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On fractional Choquard equations

We investigate a class of nonlinear Schrodinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.


Introduction
Given ω > 0, N ≥ 3, α ∈ (0, N ), p > 1 and s ∈ (0, 1), we consider the nonlocal problem where K α (x) = |x| α−N and the Hilbert space H s (R N ) is defined as with scalar product and norm given by The fractional Laplacian operator (−∆) s is defined by where C(N, s) is a suitable normalization constant. Thus, problem (P ω ) presents nonlocal characteristics in the nonlinearity as well as in the (fractional) diffusion. We point out that when s = 1, p = 2 and α = 2, then (P ω ) boils down to the so-called Choquard or nonlinear Schrödinger-Newton equation (1.1) − ∆u + ωu = (K 2 * u 2 )u, u ∈ H 1 (R N ).
This equation was elaborated by Pekar [22] in the framework of quantum mechanics. Subsequently, Choquard [15] adopted the equation as an approximation of the Hartree-Fock theory. More recently, Penrose [23] settled it as a model of self-gravitating matter. The first investigations for existence and symmetry of the solutions to (1.1) go back to the works of Lieb and Lions [15,17]. On this basis, we will refer to (P ω ) as to the generalized nonlinear Choquard equation. In the last few years, the study of equations involving pseudodifferential operators has steadily grown. In [18,19] the authors discuss recent developments in the description of anomalous diffusion via fractional dynamics and various fractional equations are derived asymptotically from Lévy random walk models, extending Brownian walk models in a natural way. In particular, in [14], a fractional Schrödinger equation with local power type nonlinearity was studied. This extends to a Lévy framework the classical statement that path integral over Brownian trajectories leads to the standard Schrödinger equation −∆u + ωu = f (u), see e.g. [6] and references therein. In the case s = 1/2, problem (P ω ) has been used to model the dynamics of pseudo-relativistic boson stars, see [12]. We shall say that u ∈ H s (R N ) is a weak solution of (P ω ) if (−∆) s/2 u (−∆) s/2 v + ω uv = (K α * |u| p )|u| p−2 uv, for all v ∈ H s (R N ). Let and introduce the Nehari manifold N ω := u ∈ H s (R N ) \ {0} : (−∆) s/2 u 2 2 + ω u 2 2 − (K α * |u| p )|u| p = 0 , and the C 1 functional E ω : H s (R N ) → R defined by A ground state of (P ω ) is a solution with minimal energy E ω and can be characterized as The main result of the paper is the following.
Under some restrictions on the values of p, there exist different ways of obtaining ground state solutions, via minimization problems which turn out to be equivalent up to a suitable change of scale, as shown in Propositions 4.1 and 4.6. In particular, in the range (1.4) 1 + α N < p < 1 + 2s + α N the ground states can be found by minimizing the functional on L 2 -spheres, which allows to obtain the additional information about the Morse index of solutions. The information provided in Proposition 4.6 is also useful when studying the orbital stability property of the family of ground states for the equation This topic was recently investigated in [28] in the case p = 2 and with α ∈ (N − 2s, N ), see the introduction therein for the physical motivations. We plan to investigate (1.6) -in presence of a parameter ε of singular perturbation -from the point of view of soliton dynamics by following an approach used in [4] to study the local case s = 1 and motivated by the absence of general results about the nondegeneracy of ground states. We point out that, contrary to the local case s = 1, the solutions can only decay at the polynomial rate |x| −N −2s . We refer the reader to [21] for sharp results about the exponential decay of ground state solutions in the case s = 1.
Moreover, we have the following multiplicity result. Next, we have the following nonexistence result.
As a consequence, the range of p detected in (1.2) is optimal for the existence of nontrivial solutions. The first complete study of Pohožaev identities and nonexistence results in star-shaped bounded domains for equations involving the fractional Laplacian and a local nonlinearity was done in [24,25]. Then, more recently, for fractional equations set on the whole R N , in [7], the authors obtained a Pohožaev identity for power type nonlinearities. Theorem 1.3 is based upon Pohožaev identity (6.1) which is obtained, as in [7], by the localization procedure due to Caffarelli and Silvestre [5].
Next, we denote byḢ s (R N ) the completion of C ∞ c (R N ) with respect to the seminorm (−∆) s/2 · 2 , known as Gagliardo seminorm, and consider the problem We have the following result.
Theorem 1.4. The following assertions hold: Then the problem writes as and any of its solutions of fixed sign have the form for some x 0 ∈ R N , C > 0 and t > 0.
The classification of the solutions to problem (1.7) is reminiscent of that for the fixed-sign solutions to In [8] the authors proved that any positive to this problem has the form of (1.8).
The plan of the paper is as follows.
In Section 2, we collect some preliminary notions and results. In Section 3 we investigate the Hölder regularity and the asymptotic behaviour of weak solutions. In Section 4 we prove the existence of least energy solutions (ground states) determining equivalent ways of characterizing them. Here we also get their symmetry and monotonicity properties and we investigate the Morse index of ground states in the particular ranges 2 ≤ p < 1 + (2s + α)/N and s ≥ 1/2. In Section 5 we get the existence of infinitely many solutions, symmetric under the action of some group. In Section 6, we obtain a general Pohǒzaev identity and we prove Theorem 1.4. In the paper, C will always denote a generic constant which may vary from line to line. Unless expressly specified, the integral are meant to be extended to R N .

Preliminaries
First of all, let us recall the following properties which follow from the fractional Sobolev embedding the Hardy-Littlewood inequality and the fractional version of the Gagliardo-Niremberg inequality In particular (2.5) defines a function r = r(q) which is strictly increasing and maps Proof. Property (i) is trivial. In order to prove (2.5), let q be as in (2.4) and u ∈ L q (R N ). Using Hardy-Littlewood-Sobolev inequality we have that Since q < N p/α, then t > 0. Moreover, since p > 1, then N + α and so t > 1. Hence, since for p > 1, by using Hölder inequality we get (2.5). Finally (iii) easily follows from by (2.3) and (2.1).
The next result is an adaptation of a classical lemma of Lions and it is crucial in the proofs of the existence theorems.
Proof. If q = 2 it is obvious. Let now q ∈ (2, 2 * s ]. Since r := N (q − 2)/2s ≤ q, for a.e. x ∈ R N , by [10, Theorem 6.7], we have where u 2 H s (B1(x)) is defined in [10,Equation (2.2)]. Hence, we cover R N with balls of radius 1 in such a way that each point of R N is contained in at most N + 1 balls. This procedure works even if in the H s (B 1 (x))-norm there is a nonlocal term (the Gagliardo seminorm) and so we conclude.

Regularity and asymptotics
In this section we want to show that any H s (R N )-solution of (P ω ) is indeed regular as well as the asymptotic profile. Let us recall the definition of the fractional Sobolev spaces for q ≥ 1 and β ≥ 0: (see [27] for more details) and the following results [11, Theorem 3.2].
Theorem 3.1. We have: We prove the following be a solution of (P ω ). Then for every q ≥ 1 such that we have that u ∈ L q (R N ). Moreover, for every r > 1 such that Proof. Let us consider Thus, since the Bessel operator preserves the Lebesgue spaces (see [27]) and by (3.1) we have that u ∈ W 2s,r0 . Then, by Sobolev embedding in (i) of Theorem 3.1, u ∈ L q (R N ) for every q ∈ [r 0 , (r 0 ) * 2s ], i.e. for every q such that Hence let us define It is easy to see that q 0 ∈ [q 1 , q 1 [. Moreover, since for every q ∈ [q 1 , q 1 [ we have u ∈ L q (R N ), then (K α * |u| p )|u| p−2 u ∈ L r (R N ) and so u ∈ W 2s,r for every r ∈ [r(q 1 ), r(q 1 )[, where the map r = r(q) has been defined in (ii) of Lemma 2.1. Hence by Sobolev embeddings and again by (ii) of Lemma 2.1, u ∈ L q (R N ) for every q ∈ [r(q 1 ), (r(q 1 )) * 2s [. If r(q 1 ) = 1, namely q 1 = N (2p − 1)/(N + α), we stop here from the left hand side of the interval of q's. Analogously, if 1/(r(q 1 )) * 2s = α/N (1 − 1/p) − 2s/N , namely q 1 = N p/α we stop here from the right hand side of the interval of q's. Otherwise we iterate the procedure. We take We have that Indeed, by induction, if we assume that q i < q i−1 then We can conclude this procedure after a finite number of steps; indeed, Lemma 3.4. For every r > 1, the solution u of (P ω ) is in W 2s,r .
The proof of the regularity in Theorem 1.1 is thereby completed. We note also the following result on the summability property of the fixed sign solutions which we will need in studying the Morse index. In this context we need the functional to be C 2 , and this is achieved for p ≥ 2.
Before to proceed with the proof, we show the following general fact.
, it is possible to choose a small positive ε in such a way that 1 B1 K α ∈ L 1+ε (R N ) and |u| p ∈ L 1+1/ε (R N ) and we conclude that Here C 0 (R N ) the space of continuous functions vanishing at infinity. Analogously, we can choose a small positive ε such that |u| p ∈ L 1+ε (R N ) and 1 B c . Hence, by the usual properties of the convolution, K α * u p is C 1 with derivatives given by and then The proof is thereby complete. Remark 3.7. Under the hypotheses of Proposition 3.5, we have u ∈ C 2 (R N ). Indeed by Theorem 3.2 and (3.4) we know that u ∈ C 1,µ and, by [26, Proposition 2.1.11], we conclude that ∂ i u ∈ C 1 (R N ).
We conclude this section by showing the asymptotic profile of the solutions of (P ω ). For the sake of simplicity we set V := −(K α * |u| p )|u| p−2 .
We get the following Theorem 3.8. Let p ≥ 2 and u be a solution of (P ω ). Then there exist two positive constants C 1 , C 2 such that, for any x ∈ R N , Proof. For a solution u of (P ω ), we have By Lemma 3.6 we have V ∈ L ∞ (R N ) and V (x) → 0 for |x| → ∞. Then, for every τ ∈ (0, 1) there exists R > 0 such that V (x) ≥ −τ , whenever |x| ≥ R. Then, we are in a position to apply [13, Lemma C.2] to obtain the conclusion.
As it can be seen in [13,Lemma C.2], the constants C 1 , C 2 depend on the solutions by their L 2 −norm. The decay estimate in Theorem 1.1 is proved.
By arguing as in [11, end of Section 3], if u(x 0 ) = 0 for some x 0 ∈ R N , then one obtains yielding u = 0, a contradiction. Whence |u| > 0 and u does not change sign. We shall assume u > 0. Given where σ H (x) is the reflected of x with respect to ∂H. Then, v H 2 2 = v 2 2 and, by (2.2) and [1, In turn, since S(u) ≤ S(u H ), we conclude that Then, by combining [21, Lemma 5.3 and Lemma 5.4], we conclude the proof.
As we said in the Introduction, here we are particularly interested into the precompactness properties of the minimizing sequences of E 0 on Σ ρ . In this case we have to assume that p satisfies (1.4). Indeed, if p ≥ 1 + 2s+α N , using the same rescaling τ N/2 u τ as in Proposition 4.1, we deduce that E 0 is unbounded from below. In the following lemma we collect some basic facts. (ii) m ρ 2 := inf u∈Σρ E 0 (u) < 0; (iii) every minimizing sequence for E 0 in Σ ρ is bounded and can be assumed non-negative, radially symmetric and decreasing.
Hence we have the following compactness result. |u n | 2 ≥ δ.
We infer that there exists {y n } ⊂ R N such that BR(yn) |u n | 2 ≥ δ.
Finally, following step by step [9, Proof of Lemma 2.6], we get the following relations between the ground states (as minima of E ω on N ω ) and the minima of E 0 on Σ ρ . are equivalent. Moreover the L 2 -norm ρ of any ground state u of (P ω ) satisfies and min u∈Σρ E ω (u) = min u∈Nω E ω (u).
We have that w solves .
We have that u ∈ Σ ρ , (4.7) holds for By Pohozaev identity (6.1) and since w ∈ N ω and E ω (w) = c we get the system Thus (4.10) becomes and, for Hence the conclusions easily follow.
Finally we study the Morse index of the ground state. In the last part of this Section we assume 2 ≤ p < 1 + (2s + α)/N to have that the functional E ω is C 2 and s > 1/2. If u is the minimum of E 0 on Σ ρ we have with λ > 0 (by (4.9)). Now consider To obtain information on the Morse index, we need to study ker E ′′ λ (u). Since the problem is invariant for the group of translations, the solutions of (P ω ) will never be isolated: in other words ker E ′′ λ (u) = {0} and in particular (4.14) span{∇u} ⊂ ker E ′′ λ (u). Indeed, for every a ∈ R N , consider the action of the group of the translations in R N induced on H s (R N ), that is which is linear and isometric. Since , for every u, v ∈ H s (R N ). For every u ∈ H s (R N ) it is also convenient to introduce the following map s u : a ∈ R N −→ u(· + a) ∈ H s (R N ).
Of course, for a generic fixed u ∈ H s (R N ), the map s u does not need to be differentiable but (for example) whenever u ∈ H s (R N ) is a solution of (P ω ) as in Proposition 3.5 it does, and the differential in 0 given by Hence, in this case, by differentiating in 0 the map , ·] = 0 for all b ∈ R N and this gives (4.14). It would be interesting to understand if the ground state is nondegenerate in the sense that span{∇u} = ker E ′′ λ (u). We define the Morse index i Morse (u) as the maximal dimension of subspaces of H s (R N ) on which E ′′ λ (u) is negative definite. We have the following result which completes the proof of Theorem 1.1.

Multiplicity
We begin with some geometric properties of the functional E ω in (1.3). The assumption (1.2) will be tacitly assumed in the whole section.
. To ensure existence of critical points of E ω , a compactness condition is necessary. To this aim some preliminaries are in order.
In particular for ℓ = 1 we have the radial functions, u(x) = u(|x|). In this we say that the functions in X are "symmetric". Then X is exactly the closed and infinite dimensional subspace of fixed points for the action (5.1). The importance of this setting is twofold. Indeed the functional E ω is G−invariant, i.e. for every g ∈ G, E ω • T g = E ω and the space X has compact embedding into L q (R N ), q ∈ (2, 2 * s ), see [16].
Secondly, for every fixed u ∈ H s (R N ), consider the problem (where Γ is the gamma function) whose weak formulation is the following one: we say that ϕ ∈Ḣ α/2 (R N ) is a weak solution if for every ξ ∈Ḣ α/2 (R N ) Recall that for every α ∈ (0, N ), (−∆) α/2 u is defined via the Fourier transform andḢ α/2 (R N ) is defined as the completion of C ∞ c (R N ) with respect to the associated Gagliardo seminorm (these notions coincide with that given in the Introduction for α ∈ (0, 2)). Observe now that, under the assumption on p, the right hand side in (5.3) defines the map which is linear and continuous; indeed . By the Riesz Representation Theorem there exists a unique weak solution ϕ of (5.2), represented as a convolution with the kernel K α /γ(α), i.e. ϕ = K α * |u| p and K α * |u| p Ḣα/2 = L ≤ C u p . As a consequence of the above setting we can prove the following result, which will help us to recover compactness.
Proof. Define the linear and continuous maps L n , L :Ḣ α/2 (R N ) → R such that By the compact embedding we may assume u n − u q → 0, for all q ∈ (2, 2 * s ). Then, denoting with (2 * α/2 ) ′ the conjugate exponent of 2 * α/2 , , which proves the convergence of L n to L in the operator norm, yielding (i).
We now observe that Hence by Young Inequality we have as well as The Dominated Convergence Theorem allows to obtain (ii) and (iii).
Theorem 5.3. The functional E ω satisfies the Palais-Smale condition in X.
Proof. Let {u n } ⊂ X be a Palais-Smale sequence, that is, . Then we deduce in a standard way the boundedness of {u n } in H s (R N ). Hence, there exists u ∈ X such that, up to subsequences, u n ⇀ u in H s (R N ). By Lemma 5.2 we have the convergences from which we deduce that u n → u . This gives the desired conclusion.

It follows
Theorem 5.4. The functional E ω possesses infinitely many critical points {u n } ⊂ X such that E ω (u n ) → ∞, and u n → ∞. In paticular, problem (P ω ) has infinitely many solutions in X.
Proof. All the hypotheses (geometry and compactness) of the Symmetric Mountain Pass Theorem on the space X are satisfied, so that the existence of infinitely many critical points {u n } ⊂ X with E ω (u n ) → ∞ is guaranteed. Then, since (K α * |u n |)|u n | p ≤ C u n 2p , it has to be u n → ∞. By the Palais Principle of Symmetric Criticality, the constrained critical points {u n } ⊂ X for E ω are indeed "true" critical points and hence solutions of (P ω ).
Observe that Proposition 5.1 holds also in the limit cases p = 1 + α/N and p = (N + α)/(N − 2s). Due to the nonexistence result (see Section 6), we see that the Palais-Smale condition cannot be satisfied for these values.
To obtain nonradial solutions we need a slight modification in the above setting, as introduced in [3]. Let N = 4 or N ≥ 6 and choose an integer m = (N − 1)/2 such that 2 ≤ m ≤ N/2. Let us define whose induced action on H s (R N ) is as usual We know that X, associated to the action (5.4), has compact embedding into L q (R N ), q ∈ (2, 2 * s ). Consider the involution in R N τ · x = (x 2 , x 1 , x 3 ) and the action (Iu)(x) = u(x) , . Define also the group via the group homomorphism ψ : H → Aut(G) given by ψ(ι H )g = g, ψ(τ )g = g −1 , g ∈ G.
Moreover, if π : K → {+1, −1} such that π(g, ι H ) = 1, π(g, τ ) = −1 denotes the canonical epimorphism, we define the action of K on H s (R N ) by Of course, this action is linear and isometric and in particular if, k = (g, ι H ) then ( and note that the unique radial function in Y is u ≡ 0. Since E ω is K−invariant and Y ⊂ X is closed and infinite dimensional, we can argue as before obtaining the following multiplicity result. Theorem 5.5. Assume N = 4 or N ≥ 6. The functional E ω possesses infinitely many critical points {u n } ⊂ Y such that E ω (u n ) → ∞ and u n → ∞. In particular, problem (P ω ) has infinitely many solutions in Y .
Hence the proof of Theorem 1.2 is completed.

Nonexistence
As known, in order to formally deduce a Pohožaev identity, one can compute where γ u (ϑ) := u(ϑx) and u is a solution to problem (P ω ). We find We shall rigorously justify this identity. We follow the localization argument developed in [7] by defining the space X s (R N +1  Therefore, our nonlocal equation (P ω ) can be restated into a local form as −div(y 1−2s ∇w) = 0 on R N +1 + ∂ s ν w(x, 0) = −ωu + (K α * |u| p )|u| p−2 u on R N , where we have set ∂ s ν w(x, 0) := − 1 ̟ s lim y→0 + y 1−2s ∂ y w(x, y). Without loss of generality, we shall set ̟ s = 1. Of course, if w is a weak solution to this problem, then u(x) = w(x, 0) is a weak solution to (P ω ).
We have the following result. Theorem 6.1. Let u ∈ C 2 (R N ) ∩ H s (R N ) ∩ L 2N p N +α (R N ) be a weak solution to (P ω ). Then (6.1) holds.
Taking into account Remark 3.7, it is not restrictive to assume u ∈ C 2 (R N ).
Moreover by the Dominated Convergence Theorem we have