα+2s,k=2Nα,λ=1,
    q=r=2NN2s,h=k=2Nα+2s,λ=0,

    in Lemma 4.1, to obtain

    RN(Iα(H|u|θ))K|u|2θdxH2NαK2Nαu22+H2NαK2Nα+2su2u2NN2s++H2Nα+2sK2Nαu2u2NN2s+H2Nα+2sK2Nα+2su22NN2s.

    Recalled that 2NN2s=2s and the Sobolev embedding (2.1), we obtain

    RN(Iα(H|u|θ))K|u|2θdx(H2NαK2Nα)u22+(H2Nα+2sK2Nα+2s)(Δ)s/2u22++(H2NαK2Nα+2s+H2Nα+2sK2Nα)u2(Δ)s/2u2, (4.1)

    where denotes an inequality up to a constant. We want to show now that, since 2Nα>2Nα+2s, we can choose the decomposition of H and K such that the L2Nα+2s-pieces are arbitrary small (see [[8], Lemma 2.1]). Indeed, let

    H=H1+H2L2Nα(RN)+L2Nα+2s(RN)

    be a first decomposition. Let M>0 to be fixed, and write

    H=(H1+H2χ{|H2|M})+H2χ{|H2|>M}.

    Since H2χ{|H2|M}L2Nα+2s(RN)L(RN) and 2Nα(2Nα+2s,), we have H2χ{|H2|M}L2Nα(RN), and thus

    H:=H1+H2χ{|H2|M}L2Nα(RN),H:=H2χ{|H2|>M}L2Nα+2s(RN).

    On the other hand

    H2Nα+2s=(|H2|>M|H2|2Nα+2sdx)α+2s2N

    which can be made arbitrary small for M0. In particular we choose the decomposition so that

    (H2Nα+2sK2Nα+2s)ε2

    and thus

    C(ε):≈(H2NαK2Nα).

    In the last term of (4.1) we use the generalized Young's inequality abδ2a2+12δb2, with

    δ:=ε2(H2NαK2Nα+2s+H2Nα+2sK2Nα)1

    so that

    (H2NαK2Nα+2s+H2Nα+2sK2Nα)u2(Δ)s/2u212ε2u22+C(ε)(Δ)s/2u22.

    Merging the pieces, we have the claim.

    The following technical result can be found in [[35], Lemma 3.5].

    Lemma 4.3 ([35]). Let a,bR, r2 and k0. Set Tk:R[k,k] the truncation in k, that is

    Tk(t):={kiftk,tift(k,k),kiftk,

    and write ak:=Tk(a), bk:=Tk(b). Then

    4(r1)r2(|ak|r/2|bk|r/2)2(ab)(ak|ak|r2bk|bk|r2).

    Notice that the (optimal) Sobolev embedding tells us that Hs(RN)L2s(RN). In the following we show that u belongs to some Lr(RN) with r>2s=2NN2s; we highlight that we make no use of the Caffarelli-Silvestre s-harmonic extension method, and work directly in the fractional framework.

    Proposition 4.4. Let H,KL2Nα(RN)+L2Nα+2s(RN). Assume that uHs(RN) solves

    (Δ)su+u=(Iα(Hu))K,inRN

    in the weak sense. Then

    uLr(RN)forallr[2,Nα2NN2s).

    Moreover, for each of these r, we have

    urCru2

    with Cr>0 not depending on u.

    Proof. By Lemma 4.2 there exists λ>0 (that we can assume large) such that

    RN(Iα(H|u|))K|u|dx12(Δ)s/2u22+λ2u22. (4.2)

    Let us set

    Hn:=Hχ{|H|n},Kn:=Kχ{|K|n},fornN

    and observe that

    Hn,KnL2Nα(RN),
    HnH,KnKalmosteverywhere,asn+

    and

    |Hn||H|,|Kn||K|foreverynN. (4.3)

    We thus define the bilinear form

    an(φ,ψ):=RN(Δ)s/2φ(Δ)s/2ψdx+λRNφψdxRN(Iα(Hnφ))Knψdx

    for every φ,ψHs(RN). Since, by (4.3) and (4.2), we have

    an(φ,φ)12(Δ)s/2φ22+λ2φ2212φ2Hs(RN) (4.4)

    for each φHs(RN), we obtain that an is coercive. Set

    f:=(λ1)uHs(RN)

    we obtain by Lax-Milgram theorem that, for each nN, there exists a unique unHs(RN) solution of

    an(un,φ)=(f,φ)2,φHs(RN),

    that is

    (Δ)sun+λun(Iα(Hnun))Kn=(λ1)u,inRN (4.5)

    in the weak sense; moreover the theorem tells us that

    unHsf21/2=2(λ1)u2

    (since 1/2 appears as coercivity coefficient in (4.4)), and thus un is bounded. Hence unˉu in Hs(RN) up to a subsequence for some ˉu. This means in particular that unˉu almost everywhere pointwise.

    Thus we can pass to the limit in

    RN(Δ)s/2un(Δ)s/2φdx+λRNunφdxRN(Iα(Hnun))Knφdx=(λ1)RNuφdx;

    we need to check only the Choquard term. We first see by the continuous embedding that unˉu in Lq(RN), for q[2,2s]. Split again H=H+H, K=K+K and work separately in the four combinations; we assume to work generally with ˜H{H,H}, ˜HLβ(RN) and ˜K{K,K}, ˜KLγ(RN), where β,γ{2Nα,2Nα+2s}. Then one can easily prove that ˜Hnun˜Hˉu in Lr(RN) with 1r=1β+1q. By the continuity and linearity of the Riesz potential we have Iα(Hnun)Iα(Hˉu) in Lh(RN), where 1h=1rαn. As before, we obtain (Iα(Hnun))Kn(Iα(Hˉu))K in Lk(RN), where 1k=1γ+1h. Simple computations show that if β=γ=2Nα and q=2, then k=2; if β=2Nα, γ=2Nα+2s (or viceversa) and q=2, then k=2s; if β=γ=2Nα+2s and q=2s, then k=2s. Therefore Hs(RN)Lk(RN) and we can pass to the limit in all the four pieces, obtaining

    RN(Iα(Hnun))KnφdxRN(Iα(Hˉu))Kφdx.

    Therefore, ˉu satisfies

    (Δ)sˉu+λˉu(Iα(Hˉu))K=(λ1)u,inRN

    as well as u. But we can see this problem, similarly as before, with a Lax-Milgram formulation and obtain the uniqueness of the solution. Thus ˉu=u and hence

    unuinHs(RN),asn+

    and almost everywhere pointwise. Let now k0 and write

    un,k:=Tk(un)L2(RN)L(RN)

    where Tk is the truncation introduced in Lemma 4.3. Let r2. We have |un,k|r/2Hs(RN), by exploiting (2.2) and the fact that h(t):=(Tk(t))r/2 is a Lipschitz function with h(0)=0. By (2.2) and by Lemma 4.3 we have

    4(r1)r2RN|(Δ)s/2(|un,k|r/2)|2dx=C(N,s)R2N4(r1)r2(|un,k(x)|r/2|un,k(y)|r/2)2|xy|N+2sdxdyC(N,s)R2N(un(x)un(y))(un,k(x)|un,k(x)|r2un,k(y)|un,k(y)|r2)|xy|N+2sdxdy.

    Set

    φ:=un,k|un,k|r2

    it results that φHs(RN), since again h(t):=Tk(t)|Tk(t)|r2 is a Lipschitz function with h(0)=0. Thus we can choose it as a test function in (4.5) and obtain, by polarizing the identity (2.2),

    4(r1)r2RN|(Δ)s/2(|un,k|r/2)|2dxC(N,s)R2N(un(x)un(y))(φ(x)φ(y))|xy|N+2sdxdy=λRNunφdx+RN(Iα(Hnun))Knφdx+(λ1)RNuφdx

    and since unφ|un,k|r we gain

    4(r1)r2N|(Δ)s/2(|un,k|r/2)|2dxλN|un,k|rdx+N(Iα(Hnun))Knφdx+(λ1)Nuφdx. (4.6)

    Focus on the Choquard term on the right-hand side. We have

    RN(Iα(Hnun))Knφdx (4.7)
    RN(Iα(|Hn||un|χ{|un|k}))|Kn||un,k|r1dx+RN(Iα(|Hn||un|χ{|un|>k}))|Kn||un,k|r1dxRN(Iα(|Hn||un,k|))|Kn||un,k|r1dx+RN(Iα(|Hn||un|χ{|un|>k}))|Kn||un|r1dx(4.3)RN(Iα(|H||un,k|))|K||un,k|r1dx+RN(Iα(|Hn||un|χ{|un|>k}))|Kn||un|r1dx=(I)+(II). (4.8)

    Focus on (I). Consider r[2,2Nα), so that θ:=2r(αN,2αN). Choose moreover v:=|un,k|r/2Hs(RN) and ε2:=2(r1)r2>0. Thus, observed that if a function belongs to a sum of Lebesgue spaces then its absolute value does the same ([[3], Proposition 2.3]), by Lemma 4.2 we obtain

    (I)2(r1)r2(Δ)s/2(|un,k|r/2)22+C(r)|un,k|r/222. (4.9)

    Focus on (II). Assuming r<min{2Nα,2NN2s}, we have unLr(RN) and HnL2Nα(RN), thus

    |Hn||un|La(RN),with1a=α2N+1r

    for the Hölder inequality. Similarly

    |Kn||un|r1Lb(RN),with1b=α2N+11r.

    Thus, since 1a+1b=N+αN, we have by the Hardy-Littlewood-Sobolev inequality (see Proposition 2.1) that

    RN(Iα(|Hn||un|χ{|un|>k}))|Kn||un|r1dxC({|un|>k}||Hn||un||adx)1/a(RN||Kn||un|r1|bdx)1/b.

    With respect to k, the second factor on the right-hand side is bounded, while the first factor goes to zero thanks to the dominated convergence theorem, thus

    (II)=ok(1),ask+. (4.10)

    Joining (4.6), (4.8), (4.9), (4.10) we obtain

    2(r1)r2RN|(Δ)s/2(|un,k|r/2)|2dxλRN|un,k|rdx+C(r)RN|un,k|rdx+(λ1)RNuφdx+ok(1).

    That is, by Sobolev inequality (2.1)

    C(r)(RN|un,k|r22sdx)2/2s(C(r)λ)RN|un,k|rdx+(λ1)RN|u||un,k|r1dx+ok(1).

    Letting k+ by the monotone convergence theorem (since un,k are monotone with respect to k and un,kun pointwise) we have

    C(r)(RN|un|r22sdx)2/2s(C(r)λ)RN|un|rdx+(λ1)RN|u||un|r1dx (4.11)

    and thus unLr22s(RN). Notice that r2[1,min{Nα,NN2s}). If N2s<α we are done. Otherwise, set r1:=r, we can now repeat the argument with

    r2(2NN2s,min{2Nα,2(NN2s)2}).

    Again, if 2Nα<2(NN2s)2 we are done, otherwise we repeat the argument. Inductively, we have

    (NN2s)m+,asm+

    thus 2Nα<2(NN2s)m after a finite number of steps. For such r=rm, consider again (4.11): by the almost everywhere convergence of un to u and Fatou's lemma

    C(r)(RN|u|r22s)2/2sdxlim infnC(r)(RN|un|r22sdx)2/2slim infn((C(r)λ)RN|un|rdx+(λ1)RN|u||un|r1dx)(C(r)λ)lim supnRN|un|rdx+(λ1)lim supnRN|u||un|r1dx.

    Being un equibounded in Hs(RN) and thus in L2s(RN), by the iteration argument we have that it is equibounded also in Lr(RN); in particular, the bound is given by u2 times a constant C(r). Thus the right-hand side is a finite quantity, and we gain uLr22s(RN), which is the claim.

    The following Lemma states that IαgL(RN) whenever g lies in Lq(RN) with q in a neighborhood of Nα (in particular, it generalizes Proposition 2.1 to the case h= and rNα).

    In addition, it shows the decay at infinity of the Riesz potential, which will be useful in Section 5.

    Proposition 4.5. Assume that (f1)–(f2) hold. Let uHs(RN) be a solution of (3.1). Then uLq(RN) for q[2,Nα2NN2s), and

    IαF(u)C0(RN),

    that is, continuous and zero at infinity. In particular,

    IαF(u)L(RN)

    and

    (IαF(u))(x)0as|x|+.

    Proof. We first check to be in the assumptions of Proposition 4.4. Indeed, by (f1)–(f2) and the fact that uHs(RN)L2(RN)L2s(RN) we obtain that

    H:=F(u)u,K:=f(u)

    lie in L2Nα(RN)+L2Nα+2s(RN), since bounded by functions in this sum space (see e.g., [[3], Proposition 2.3]). Now by Proposition 4.4 we have uLq(RN) for q[2,Nα2NN2s).

    To gain the information on the convolution, we want to use Young's Theorem, which states that if g,h belong to two Lebesgue spaces with conjugate (finite) indexes, then ghC0(RN). We first split

    IαF(u)=(IαχB1)F(u)+(IαχBc1)F(u)

    where

    IαχB1Lr1(RN),forr1[1,NNα),
    IαχBc1Lr2(RN),forr2(NNα,].

    We need to show that F(u)Lq1(RN)Lq2(RN) for some qi satisfying

    1qi+1ri=1,i=1,2

    that is

    q1q11[1,NNα),q2q21(NNα,]

    or equivalently q2<Nα<q1. Recall that

    |F(u)|C(|u|N+αN+|u|N+αN2s).

    Note that uLq(RN) for q[2,Nα2NN2s) implies

    |u|N+αN,|u|N+αN2sLq1(RN)Lq2(RN)

    for some q2<Nα<q1. Thus we have the claim.

    Once obtained the boundedness of the Choquard term, we can finally gain the boundedness of the solution.

    Proposition 4.6. Assume that (f1)–(f2) hold. Let uHs(RN) be a positive solution of (3.1). Then uL(RN).

    Proof. By Lemma 4.5 we obtain

    a:=IαF(u)L(RN).

    Thus u satisfies the following nonautonomous problem, with a local nonlinearity

    (Δ)s/2u+μu=a(x)f(u),inRN

    with a bounded. In particular

    (Δ)s/2u=g(x,u):=μu+a(x)f(u),inRN

    where

    |g(x,t)|μ|t|+Ca(|t|αN+|t|α+2sN2s).

    Set γ:=max{1,α+2sN2s}[1,2s), we thus have

    |g(x,t)|C(1+|t|γ).

    Hence we are in the assumptions of [[26], Proposition 5.1.1] and we can conclude.

    We observe that a direct proof of the boundedness for generally signed solutions, but assuming also (f5), can be found in Appendix A.

    Gained the boundedness of the solutions, we obtain also some additional regularity, which will be implemented in some bootstrap argument for the L1-summability.

    Proposition 4.7. Assume that (f1)–(f2) hold. Let uHs(RN)L(RN) be a weak solution of (3.1). Then uH2s(RN)C0,γ(RN) for any γ(0,min{1,2s}). Moreover u satisfies (3.1) almost everywhere.

    Proof. By Proposition 4.6, Proposition 4.5 and (f2) we have that uL(RN) satisfies

    (Δ)su=gL(RN)

    where g(x):=μu(x)+(IαF(u))(x)f(u(x)). We prove first that uH2s(RN). Indeed, we already know that f(u), F(u) and IαF(u) belong to L(RN). By Remark 3.1, we obtain

    f(u)L2Nα+2s(RN)L(RN),F(u)L2NN+α(RN)L(RN),
    IαF(u)L2NN2s(RN)L(RN),(IαF(u))f(u)L2(RN)L(RN).

    In particular,

    g:=(IαF(u))f(u)μuL2(RN).

    Since u is a weak solution, we have, fixed φHs(RN),

    RN(Δ)s/2u(Δ)s/2φdx=RNgφdx. (4.12)

    Since gL2(RN), we can apply Plancharel theorem and obtain

    RN|ξ|2sˆuˆφdξ=RNˆgˆφdξ. (4.13)

    Since Hs(RN)=F(Hs(RN)) and φ is arbitrary, we gain

    |ξ|2sˆu=ˆgL2(RN).

    By definition, we obtain uH2s(RN), which concludes the proof. Observe moreover that F1((1+|ξ|2s)ˆu)=u+gL2(RN)L(RN), thus by definition uH2s(RN)W2s,(RN). By the embedding [[28], Theorem 3.2] we obtain uC0,γ(RN) if 2s<1 and γ(0,2s), while uC1,γ(RN) if 2s>1 and γ(0,2s1) (see also [[68], Proposition 2.9]).

    It remains to show that u is an almost everywhere pointwise solution. Thanks to the fact that uH2s(RN), we use again (4.13), where we can apply Plancharel theorem (that is, we are integrating by parts (4.12)) and thus

    RN(Δ)suφdx=RNgφdx.

    Since φHs(RN) is arbitrary, we obtain

    (Δ)su=galmosteverywhere.

    This concludes the proof.

    We observe, by the proof, that if s(12,1), then uC1,γ(RN) for any γ(0,2s1), and u is a classical solution, with (Δ)suC(RN) and equation (3.1) satisfied pointwise.

    We end this section by dealing with the summability of u in Lebesgue spaces Lr(RN) for r<2.

    Remark 4.8. We start noticing that, if a solution u belongs to some Lq(RN) with q<2, then uL1(RN). Assume thus q(1,2) and let uLq(RN)L(RN), then we have

    f(u)LqNα(RN)L(RN),F(u)LqNN+α(RN)L(RN),
    IαF(u)LqNN+α(1q)(RN)L(RN),(IαF(u))f(u)LqNN+α(2q)(RN)L(RN).

    Thanks to Proposition 4.7, u satisfies (3.1) almost everywhere, thus we have

    F1((|ξ|2s+μ)ˆu)=(Δ)su+μu=(IαF(u))f(u)LqNN+α(2q)(RN)

    which equivalently means that the Bessel operator verifies

    F1((|ξ|2+1)sˆu)LqNN+α(2q)(RN).

    Thus by [[2], Theorem 1.2.4] we obtain that u itself lies in the same Lebesgue space, that is

    uLqNN+α(2q)(RN).

    If qNN+α(2q)<1, we mean that (IαF(u))f(u)L1(RN)L(RN), and thus uL1(RN)L(RN). We convey this when we deal with exponents less than 1.

    If q<2, then

    qNN+α(2q)<q

    and we can implement a bootstrap argument to gain uL1(RN). More precisely

    {q0[1,2)qn+1=qnNN+α(2qn)

    where qn0 (but we stop at 1). Thus, in order to implement the argument, we need to show that uLq(RN) for some q<2.

    We show now that uL1(RN). It is easy to see that, if the problem is (strictly) not lower-critical, i.e., (f2) holds together with

    limt0F(t)|t|β=0

    for some β(N+αN,N+αN2s), then uL1(RN). Indeed uHs(RN)L(RN)L2(RN)L(RN) and

    (IαF(u))f(u)Lq(RN),

    where 1q=β2α2N; noticed that q<2, we can implement the bootstrap argument of Remark 4.8.

    We will show that the same conclusion can be reached by assuming only (f2).

    Proposition 4.9. Assume that (f1)(f2) hold. Let uHs(RN)L(RN) be a weak solution of (3.1). Then uL1(RN).

    Proof of Proposition 4.9. For a given solution uHs(RN)L(RN) we set again

    H:=F(u)u,K:=f(u).

    Since uL2(RN)L(RN), by (f2) we have H, KL2Nα(RN). For nN, we set

    Hn:=Hχ{|x|n}.

    Then we have

    Hn2Nα0asn. (4.14)

    Since supp(HHn){|x|n} is a bounded set, we have for any \beta \in [1, {2N\over \alpha}]

    \begin{equation} H-H_n \in L^\beta( \mathbb{R}^N) \quad {\rm{for all}}\; n\in \mathbb{N}. \end{equation} (4.15)

    We write our equation (3.1) as

    (-\Delta)^s u+\mu u = (I_\alpha*H_nu)K +R_n \quad {\rm{in}}\;{\mathbb{R}^N},

    where we introduced the function R_n by

    R_n: = (I_\alpha*(H-H_n)u)K.

    Now we consider the following linear equation:

    \begin{equation} (-\Delta)^s v+\mu v = (I_\alpha*H_nv)K +R_n \quad {\rm{in}}\;{\mathbb{R}^N}. \end{equation} (4.16)

    We have the following facts:

    (i) The given solution u solves (4.16).

    (ii) By the property (4.15) with \beta \in (\frac{2N}{N+\alpha}, {2N\over \alpha}) , there exists q_1 \in (1, 2) , namely {1\over q_1} = {1\over\beta}+{{1\over 2}}-{\alpha\over 2N} , such that R_n\in L^{q_1}(\mathbb{R}^N)\cap L^2(\mathbb{R}^N) .

    (iii) By the property (4.14), for any r\in (\frac{2N}{2N-\alpha}, 2] \subset (1, 2]

    v\in L^r( \mathbb{R}^N) \mapsto A_n(v): = (I_\alpha*H_n v)K\in L^r( \mathbb{R}^N)

    is well-defined and verifies

    \begin{equation} \|{A_n(v)}\|_r \leq C_{r, n}\|v_r\|. \end{equation} (4.17)

    Here C_{r, n} satisfies C_{r, n}\to 0 as n\to\infty .

    We show only (iii). Since v\in L^r(\mathbb{R}^N) , by Hardy-Littlewood-Sobolev inequality and Hölder inequality we obtain

    \|{A_n(v)}\|_r \leq C_r\|{H_n}\|_{2N\over \alpha} \|K_\|{2N\over \alpha} \|v_r\|,

    where C_r > 0 is independent of n , v . Thus by (4.14) we have C_{r, n}: = C_r\|{H_n}\|_{2N\over \alpha}\|K_\|{2N\over \alpha}\to 0 as n\to\infty .

    Now we show u\in L^{q_1}(\mathbb{R}^N) , where q_1\in (1, 2) is given in (ii). Since ((-\Delta)^s+\mu)^{-1}:\, L^r(\mathbb{R}^N)\to L^r(\mathbb{R}^N) is a bounded linear operator for r\in(1, 2] (see [[2], Theorem 1.2.4]), (4.16) can be rewritten as

    v = T_n(v),

    where

    T_n(v): = ((-\Delta)^s+\mu)^{-1}\big(A_n(v)+R_n\big).

    By choosing \beta \in (2, \frac{2N}{\alpha}) we have q_1 \in (\frac{2N}{2N-\alpha}, 2)\subset(1, 2) , thus we observe that for n large, T_n is a contraction in L^2(\mathbb{R}^N) and in L^{q_1}(\mathbb{R}^N) . We fix such an n .

    Since T_n is a contraction in L^2(\mathbb{R}^N) , we can see that u\in H^s(\mathbb{R}^N) is a unique fixed point of T_n . In particular, we have

    u = \lim\limits_{k\to \infty} T_n^k (0) \quad {\rm{in}}\; \ L^2( \mathbb{R}^N).

    On the the other hand, since T_n is a contraction in L^{q_1}(\mathbb{R}^N) , (T_n^k (0))_{k = 1}^\infty also converges in L^{q_1}(\mathbb{R}^N) . Thus the limit u belongs to L^{q_1}(\mathbb{R}^N) .

    Since q_1 < 2 we can use the bootstrap argument of Remark 4.8 to get u\in L^1(\mathbb{R}^N) , and reach the claim.

    We prove now the polynomial decay of the solutions. We start from two standard lemmas, whose proofs can be found for instance in [[15], Lemma A.1 and Lemma A.3].

    Lemma 5.1 (Maximum Principle). Let \Sigma \subset \mathbb{R}^N , possibly unbounded, and let u\in H^s(\mathbb{R}^N) be a weak subsolution of

    (-\Delta)^s u + a u \leq 0 \quad {\rm{in}}\;\mathbb{R}^N\setminus \Sigma

    with a > 0 , in the sense that

    \int_{ \mathbb{R}^N} (-\Delta)^{s/2} u \, (-\Delta)^{s/2} \varphi \, dx+ a \int_{ \mathbb{R}^N} u \varphi \, dx\leq 0

    for every positive \varphi \in H^s(\mathbb{R}^N) with {\mathop{{\rm{supp}}}}(\varphi) \subset \mathbb{R}^N \setminus \Sigma . Assume moreover that

    u\leq 0, \quad for\;a.e.\;x \in \Sigma .

    Then

    \begin{equation} u\leq 0, \quad for\;a.e.\;x\;\in \mathbb{R}^N . \end{equation} (5.1)

    Lemma 5.2 (Comparison function). Let b > 0 . Then there exists a strictly positive continuous function W\in H^s(\mathbb{R}^N) such that, for some positive constants C', C'' (depending on b ), it verifies

    (-\Delta)^s W + b W = 0 \quad {\rm{in}}\;\mathbb{R}^N\setminus B_{r}

    pointwise, with r: = b^{-1/2s} , and

    \begin{equation} \frac{C'}{|x|^{N+2s}} < W(x) < \frac{C''}{|x|^{N+2s}}, \quad for\;|x| > 2 r . \end{equation} (5.2)

    We show first some conditions which imply the decay at infinity of the solutions.

    Lemma 5.3. Assume that \text{(f1)–(f2)} hold. Let u be a weak solution of (3.1). Assume

    u \in L^{\frac{N}{2s}}( \mathbb{R}^N)\cap L^{\infty}( \mathbb{R}^N)

    and

    (I_{\alpha}*F(u))f(u)\in L^{\frac{N}{2s}}( \mathbb{R}^N)\cap L^{\infty}( \mathbb{R}^N).

    Then we have

    \begin{equation} u(x) \to 0 \quad {\rm{as}}\;|x|\to +\infty. \end{equation} (5.3)

    Proof. Being u solution of

    (-\Delta)^s u + u = (1-\mu) u + \big(I_{\alpha}*F(u)\big) f(u) = : \chi \quad {\rm{in}}\;{\mathbb{R}^N},

    where \chi \in L^{\frac{N}{2s}}(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N) , we have the representation formula

    u = \mathcal{{K}} * \chi

    where \mathcal{{K}} is the Bessel kernel; we recall that \mathcal{{K}} is positive, it satisfies \mathcal{{K}}(x) \leq \frac{C}{|x|^{N+2s}} for |x| \geq 1 and \mathcal{{K}} \in L^q(\mathbb{R}^N) for q \in [1, 1 + \tfrac{2s}{N-2s}) (see [[28], page 1241 and Theorem 3.3]). Let us fix \eta > 0 ; we have, for x \in \mathbb{R}^N ,

    \begin{align*} u(x) = & \int_{ \mathbb{R}^N} \mathcal{{K}}(x-y) \chi(y) dy \\ = & \int_{|x-y|\geq 1/\eta} \mathcal{{K}}(x-y) \chi(y)dy +\int_{|x-y| < 1/\eta} \mathcal{{K}}(x-y) \chi(y)dy. \end{align*}

    As regards the first piece

    \int_{|x-y|\geq 1/\eta} \mathcal{{K}}(x-y) \chi(y)dy \leq \|{\chi}\|_{\infty} \int_{|x-y|\geq 1/\eta} \frac{C}{|x-y|^{N+2s}} dy \leq C \eta^{2s}

    while for the second piece, fixed a whatever q \in (1, 1 + \tfrac{2s}{N-2s}) and its conjugate exponent q' > \frac{N}{2s} , we have by Hölder inequality

    \int_{|x-y| < 1/\eta} \mathcal{{K}}(x-y) \chi(y)dy \leq \|{\mathcal{{K}}}\|_q \|{\chi}\|_{L^{q'}(B_{1/\eta}(x))}\\

    where the second factor can be made small for |x| \gg 0 . Joining the pieces, we have (5.3).

    We observe that the assumptions of the Lemma are fulfilled by assuming that u is bounded thanks to Proposition 4.9. We are now ready to prove the polynomial decay of the solutions.

    Conclusion of the proof of Theorem 1.3. Observe that, by (f5) and Lemma 5.3, we have

    \begin{equation} \frac{f(u)}{u} \in L^{\infty}( \mathbb{R}^N). \end{equation} (5.4)

    Thus we obtain, by applying Proposition 4.5, that

    \begin{equation} (I_{\alpha}*F(u))(x) \frac{f(u(x))}{u(x)} \to 0 \quad {\rm{as}}\; |x| \to + \infty . \end{equation} (5.5)

    Thus, by (5.5) and the positivity of u , we have for some R'\gg 0

    (-\Delta)^s u + \tfrac{1}{2} \mu u = (I_{\alpha}*F(u))f(u) - \tfrac{1}{2} \mu u = \left( (I_{\alpha}*F(u))\tfrac{f(u)}{u} - \tfrac{1}{2} \mu \right) u \leq 0 \quad {\rm{in}}\; \mathbb{R}^N\setminus B_{R'} .

    Similarly

    (-\Delta)^s u + \tfrac{3}{2} \mu u = (I_{\alpha}*F(u))f(u) + \tfrac{1}{2} \mu u = \left( (I_{\alpha}*F(u))\tfrac{f(u)}{u} + \tfrac{1}{2} \mu \right) u \geq 0 \quad {\rm{in}}\; \mathbb{R}^N\setminus B_{R'} .

    Notice that we always intend differential inequalities in the weak sense, that is tested with functions in H^s(\mathbb{R}^N) with supports contained in the reference domain (e.g., \mathbb{R}^N \setminus B_{R'} ).

    In addition, by Lemma 5.2 we have that there exist two positive functions \underline{W}' , \overline{W}' and three positive constants R'' , C' and C'' depending only on \mu , such that

    \left\{ \begin{aligned} (-\Delta)^s \underline{W}' + \frac{3}{2}\mu \, \underline{W}' = 0 \quad {\rm{in}}\; \mathbb{R}^N \setminus B_{R''} , \\ \frac{C'}{|x|^{N+2s}} < \underline{W}' (x), \quad {\rm{ for }}\;|x| > 2R'' .\end{aligned}\right.

    and

    \left\{ \begin{aligned} (-\Delta)^s \overline{W}' + \frac{1}{2}\mu \, \overline{W}' = 0 \quad {\rm{in}}\; \mathbb{R}^N \setminus B_{R''} , \\ \overline{W}'(x) < \frac{C''}{|x|^{N+2s}}, \quad {\rm{ for}} \;|x| > 2R'' .\end{aligned}\right.

    Set R: = \max\{ R', 2R''\} . Let \underline{C}_1 and \overline{C}_1 be some lower and upper bounds for u on B_R , \underline{C}_2: = \min_{B_R} \overline{W}' and \overline{C}_2: = \max_{B_R} \underline{W}' , all strictly positive. Define

    \underline{W}: = \underline{C}_1 \overline{C}_2 ^{-1} \underline{W}', \quad \overline{W}: = \overline{C}_1 \underline{C}_2^{-1} \overline{W}'

    so that

    \underline{W}(x)\leq u(x) \leq \overline{W}(x), \quad {\rm{ for}}\; |x|\leq R .

    Thanks to the comparison principle in Lemma 5.1, and redefining C' and C'' , we obtain

    \frac{C'}{|x|^{N+2s}} < \underline{W}(x) \leq u(x) \leq \overline{W}(x) < \frac{C''}{|x|^{N+2s}}, \quad \rm{ for |x| > R }.

    By the boundedness of u , we obtain the claim.

    We see that, for non sublinear f (that is, (f5)), the decay is essentially given by the fractional operator. It is important to remark that, contrary to the limiting local case s = 1 (see [60]), the Choquard term in case of linear f does not affect the decay of the solution.

    Remark 5.4. We observe that the conclusion of the proof of Theorem 1.3 can be substituted by exploiting a result in [30]. Indeed write V: = -(I_{\alpha}*F(u)) \frac{f(u)}{u} , which is bounded and zero at infinity as observed in (5.4)–(5.5), and gain

    (-\Delta)^s u + V(x) u = - \mu u \quad {\rm{in}}\;{\mathbb{R}^N}.

    Up to dividing for \|{u}\|_2 , we may assume \|{u}\|_2 = 1 . Thus we are in the assumptions of [[30], Lemma C.2] and obtain, even for changing-sign solutions of (3.1),

    |u(x)| \leq \frac{C_1}{(1 + |x|^2)^{\frac{N+2s}{2}}}

    together with

    |u(x)| = \frac{C_2}{|x|^{N+2s}} + o\left( \frac{1}{|x|^{N+2s}}\right) \quad {\rm{as}}\; |x| \to +\infty

    for some C_1, C_2 > 0 .

    The first and second authors are supported by PRIN 2017JPCAPN "Qualitative and quantitative aspects of nonlinear PDEs'' and by INdAM-GNAMPA. The third author is supported in part by Grant-in-Aid for Scientific Research (19H00644, 18KK0073, 17H02855, 16K13771) of Japan Society for the Promotion of Science.

    All authors declare no conflicts of interest in this paper.



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