Liouville theorems for an integral equation of Choquard type

We establish sharp Liouville theorems for the integral equation \begin{document}$ u(x) = \int_{\mathbb{R}^n} \frac{u^{p-1}(y)}{|x-y|^{n-\alpha}} \int_{\mathbb{R}^n} \frac{u^p(z)}{|y-z|^{n-\beta}} dz dy, \quad x\in\mathbb{R}^n, $\end{document} where \begin{document}$ 0 and \begin{document}$ p>1 $\end{document} . Our results hold true for positive solutions under appropriate assumptions on \begin{document}$ p $\end{document} and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive \begin{document}$ H^{\frac{\alpha}{2}}(\mathbb{R}^n) $\end{document} solutions of the higher fractional order Choquard type equation \begin{document}$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * u^p\right) u^{p-1} \quad\text{ in } \mathbb{R}^n. $\end{document}


(Communicated by Wenxiong Chen)
Abstract. We establish sharp Liouville theorems for the integral equation where 0 < α, β < n and p > 1. Our results hold true for positive solutions under appropriate assumptions on p and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive H α 2 (R n ) solutions of the higher fractional order Choquard type equation 1. Introduction. In this paper, we study Liouville theorems, i.e., nonexistence of positive solutions to the following integral equation where 0 < α, β < n and p > 1. This equation is closely related to the following higher fractional order Choquard type equation One should observe that both the fractional Laplacian (−∆) α 2 u and the Choquard type nonlinearity 772 P. LE understood as the infinitesimal generator of a stable Lévy diffusion process and is used to model American options in finance (see [1,4] and the references therein).
Equation of type (2) is analogous to the well known Choquard-Pekar equation which can date back to [25] in 1954, where the equation was used to describe a polaron at rest in quantum theory. In 1977, Lieb [19] built the model of an electron trapped in its own hole. Equation (3) was also proposed to characterize the particle moving in its own gravitational field (see [22], where it is called Schrödinger-Newton equation). Since then, more and more researchers have focused their attention on the existence of solutions for equation (2) and related problems using variational methods. Without any intention to provide a survey about the subject, we would like to refer the reader to the papers [2,3,21,23,29] and the references therein. Liouville theorems for equation (2) were also studied by several authors in the last decade. Moroz and J.V. Schaftingen [24] studied nonexistence of weak supersolutions of (2) in exterior domains when α = 2 and various sufficient conditions were listed. However, it seems difficult to investigate directly the nonexistence of positive solutions of (2) and (1) in view of the convolution term. By setting The idea of considering the equivalent systems of integral equations like this was initially used by Ma-Zhao [21]. In the critical case p = n+β n−α , one may use the method of moving planes in integral forms for this system to classify all positive L 2n n−α (R n ) solutions of (1) and all positive H α 2 (R n ) solutions of (2). This classification result was proved in [11,16,28] for α = β, in [7,13,20] for β = n − 2α and in [14,15] for more general cases.
Liouville theorems for positive solutions were also established by some authors in the subscritical case p < n+β n−α under additional assumptions on α and β. By using a Pohozaev identity in integral forms, Xu-Lei [28] proved that (1) has no positive L n(p−1) α (R n ) solutions if β = α ∈ (1, n) and p < n+α n−α . Similar result was established by Lei [16] for α = β = 2 and by Cao-Dai [7] for n − β = 2α = 8 using the moving plane method. According to [17], if β = α ∈ (1, n), then the solutions u of (1) have better regularity, i.e., u ∈ C 1 (R n ). This smoothness of u is essential for applying the Pohozaev identity to deduce nonexistence result for (1).
Motivated by above works, in this paper, we prove some Liouville theorems for positive solutions of (1) and (2) in the general case 0 < α, β < n. Our first nonexistence result is the following one, which can be applied to positive supersolutions of more general equations.
For any y ∈ R n \ {0}, we have From (5) and by a similar argument, we deduce for any x ∈ R n \ {0}, Inequality (6) implies that u satisfies (4) for λ > 0 sufficiently small.
As a consequence of Theorem 1, we have the following corollary.

Remark 2.
The assumption p < n+β n−α in Theorem 2 is sharp in the sense that (1) may have positive solutions in L n−α and all positive solutions u ∈ L 2n n−α (R n ) of (1) have been classified in [14] as where c, t > 0 and x 0 ∈ R n . Now we deal with Liouville theorems for equation (2). The positive solutions u of (2) are defined in the distributional sense, i.e., u ∈ H α 2 (R n ) and satisfies for any φ ∈ C ∞ 0 (R n ), where the fractional Laplacian is defined by Fourier transform as in [26] (−∆) while the homogeneous Sobolev norm Here, as usual, denotes the Fourier transform of u. By using Corollary 1, Theorem 2 and exploiting the relation between equations (1) and (2), we establish the following Liouville theorem for (2).
Theorem 3. Assume that 0 < α, β < n. Then equation (2) has no positive solution in H α 2 (R n ) provided that one of the following conditions holds The rest of our paper is organized as follows. In Section 2, we prove Theorem 1 via the method of integral estimates. In Section 3, we employ the method of moving planes in integral forms to prove Theorem 2. The proof of Theorem 3 is given in Section 4. Throughout the paper, we denote by C the generic positive constant whose value may change from line to line or even in the same line.

2.
Liouville theorem for positive super-solutions. In this section, we prove Theorem 1 by following the ideas in [8].
Proof of Theorem 1. Suppose that such function u exists. Let us define Choosing any R > 1. Then we have for a.e. x ∈ R n , From (8), we deduce that This indicates u q v ∈ L 1 loc (R n ). Since R > 1, one can observe that (8) also implies Using (11) and (10), we also deduce From (9) and (12), we have If p + q < n+β n−α , we reach a contradiction by letting R → ∞ in (13). It remains to consider the case p + q = n+β n−α . In this case, (13) Letting R → ∞ and noting that u q v ∈ L 1 (R n ), we obtain u p v L 1 (R n ) = 0, which is impossible.

3.
Liouville theorem for positive integrable solutions. Throughout this section, we always denote γ = n + β − p(n − α). If u is a positive solution of (1), we denote by u the Kelvin transformation of u centered at the origin. That is, Lemma 4. The Kelvin transformation u satisfies the integral equation Proof. Using the fact we have That is, (14).
To prove Theorem 2, we exploit the method of moving planes in integral forms (see [9,10]) to integral equation (1) in the x 1 direction. For this purpose, we need some definitions. For any λ ∈ R, let be the moving plane, Σ λ = {x ∈ R n | x 1 < λ} be the region to the left of the plane and be the reflection of the point x = (x 1 , x 2 , . . . , x n ) about the plane T λ .
For each positive solution u of (1), we also define Proof. One can observe from Lemma 4 that, for any Using the mean value theorem, we get from (15) that, for any λ < 0 and x ∈ Σ − λ , Let us recall the following Hardy-Littlewood-Sobolev inequality that will be used in the method of moving planes.
One can easily check that

Now we observe that
That is, From Lemma 5 and the Hardy-Littlewood-Sobolev inequality, we have .
By Hölder inequality, we obtain for p > 2 While for p = 2, we have On the other hand, Substituting (18), (19) and (20) into (17), we arrive at the following estimate With the aid of (21), we are able to start moving the plane T λ from near λ = −∞ to the right until it reaches the limiting position in order to derive symmetry. This procedure contains two steps.
Step 1. We show that, for λ sufficiently negative,
It can be clearly seen from (21) that, our primary task is to prove that, one can choose ε > 0 sufficiently small such that for all λ ∈ [λ 0 , λ 0 + ε), where the constant C is the same as in (21). From (16) and p > 1, there exists R > 0 large enough such that Now fix this R, in order to derive (26), we only need to show that To prove this, we define and Therefore, for an arbitrarily fixed η > 0, one can choose δ > 0 small enough such that |F δ | ≤ η. For this fixed δ, we will point out that Indeed, for all where e 1 = (1, 0, . . . , 0). Hence lim λ→λ + 0 |G λ δ | = 0, from which (31) follows.
We may repeat above arguments for any Kelvin transformation of u to deduce that u is radially symmetric about any point in R n . This only occurs if u is constant. However, this is absurd since a positive constant function does not satisfy equation (1).
Therefore, equation (1) has no positive solution in L s (R n ).
Then (−∆) α 2 ψ = φ and ψ ∈ H α (R n ) ⊂ H α 2 (R n ). Testing (7) with this ψ, we have  Integrating by parts of the left hand side and exchanging the order of integration of the right hand side, we obtain R n u(x)φ(x) dx = R n,α R n R n u p−1 (y) |x − y| n−α R n u p (z) |y − z| n−β dzdy φ(x) dx.
Since this formula holds for any φ ∈ C ∞ 0 (R n ), we deduce that u(x) = R n,α R n u p−1 (y) |x − y| n−α R n u p (z) |y − z| n−β dzdy, x ∈ R n , which implies that v satisfies the integral equation (1).
It follows that, for any φ ∈ C ∞ 0 (R n ), we have which implies that u is also a weak solution to equation (2).