Ground state solutions for asymptotically periodic fractional Choquard equations

This paper is dedicated to studying the following fractional Choquard equation (−4)su + V(x)u = (∫ RN Q(y)F(u(y)) |x− y|μ dy ) Q(x) f (u), u ∈ Hs(RN), where s ∈ (0, 1), N ≥ 3, μ ∈ (0, N), V(x) and Q(x) are periodic or asymptotically periodic, and F(t) = ∫ t 0 f (s)ds. By combining the non-Nehari manifold approach with some new inequalities, we establish the existence of Nehari type ground state solutions for the above problem in the periodic and asymptotically periodic cases under mild assumptions on f . Our results generalize and improve the ones in [Y. H. Chen, C. G. Liu, Nonlinearity 29(2016), 1827–1842] and some related literature.

Problem (1.1) presents nonlocal characteristics in the nonlinearity as well as in the (fractional) diffusion.Such a problem has a strong physical meaning because the fractional Laplacian appears in anomalous diffusions in plasmas, flames propagation and chemical reactions in liquids, population dynamics, geographical fluid dynamics, and American options in finance, see [3,5,21]; and the nonlocal nonlinearities were also used to model the dynamics of pseudo-relativistic boson stars, see [15,18].
If s = 1, then (1.1) formally reduces to the following generalized Choquard equation: which goes back to the description of the quantum theory of a polaron at rest by Pekar [30] in the case N = 3, µ = 1 and f (u) = u, see [22,31] for more details in the physical aspects.In the last decades, there have been many results on nontrivial solutions, ground state solutions and multiple solutions for (1.3), see e.g.[7,22,24,[26][27][28] for the case where V = Q = 1; see e.g.[1,2,19,40] for V or Q is nonconstant.In particular, when N = 3, Q = 1, µ = 1, f (u) = u and V is a continuous periodic function, Ackermann [2] proved the existence of nontrivial solutions by reduction methods.When V and Q are asymptotically periodic, Zhang, Xu and Zhang [40] proved that (1.3) has a ground state solution, based on the generalized Nehari manifold method, developed by Szulkin and Weth in [34], if f satisfies (F1), (F3) and the following monotonicity assumption: (Ne) f (t) is increasing on R.
We point out that (Ne) is very crucial in the arguments of [40].In fact, the starting point of their approach is to show that for each u ∈ H 1 (R N ) \ {0}, the Nehari manifold N intersects H 1 (R N ) in exactly one point m(u) = t u u with t u > 0. The uniqueness of m(u) enables one to define a map u → m(u), which is important in the remaining arguments.
Recently, many researchers began to focus on problems like (1.1).The greatest part of the literature focuses on the study of (1.1) with V = Q = 1, see e.g.[17,22,25] for the case where N = 3, s = 1/2, µ = 1 and f (u) = u; see [13] for the case where N ≥ 3, s ∈ (0, 1) and ; see [33] for the case where N ≥ 3, s ∈ (0, 1) and f satisfies the assumption of Berestycki-Lions type [4,29].Since (1.1) with V = Q = 1 is autonomous, d'Avenia, Siciliano and Squassina [13] showed that the following two minimizing problems: with ρ > 0 are equivalent; Shen, Gao and Yang [33], inspired by Jeanjean [20] and Moroz and Van Schaftingen [29], constructed a Pohozaev-Palais-Smale sequence.With these facts in hand, they can easily prove that (1.1) has a ground state solution.However, the methods used in [13,33] are invalid for (1.1) when ) and lim |x|→∞ a(x) = 0; a(x) ≥ 0 and a(x) > 0 on a positive measure set, Chen and Liu [8] proved that (1.1) has a Nehari type ground state solution by using the Nehari manifold method and comparing the critical level with the one of the problem at infinity.The main idea comes from Cerami and Vaira [6].Note that this approach relies heavily on the special form V = 1 and f (u) = |u| p−2 u.Moreover the assumption a ∈ L 2N/(2N−µ−N p+2sp) (R N ) also plays an important role.When V(x) and Q(x) are asymptotically periodic and f is continuous but not differentiable, the approach used in [8] is no longer applicable for (1.1).To the best of our knowledge, there seems to be no paper dealing with this case.Motivated by the above works and [9,10], in the present paper, by combining the non-Nehari manifold approach used in [35,38,39] with some new inequalities, we shall establish the existence of Nehari type ground state solutions for (1.1) under (F1)-(F3) in the periodic and asymptotically periodic cases.
To state our results, we first introduce a notation and some assumptions on V and Q.Let Based on the mountain pass theorem due to Rabinowitz [32], we shall prove the above results by applying the non-Nehari manifold approach, which lies on finding a minimizing Cerami sequence for Φ outside N by using the diagonal method (see Lemma 2.8), different from the Nehari manifold method and the generalized Nehari manifold method used in [8,13,33,40].To this end, we establish some new inequalities (see Lemmas 2.3 and 2.4).With these inequalities in hand, we verify the boundedness of Cerami sequences (see Lemma 2.9), and overcome the difficulties caused by the lose of Z N -translation invariance in the asymptotically periodic case.
Remark 1.4.Our results are available for Choquard equation (1.3) with slight modification.From this point of view, we give an extension of the corresponding result in [40].
The paper is organized as follows.In Section 2, we give some preliminaries.We complete the proofs of Theorems 1.1 and 1.2 in Sections 3 and 4 respectively.
Throughout this paper, we denote the norm of L q (R N ) by u q = R N |u| q dx 1/q for q ∈ [2, ∞), B r (x) = {y ∈ R N : |y − x| < r}, and positive constants possibly different in different places, by C 1 , C 2 , . . .

Preliminaries
In this section, we give some preliminaries which are crucial for proving our results.Firstly, to establish the variational setting, we present the following Hardy-Littlewood-Sobolev inequality.
Secondly, we state a version of Lions' concentration-compactness lemma for fractional Laplacian, which is an adaptation of a classical lemma of Lions [26].
Now, inspired by [11,12,36,37], we establish some new inequalities, which are key points in the present paper.
In the following, based on the mountain pass theorem due to P. H. Rabinowitz [32] in 1992, we will find a minimizing Cerami sequence for Φ outside N by the diagonal method, this idea goes back to [35,38], which is essential in the proofs of Theorems 1.1 and 1.2.Lemma 2.8.Assume that (VQ) and (F1)-(F3) hold.Then there exist a constant c * ∈ (0, m] and a sequence {u n } ⊂ H s (R N ) satisfying (2.12) Proof.In view of the definition of m, we choose By (F3) and (2.10), we have Now, we can choose a sequence {n k } ⊂ N such that Then, going if necessary to a subsequence, we have Lemma 2.9.Assume that (VQ) and (F1)-(F3) hold.Then any sequence {u n } ⊂ H s (R N ) satisfying Proof.To prove the boundedness of { u n }, arguing by contradiction, suppose that u n → ∞.
then by Lemma 2.2, one has v n → 0 in L q (R N ) for 2 < q < 2 * s , and so v n rp 1 → 0 and This contradiction shows that δ > 0.

The periodic case
In this section, we give the proof of Theorem 1.1.
Going if necessary to a subsequence, we may assume the existence of k n ∈ Z N such that Since V(x) and Q(x) are periodic on x, we have Passing to a subsequence, we have ūn ū in H s (R N ), ūn → ū in L p loc (R N ) for 2 ≤ p < 2 * s , and ūn → ū a.e. in R N .Thus, (3.3) implies that ū = 0.It is easy to verify that Φ ( ū) = 0. Since f (t) = 0 for all t ≤ 0, as in [8], by minor modification of [13, Theorem 3.2] and using the maximum principle for fractional Laplacian in [14], we have ū > 0. This shows that ū ∈ N is a solution of (1.1) and so Φ( ū) ≥ m.From (2.3), (2.4), (2.6), (3.4) and Fatou's lemma, we have This shows that Φ( ū) ≤ m and so Φ( ū) = m = inf N Φ > 0.

The asymptotically periodic case
In this section, we have By (VQ2), (F1) and Proposition 2.1, we have In view of Lemmas 2.8 and 2.9, there exists a bounded sequence {u n } ⊂ H s (R N ) such that (2.12) holds.Passing to a subsequence, we may assume that u n ū in H s (R N ), u n → ū in L p loc (R N ) for 2 ≤ p < 2 * s and u n → ū a.e. in R N .Next, we prove that ū = 0. Arguing by contradiction, suppose that ū = 0. Then u n 0 in H s (R N ), u n → 0 in L p loc (R N ) for 2 ≤ p < 2 * s and u n → 0 a.e. in R N .From (1.2), (2.4), (2.12), (4.1), (4.2) and Lemma 4.1, we deduce Analogous to the proof of (3.2), there exists k n ∈ Z N , going if necessary to a subsequence, such that Since V 0 (x) and Q 0 (x) are periodic in x, it follows from (4.3) that Passing to a subsequence, we have v n v in H s (R N ), v n → v in L p loc (R N ) for 2 ≤ p < 2 * s and v n → v a.e. in R N .Thus, (4.4) implies that v = 0. Arguing as in Theorem 1.1, we can prove that Φ 0 ( v) = 0, Φ 0 ( v) ≤ c * and v > 0. In view of Lemma 2.6, there exists t > 0 such that t v ∈ N and so Φ( t v) ≥ m.Noting that the conclusion of Lemma 2.4 holds for Φ 0 , from (VQ2), (1.2), (4.1), (4.2) and (4.5), we derive This contradiction shows that ū = 0.In the same way as the last part of the proof of Theorem 1.1, we can prove that that ū ∈ H s (R N ) is a ground state solution for (1.1) with Φ( ū) = m = inf N Φ > 0.
Thus, from (2.13) and (2.14), one has .2) Similar to the proof of [40, Lemma 4.3], we can obtain the following lemma.