Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity

In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016).


Introduction
In this paper, we deal with the following nonlocal equation: where ) and F(t) = t 0 f (s)ds.The fractional Laplacian (−∆) s is defined as |x − y| N+2s dy, u ∈ S(R N ), where P.V. denotes the principal value of the singular integral, S(R N ) is the Schwartz space of rapidly decaying C ∞ functions in R N , and (−∆) s is a pseudo-differential operator, and can be equivalently defined via Fourier transform as where F is the Fourier transform, that is, optimization, finance, phase transitions, crystal dislocation, multiple scattering, and materials science, see [1][2][3][4][5] and their references.
Recently, a great deal of work has been devoted to the study of the Choquard equations, see [6][7][8][9][10][11][12][13][14] and their references.For instance, Alves, Cassani, Tarsi, and Yang [7] studied the following singularly perturbed nonlocal Schrödinger equation: where 0 < µ < 2 and ε is a positive parameter, the nonlinearity f has critical exponential growth in the sense of Trudinger-Moser.By using variational methods, the authors established the existence and concentration of solutions for the above equation.
In [6], Alves, Figueiredo and Yang studied the following Choquard equation: ( Under the assumption V(x) → 0 as |x| → ∞, the authors obtained a nontrivial solution for (2) by using a penalization method.
In the physical case (2) is also known as the stationary Hartree equation [15].It dates back to the description of the quantum mechanics of a polaron at rest by Pekar in 1954 [16].In 1976, Choquard used (2) to describe an electron trapped in its own hole, in a certain approximation to the Hartree-Fock theory of one-component plasma [11].In 1996, Penrose proposed (2) as a model of self-gravitating matter, in a programme in which quantum state reduction is understood as a gravitational phenomenon [15].
In addition, there is little literature on the fractional Choquard equations.Frank and Lenzmann [17] established the uniqueness and radial symmetry of ground state solutions for the following equation: D'Avenia, Siciliano, and Squassina [18] obtained the existence, regularity, symmetry, and asymptotic of the solutions for the nonlocal problem In [19], Shen, Gao, and Yang studied the following fractional Choquard equation: where N ≥ 3, s ∈ (0, 1), and µ ∈ (0, N).Under the general Berestycki-Lions-type conditions [20], the authors obtained the existence and regularity of ground states for (3).The authors also established the Pohozaev identity for (3): Motivated by the above works, in the first part of this article, we study the ground state solution for (1).We assume a sequence of Borel sets such that meas{A n } ≤ δ for all n and some δ > 0, then (III) one of the below conditions occurs: or there exists p ∈ (2, 2 * s ) such that where ) as t → 0 if (5) holds; (F2) It is necessary for us to point out that the original of assumptions (I)-(III) come from [21][22][23].The assumptions can be used to prove that the work space E is compactly embedded into the weighted Lebesgue space L q K (R N ), see Section 2 and Lemma 1. Now, we can state the first result of this article.
Remark 1.Since the Nehari-type monotonicity condition for f is not satisfied, the Nehari manifold method used in [24] no longer works in our setting.To prove Theorem 2, we use the non-Nehari manifold method developed by Tang [25], which relies on finding a minimizing sequence outside the Nehari manifold by using the diagonal method (see Lemma 8).
In the second part of this article, we consider the following fractional Choquard equation with zero mass case: where N ≥ 3, 0 < s < 1, 0 < µ < min{N, 4s}.The homogeneous fractional Sobolev space D s,2 (R N ), also denoted by Ḣs (R N ), can be characterized as the space ) satisfy the following Berestycki-Lions-type condition [19,20]: there exists C > 0 such that for every t ∈ R, The second result of this paper is as follows.
Remark 2. Notice that the method used in [13] is no longer applicable for (6), because it relies heavily on the constant potentials.In the zero mass case, we use the perturbation method and the Pohozaev identity established in [19] to overcome this difficulty.
In this article, we make use of the following notation: denotes the infinitesimal as n → +∞.

Ground State Solutions for (1)
From [5], we have the following identity: Then, we can define the best constant S > 0 as . Let Under the assumptions (I)-(III), following the idea of ([21], Proposition 2.1) or ( [22], Proposition 2.2), we can prove that the Hilbert space E endowed with scalar product and norm Given ε > 0 and fixed q ∈ (2, 2 * s ), there exist 0 < t 0 < t 1 and C > 0 such that Hence, where Let {v n } be a sequence such that v n v in E, then there exists a constant which implies that {W(v n )} is bounded.On the other hand, setting we have s and so sup n∈N |A n | < +∞.Therefore, from (I I), there is r > 0 such that Combining (7) and (8), we have By q ∈ (2, 2 * s ), we have from Sobolev embeddings that lim n→+∞ B r (0) Combining (9) and (10), we have Next, we suppose that (K2) holds.For each x ∈ R N fixed, we observe that the function s −2 as its minimum value, where Hence Combining this inequality with (K2), given ε ∈ (0, C p ), there exists r > 0 large enough such that Let {v n } be a sequence such that v n v in E, then there exists a constant M 2 > 0 such that and so, Since p ∈ (2, 2 * s ) and K is a continuous function, we have From (11) and (12), we have Lemma 2. (Hardy-Littlewood-Sobolev inequality, see [26]).Let 1 < r, t < ∞, and µ ∈ (0, N) with ) and ψ ∈ L t (R N ), then there exists a constant C(N, µ, r, t) > 0, such that Lemma 3. Assume that (I)-(III) and (F1)-(F3) hold.Then for u ∈ E and there exists a constant C 1 > 0 such that Furthermore, let {u n } ⊂ E be a sequence such that u n u in E, then and By (F1), (F2), Lemma 2, Hölder inequality and Sobolev inequality, we have and Applying Lemma 2 and (17), we have which yields (13) holds.Similarly, we have which, together with (17) and (18), implies that (14) holds.Similar to ([21], Lemma 2), by (F 2 ), (F 3 ), and Lemma 2, we have Combining (18), (20), and (21), we deduce that (15) and (16) hold.

Zero Mass Case
In this section, we consider the zero mass case, and give the proof of Theorem 2. In the following, we suppose that (F5)-(F7) and µ < 4s hold.Fix q ∈ (2, 2N−µ N−2s ), by (F7), for every > 0 there is C > 0 such that To find nontrivial solutions for (6), we study the approximating problem where ε ≥ 0 is a small parameter.The energy functional associated to (37) is By using (F5)-(F7) and Lemma 2, it is easy to check that Φ In view of ([19], Proposition 2), for every ε > 0, any critical point u of Φ ε in H s (R N ) satisfies the following Pohozaev identity For every ε > 0, let Φ ε (γ(t)).
The following lemma is a version of Lions' concentration-compactness Lemma for fractional Laplacian.Then u n → 0 in L q (R N ) for q ∈ (2, 2 * s ).
Passing to a subsequence, we have ũn u 0 in D s,2 (R N ).Clearly, (45) implies that u 0 = 0.By the standard argument, u 0 ∈ D s,2 (R N ) is a nontrivial solution for (6).

Conclusions
In this work, we study a class of nonlinear Choquard equation driven by the fractional Laplacian.

Lemma 9 .
Assume that (I)-(I I I) and (F1)-(F4) hold.Then, the sequence {u n } satisfying (31) is bounded in E. Proof.Arguing by contradiction, suppose that u n → ∞.Let v n = u n u n , then v n = 1.Passing to a subsequence, we have v n v in E. There are two possible cases: