ON FINITE ENERGY SOLUTIONS OF FRACTIONAL ORDER EQUATIONS OF THE CHOQUARD TYPE

. Finite energy solutions are the important class of solutions of the Choquard equation. This paper is concerned with the regularity of weak ﬁnite energy solutions. For nonlocal fractional-order equations, an integral system involving the Riesz potential and the Bessel potential plays a key role. Applying the regularity lifting lemma to this integral system, we can see that some weak integrable solution has the better regularity properties. In addition, we also show the relation between such an integrable solution and the ﬁnite energy solution. Based on these results, we prove that the weak ﬁnite energy solution is also the classical solution under some conditions. Finally, we point out that the least energy with the critical exponent can be represented by the sharp constant of some inequality of Sobolev type though the ground state solution cannot be found.

1. Introduction. In this paper, we consider the nonlocal static Schrödinger equation (id − ∆) α/2 u = u p−1 ( 1 |x| γ * u p ), u > 0 in R n (1.1) and the corresponding energy functional Here n ≥ 3, α, γ ∈ (0, n), p > 1 and id is the identity operator. We are concerned with the finite energy solutions of (1.1) and the ground state energy with some critical exponent.
Eq  [32]). In general, such a problem is often referred to as Choquard-Pekar equation, which is widely used as the self-consistent field approximation for the same physical system. In particular, it comes up with an approximation to Hartree-Fock theory of a plasma or in the Hartree theory of describing a non relativistic many-boson systems. It also appears as a Hartree equation for the helium atom. The readers can see [1], [2], [8], [9], [11], [18], [27] and [36].
In 1977, Lieb [24] proved the existence and uniqueness (up to translation) of the least energy solutions of equation (1.2). Afterwards, Lions [28] showed the existence of a sequence of radially symmetric solutions of this equation. In 2010, Ma and Zhao [31] used the method of moving planes to obtain the radial symmetry of positive solutions. Furthermore, they classified the positive solutions of (1.2) without the restriction of the least energy. For (1.2), more systematical results on the existence, the smoothness and the decay estimates can be seen in [33] and [34].
In this paper, we are concerned with the relation between the classical solutions and the weak solutions. This relation is helpful to understand the well-posedness of the fractional order PDE (1.1) and the corresponding system of integral equations.
Different from the nonlocal operator of equations in [13] and [35], the fractional order differential operator in (1.1) is related to the Bessel potential. A positive function u ∈ H α/2 (R n ) is called a H α/2 -weak solution of (1.1), if for all φ ∈ C ∞ 0 (R n ), the following equality makes sense Here, the left hand side can be defined by Re R n (1 + 4π 2 |ξ| 2 ) α/2ûφ dξ. Since C ∞ 0 (R n ) is dense in H α/2 (R n ), the test function φ can be taken in H α/2 (R n ). When α = 2, (1.3) shows that u is a usual H 1 -weak solution. A positive function u ∈ L 2 (R n ) is called a L 2 -weak solution of (1.1), if for all φ ∈ C ∞ 0 (R n ), there holds R n u(id − ∆) α/2 φdx = R n u p−1 (|x| −γ * u p )φdx. (1.4) Similarly, the left hand side can still be defined by Re R n (1 + 4π 2 |ξ| 2 ) α 2ûφ dξ, and the test function can be taken in H α (R n ). Clearly, H α/2 -weak solutions are L 2 -weak solutions.
Since some solutions of (1.2) have better regularity (cf. [10] and [34]), we can consider the classical solutions. To define the classical solutions of (1.1), we assume that α(= 2m) is even, and (1.5) is locally uniformly convergent (cf. the argument in §2). Let m > 0 be an integer. A positive function u ∈ C 2m (R n ) is called a classical solution of (1.1) with α = 2m, if u satisfies (1.5), and (1.1) holds pointwise in any compact subset of R n .
The fractional order equation arises in many branches of sciences such as phase transitions, flame propagation, stratified materials and others. In particular, it can be understood as the infinitesimal generator of a stable Levy process. The readers can see [5], [6], [22] and the references therein. A useful method to study the fractional order equation (1.1) is the integral equations method, which turns a given fractional order equation into its equivalent integral equation (cf. [12] and [31]), and then various properties of the original equation can be obtained by investigating the integral equation, see [19], [20] and references therein.
Remark 1.1. In general, it is easy to verify that the C 2 (R n )-solution of (1.6) with α = 2 satisfies (1.2). On the contrary, when the PDE is linear, the solution must be unique (cf. Lemma 9.11 in [26]). This unique solution has the integral form as (1.6) by a simple calculation. For the nonlinear PDE, it is nontrivial to verify that the solution satisfies the corresponding integral equation. For the Lane-Emden-type equation, Chen, Li and Ou [7] pointed out the equivalence between the PDE and the corresponding integral equation. In addition, [4] gave a proof for the Lane-Emden system with α = 2. In §2, we prove that if u is an L 2 -weak solution of (1.1), then there exists c > 0 such that cu satisfies (1.6). First we present a necessary condition of existence of weak solutions of (1.1).
Remark 1.2. When α = 2, Theorem 2 in [34] shows the same conclusion. Applying the integral system (1.6), we can establish the following theorem, which shows that some integrable solutions have many good regularity properties. Write when n − β < γ; Theorem 1.2. Assume that (u, v) is a pair of positive solutions of system (1.6).
If u ∈ L 2n(p−1) n−γ+β (R n ) for some p ∈ I 2 and some β ∈ (0, α], then (R1) u ∈ L s (R n ) for any s ≥ 1. Remark 1.3. These integrability and the regularity are helpful to estimate the decay rates of u when |x| → ∞ (cf. [14], [33] and [34]). For other integral system, the integrability of the solutions is also essential to estimate the asymptotic rates. The readers can see [3] and [16] for the system involving the Riesz potentials, and see [29] and [38] for the system involving the Wolff potentials.
According to Theorem 1.2, we define integrable solutions. Let β be an arbitrary given number in (0, α]. A positive solution u is called an integrable solution of (1. 1) or (1.6), if u ∈ L 2n(p−1) n−γ+β (R n ). In particular, a positive weak (or classical) solution u is called an integrable solution of (1.2), if u ∈ L 2n(p−1) n−γ+β (R n ) for some β ∈ (0, 2]. We now introduce another important class of weak solutions. A positive weak solution u is called a finite energy solution of (1.2), if u ∈ L 2 (R n ) ∪ L 2 * (R n ) and u p (|x| −γ * u p ) ∈ L 1 (R n ).
The following theorem implies the reason of introduction of finite energy solutions of (1.2). It also shows that the definition of H 1 -weak solutions makes sense. Theorem 1.3. Assume u > 0 solves (1.2) in the classical sense. Then u ∈ H 1 (R n ) if and only if u is a finite energy solution. Moreover, p ∈ I 1 with α = 2 and ∇u 2 2 + u 2 2 = u p v 1 . In general, we often use ∇u 2 + u 2 < ∞ to define the finite energy solution. Theorem 1.3 shows that the definition above is still reasonable. Noting that (1.6) has no gradient term, we generalize the way of definition of finite energy solution of (1.2) to (1.6). Namely, a positive solution u is called a finite energy solution of (1.1) or (1.6), if u ∈ L 2 (R n ) ∪ L α * (R n ) and u p v ∈ L 1 (R n ). Here α * = 2n n−α . Theorem 1.2 implies that the integrable solutions have good regularity. We are concerned naturally when the weak finite energy solutions are the integrable solutions. The following result implies the relation between the integrable solutions and the finite energy solutions. Theorem 1.4. (1) If u is an L 2 -weak integrable solution or a classical integrable solution of (1.1) for some p ∈ I 1 ∩ I 2 and some β ∈ (0, α], then u is a finite energy solution. (2) If u is an L 2 -weak finite energy solution or an H α/2 -weak solution of (1.1) with p ∈ I 1 , then it is an integrable solution. Moreover, if u is a classical finite energy solution of (1.2), then it is also an integrable solution.
Remark 1.4. It is easy to see that I 1 ∩ I 2 is not empty. Theorem 1.3 shows that the classical positive solution of (1.2) is a H 1 -weak solution as long as it is a finite energy solution. Moreover, it is also a L 2 -weak solution. On the contrary, the following theorem shows that a L 2 -weak solution u is a classical solution, as long as u is an integrable solution.
Finally, we consider the ground states of the functional The minimizers are called the least energy solutions of (1.1). Clearly, they are H α/2weak solutions.
In some subcritical case, the ground states can represent the sharp constant of the Gagliardo-Nirenberg inequality (cf. [39], or Lemma 8.4.2 in [2]). Naturally, in the critical case, we expect to shed light on the relation between the ground states and the extremal functions of some Sobolev-type inequality.
Theorem 1.6. Assume α ∈ (0, n/2) and p = 2n−γ n−α , then Theorem 1.6 shows the relation between the energy functionals involving the Riesz potential and the Bessel potential in critical case.
The proofs of Theorems 1.1 and 1.2 are given in Sections 2 and 3, respectively. In Section 4, we give the proofs of Theorems 1.3, 1.4 and 1.5. Theorem 1.6 is proved in Section 5.
Since the Bessel kernel g 2m is a fundamental solution of (id − ∆) m w = 0, we get Here δ is a Dirac function at 0. This implies that u solves (1.1) with α = 2m in the classical sense.
The argument above shows that u solves (1.1) if it is a solution of (1.6). On the contrary, we have the following result.
Theorem 2.1. Assume u ∈ L 2 (R n ) is a weak solution of (1.1), then u satisfies (1.6) up to a positive multiplicative constant. In addition, if u is a classical solution of (1.2), then it also solves (1.6) with α = 2. Proof.
Step 1. When u ∈ L 2 (R n ) solves (1.1) in the weak sense, by definition, u satisfies In fact, if u is a H α/2 -weak solution, it is still true. Let g α be the Bessel kernel. For any ψ ∈ C ∞ 0 (R n ), we set Ψ(x) = (g α * ψ)(x). Clearly, Ψ ∈ H α (R n ), and Taking the test function φ = Ψ in (2.2), we get Inserting (2.3) into the left hand side of (2.4) and using the Parseval formula (cf. Theorem 5.3 in [26]), we have Exchanging the order of the variants of the right hand side of (2.4), from (2.5) we obtain . This shows that u satisfies (1.6) almost everywhere.
Step 2. Next, we consider the case that u ∈ C 2 (R n ) is a classical solution of (1.2). Fix where δ is a Dirac function at the origin. Since Lemma 9.11 in [26] shows that the Bessel kernel g 2 is the unique fundamental solution, in arbitrary compact subset of where c 0 is a positive constant. In addition, by (2.6), we get where ν is the unit outward norm vector on ∂B r . Multiplying (1.2) by φ r and integrating on B r , we have (2.10) Letting r → ∞, using the Fatou lemma, from (2.7) and (2.9), we obtain We claim that there exists a subsequence r j → ∞ such that In fact, when x ∈ B 1 and y From this result and (2.11), we see that Hence, when R → ∞, B 2R \B R u p (y)dy |y−x0| γ → 0. Thus, we can find a subsequence r j of R such that Using the Hölder inequality, we get (2.14) By (2.6.4) in [40], we have where c, C > 0 are independent of r. Combining this result with (2.8), from (2.13) and (2.14), we deduce that I j → 0 when r j → ∞. Eq. (2.12) is verified. Once (2.12) is true, letting r = r j → ∞ in (2.10) and using (2.7), we can see that u satisfies (1.6) at x 0 up to a positive multiplicative constant. Since x 0 is an arbitrary point in R n , we can complete the proof.
. Multiplying byū and using the Parseval formula, we get Therefore, the proof is complete. Theorem 2.3. If p ∈ I 1 , then (1.1) has no H α/2 (R n )-weak solution.
Proof. Assume that u is an H α/2 (R n )-weak solution of (1.1). Taking φ = u in (1.3), we have (2.15) In addition, (1.3) implies that the Gateaux derivative of the functional the Pohozaev identity leads to Combining with (2.15), we get Noting that the left hand side of (2.16) is positive and that of (2.17) is negative, we can see p ∈ I 1 easily.
(1) If u ∈ H α/2 (R n ) is a positive solution of (1.6), Proposition 2.2 shows that (2.15) still holds. In addition, (2.1) implies that u is a H α/2 -weak solution of (1.1), and hence it is also a critical point of functional E(u). By the same argument in the proof of Theorem 2.3, p ∈ I 1 .
(2) If u is an L 2 -weak solution of (1.1) satisfying u p v ∈ L 1 (R n ), by Theorem 2.1 we can find c > 0 such that cu satisfies (1.6). Combining with u p v ∈ L 1 (R n ) we know that u ∈ H α/2 (R n ) by Proposition 2.2. Therefore, cu is a positive solution of (1.6) in H α/2 (R n ). The argument in (1) implies p ∈ I 1 .
Using Lemma 2.1 in [16], we also get (u, v) ∈ L s (R n ) × L r (R n ) for all s, r satisfying (3.6).
By finite steps, we claim that u ∈ L s (R n ) for all 1 s ∈ (0, n−β n ). Otherwise, the limit lim j→∞ 1 ξj must exist and belong to (0, n−β n ). We denote the limit by L. When 1 ξj is increasing, there holds 1 Let j → ∞, then L = (n+1) −γ n(p−2) , which implies p < 2. Noting p > 2 − 2γ n+γ−β , we see L < 1 ξ0 , which contradicts with the monotonicity as is sufficiently small. When 1 ξj is decreasing, there holds 1 This lead to L = α− n(p−2) which implies p > 2. From p < 2 + 2β n−γ−β , we deduce that the value of L contradicts with L < 1 ξ0 as is sufficiently small. Finally, similar to the proof of Theorem 3.5 in [17], we also get u ∈ L s (R n ) for all s ≥ 1.
Multiplying (1.2) by uζ 2 R and integrating on D := B 3R (0) (here R > 1), we have Integrating by parts, we obtain Applying the Cauchy inequality, we get for any δ ∈ (0, 1/2). When u ∈ L 2 (R n ), we can find C > 0 which is independent of R such that When u ∈ L 2 * (R n ), (4.5) is still true. In fact, by the Hölder inequality, The claim is verified. Now, (4.2) still holds, and hence (4.1) shows ∇u 2 2 + u 2 2 = u p v 1 . Finally, according to Theorem 2.1, if u is an classical solutions of (1.2), we can find c > 0 such that cu solves (1.6). By Theorem 2.3, we know that u ∈ H 1 (R n ) implies p ∈ I 1 .
Theorem 4.2. If u is an L 2 -weak integrable solution or a classical integrable solution of (1.1) for some p ∈ I 1 ∩ I 2 and some β ∈ (0, α], then it is a finite energy solution. Proof. First, Theorem 2.1 shows that there exists c > 0 such that cu solves (1.6). Thus, Theorem 3.1 shows that u ∈ L s (R n ) for all 1 s ∈ [0, 1], which implies u ∈ L 2 (R n ) ∪ L α * (R n ). In addition, by the Hölder inequality and the integrability results, we obtain u p v 1 ≤ u p k v r0 < ∞ by taking 1 k = 1 − 1 r0 . Here r 0 is the constant in the proof of Theorem 3.1.

Theorem 4.3.
If u is an L 2 -weak finite energy solution or an H α/2 -weak solution of (1.1) with p ∈ I 1 , then it is an integrable solution. Moreover, if u is a classical finite energy solution of (1.2), then it is also an integrable solution.
If u is a classical finite energy solution of (1.2), then by Theorem 4.1, u ∈ H 1 (R n ), and hence it is also an integrable solution. Proof. First, Theorem 2.1 implies that there exists c > 0 such that cu satisfies the integral system (1.6). Since u is an integrable solution which is implied by Theorem 4.3, Theorem 3.1 shows that (R4) holds true and u ∈ L t (R n ) for all t ∈ [1, ∞]. Therefore, Here 0 < k < n n−γ and 1 k = 1 − 1 k . Eq. (1.5) is verified.
On the other hand, the radial function is an extremal function of the following Hardy-Littlewood-Sobolev inequality where u is an arbitrary nonnegative measurable function. In addition, when α ∈ (0, n/2), according to the classification results in [7] and [23], the radial function U * is also the extremal function of the Sobolev inequality Combining (5.1) and (5.2), we have c( Thus, U * is still the extremal function of the functional The corresponding minimum can be expressed by the sharp constant of (5.3).
Inequality (5.3) implies that, when u ∈ H α/2 (R n ), the two improper integral terms in E(u) with p = 2n−γ n−α are convergent, and E(u) is bounded from below.
Therefore,m 0 =m. The argument above shows that m =m =m 0 = m 0 . Thus, Here c * is the sharp constant of (5.3). According to the classification results in [7] and [25], we know c * = E * (U * ).