A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities

In this paper, we investigate the following Choquard equation \begin{document}$ \begin{equation*} -\Delta u+V(x)u = \lambda(I_\alpha*F(u))f(u) \ \ \ \ \ \ {\rm in} \ \mathbb{R}^N, \end{equation*} $\end{document} where \begin{document}$ N\geq 3, \lambda>0, \alpha\in (0, N) $\end{document} , \begin{document}$ V $\end{document} is an asymptotically periodic potential, \begin{document}$ I_\alpha $\end{document} is the Riesz potential, the nonlinearity term \begin{document}$ F(s) = \int_{0}^{s}f(t)dt $\end{document} and \begin{document}$ f $\end{document} is only locally defined in a neighborhood of \begin{document}$ u = 0 $\end{document} and satisfies the suitable conditions. By using the Nehari manifold and the Moser iteration, we prove the existence of positive solutions for the equation with sufficiently large \begin{document}$ \lambda $\end{document} .

In general, the hypothesis p ∈ [ N +α N , N +α N −2 ] is necessary for the existence of solutions to Eq. (P), see [20,21,23]. Since we consider the case where p ≥ 2, the restriction α ∈ ((N − 4) + , N ) on α imposed is natural. The exponent N +α N −2 (or N +α N ) is called the upper (or lower) critical exponent with respect to the Hardy-Littlewood-Sobolev inequality.
Prior to our work, we recall some important results about Choquard equations. Especially, the case of Eq. (P) is as following which was used to describe the quantum theory of a polaron at rest by Pekar [24] in 1954. Eq. (1) arisen in a certain approximation to Hartree-Fock theory for a one component plasma. Ph. Choquard [12] proposed it for investigation at the Symposium on Coulomb Systems, Lausanne, July, 1976. Knowledge of the solutions of Eq. (1) has a great importance for studying standing wave solutions for the Hartree-Fock equation where h > 0, W and Φ are real-valued functions and Ω is a domain in R 3 . Eq.
(NLS) has many interesting applications in the quantum theory of large systems of non-relativistic bosonic atoms and molecules. In recent years, Eq. (P) has been researched intensively. When Eq. (P) is autonomous, it has been studied by Lieb [12],Ma and Zhao [18], Moroz and Van Schaftingen [20,21] and so on. When Eq. (P) is non-autonomous, there are many results. Among these papers, Menzala [19] have studied the case where V is spherically symmetric, decreases with r = |x| and vanishes at infinity. In [5,9], the authors have studied the case where V is a periodic function. The case where V is asymptotic to a positive constant has been considered in [17]. V is a vanishing potential, which has received attention in the paper of Alves et al. [1]. For the case where V is coercive, i.e., we would like to cite the paper of Van Schaftingen and Xia [27]. In [29], a more weak condition than coercivity on V has been assumed, more precisely, A case dealing with Eq. (P) with steep potential well is studied in [2,26]. The case where V is asymptotically periodic, that is, was considered by zhang et al. [30]. At the same time, there also have been intensive studies on semiclassical states for Choquard equations for which in Eq. (P) there is a small parameter corresponding to the Plank constant, i.e., Further results for related problems can be found in [3,10,22] and references therein. For a complete and updated discussion upon the current literature of such problems, we refer the interested reader to the guide [23].
In the above works, we also observe that many interesting conditions on f have been studied. Notice that, it seems necessary that (f 3 ) can be assumed on f , This aim is to ensure that the associated energy functional would be well defined and of class C 1 from the Hardy-Littlewood-Sobolev inequality and the Sobolev inequality. Classically, it follows from variational methods that critical points of I are precisely the solutions of Eq. (P) (see [28]). If we don't make any assumptions on f at infinity, we can prove that there exists a nontrivial solution for Eq. (P). Mathematically this problem is new and interesting.
Here, the assumptions (f 1 ) − (f 2 ) we make on the nonlinearity f only refer solely to its behavior in a neighborhood of u = 0, and we will show that they suffice for the existence of a positive solution of Eq. (P) when λ is large enough. Exactly we give our main result.

Remark 1.
For the case where the nonlinear term is only locally defined for |u| small, we should point out that we refer [6,7,8,11] for references in this direction. Chu and Liu [6] investigated quasi-linear Schrödinger equations in the radial space. Costa and Wang [7] considered Schrödinger equations in bound domain. doÓ et al. [8] considered Schrödinger equations when V was coercive potential or satisfied that [V (x)] −1 belongs to L 1 (R N ). Li and Zhong [11] studied the Kirchhoff equation when the nonlinearity term was sub-linear growth. In these papers, the compactness is obtained obviously, then they can prove certain solutions. However, in our cases we do not have compact embedding, which is the main difficulty in this paper. And the conditions in [6,7,8,11] and ours are mutually non-inclusive and the methods are different.
[30] also studied the asymptotically periodic potential. However, in our cases the nonlinear term is only locally defined for |u| small and (V 1 ) is a reformative condition about the asymptotic processes of (V 4 ) (see [15]), which lead to the difference from [30]. These are a little surprising.

Remark 2.
For instance, let f (s) = |s| p−2 s + |s| q−2 s, s ∈ R, where p ∈ [2, N +α N −2 ) and p ≤ q. Then it is easy to check that f satisfies conditions (f 1 ) − (f 2 ). However, when N +α N −2 < q, this functional I takes the value −∞ for some u ∈ E and in particular, it is not of class C 1 . This implies that variational methods fail to our problem directly. Then we take a new technique to deal with our problem. We now make some comments on the key ingredients of the analysis in this paper. Following the idea of dealing with the elliptic problem in [6,7,8,11], we first extend the nonlinear term f and introduce a modified Choquard equation. Next, we prove that the modified Choquard equation possesses a positive ground state solution u by variational methods. Finally, inspired by the results of [7], we can show an a priori bound of the form The organization of this paper is as follows. In the next section we reserve for setting the framework and establishing some preliminary results. Theorem 1.1 is proved in Section 3.

2.
Preliminaries. From now on, we will use the following notations.
• meas Ω denotes the Lebesgue measure of the set Ω.
• C denotes a positive constant and is possibly various in different places. We work on the space E and recall some facts that the norms · and · H are equivalent and the embedding E → L s (R N ) for any s ∈ [2, 2 * ] is continuous. The proof can be done similarly to that in [15] and details are omitted here. We start by observing that (f 1 ) − (f 2 ) imply that the existence of a positive constant δ * ≤ min{1, δ} such that In order to prove our main results via variational methods, we need to modify and extend f (u) for outside a neighborhood of u = 0 to get f (u). We set Combining the definition of f with the proof of Lemma 2.3 in [16] (see also [30]), one can easily complete the proof of the following lemma.
(ii) there exists µ ∈ (1, min{ν + 1, p}) such that the function s → f (s) s µ−1 is strictly increasing on (0, +∞), Now let us consider the modified equation of Eq. (P) given by From (i) of Lemma 2.1, the Hardy-Littlewood-Sobolev inequality and the Sobolev inequality, we have Then the corresponding functional is well defined and of class C 1 by a standard argument and whose derivative is given by Formally, critical points of I are the solutions of Eq. ( P). We note that nonnegative critical points of I with L ∞ -norm less than or equal to δ * are also solutions of the original Eq. (P). We recall the Nehari manifold Then (a) for any u ∈ K , there exists a unique t u > 0 such that t u u ∈ N . Moreover, the maximum of I(tu) for t > 0 is achieved at t u , (b) there exists ρ > 0 such that u ≥ ρ for all u ∈ N , (c) I is bounded from below on N by a positive constant.
We could deduce Ψ(t) → −∞ as t → +∞. So there exists t u > 0 such that Ψ(t u ) = max t>0 Ψ(t) and Ψ (t u ) = 0, i.e., I(t u u) = max t>0 I(tu) and t u u ∈ N . Suppose that there exists t 1 > t 2 > 0 such that t i u ∈ N , i = 1, 2, one has which is a contradiction with (ii) of Lemma 2.1. Consequently t u is unique.
(b) For any u ∈ N , combining (i) of Lemma 2.1, the Hardy-Littlewood-Sobolev inequality and the Sobolev embedding one obtains It follows from (3) that there exists ρ > 0 independent of u such that ρ ≤ u .
If {z n } is bounded, there exists R > 0 such that which is a contradiction with v n → 0 in L 2 loc (R N ). Then {z n } is unbounded, up to a subsequence, |z n | → ∞. We set w n := v n (· + z n ), where w n satisfies Thus up to a subsequence, there exists a nonnegative function w ∈ E such that w n w in E, w n → w in L 2 loc (R N ) and w n (x) → w(x) a.e. in R N . Evidently, meas Ω 2 > 0 where Ω 2 = {x ∈ R N : w(x) > 0}. In fact w n (x) = un(x+zn) un . Also from the Fatou lemma and (ii) of Lemma 2.1, one obtains lim inf F (u n (x+z n )) F (u n (y+z n )) u n (x+z n )u n (y+z n ) w n (x+z n )w n (y+z n )dxdy = +∞.
Hence 0 = lim sup which is a contradiction. In a word, the (Ce) m sequence {u n } is bounded in E. Now, we recall the following some facts. The proof is similar to Lemmas 2.2-2.4 in [15]. Here we omit it.
Lemma 2.5. Suppose that (V 1 ) holds. If {u n } is bounded in E and |z n | → +∞, Then Eq. ( P) has a positive ground state solution.
Proof. Notice that 0 < c ≤ c ∞ . Therefore, one of the two cases occurs: Then ω is also a positive ground state solution of Eq. ( P) from Lemma 2.3. Case 2. 0 < c < c ∞ . We can see that I satisfies the mountain pass geometry. From the mountain pass theorem [25,28] and Lemma 2.4, there exists a nonnegative and bounded sequence {u n } ∈ E such that Then there exists a nonnegative function u ∈ E such that up to a subsequence, u n u in E, u n → u in L 2 loc (R N ) and u n (x) → u(x) a.e. in R N . For any ϕ ∈ C ∞ 0 (R N ), one has 0 = I (u n ), ϕ + o(1) = I (u), ϕ , i.e., u is a solution of Eq. ( P). If u = 0 in E, combining with the Fatou lemma one obtains At the same time, from the definition of c and u ∈ N one knows c ≤ I(u). We could deduce that u is a positive ground state solution of Eq. ( P) by the strongly maximum principle. We assume that u = 0 (otherwise we complete the proof). Then there exists ≥ 0 such that lim n→∞ sup z∈R N B1(z) |u n | 2 dx = .
Indeed, if = 0, applying the Lions lemma [13,28], we obtain Hence it follows from (i) in Lemma 2.1 that I(u n ) → 0 as n → ∞, which is a contradiction with c > 0. Then there exists {z n } ⊂ R N such that B1(zn) |u n | 2 dx ≥ 2 > 0. If {z n } is bounded, there exists R > 0 such that B R (0) |u n | 2 dx ≥ 2 > 0, which is a contradiction with u n → 0 in L 2 loc (R N ). Then {z n } is unbounded. After extracting a subsequence if necessary, we have From Lemma 2.5, we have Then v is a nontrivial solution of Eq. (P ∞ ). At the same time, from Lemma 2.5 we have which is a contradiction.
In conclusion, whether case 1 occurs or case 2 occurs, we can prove Proposition 1.
If u is a critical point of I, then u ∈ L ∞ (R N ). Furthermore, there exists a positive constant C independent of λ such that .
Proof. We prove this result by using the Moser iteration. For each k > 0, we define For β > 1, we use ϕ k = |u k | 2(β−1) u as a test function in I (u), ϕ k to obtain Then we use the Sobolev inequality to yield where we also have used the facts that u 2 |∇u k | 2 ≤ u 2 k |∇u| 2 and β > 1. If p > 2, from (i) in Lemma 2.1, we deduce . Combining (8) with (9), we obtain Letting k → ∞ in (10), we have Now we carry out an iteration process. Set Then we have By (11), one obtains One can easily see .
Letting m → ∞, we would conclude that u ∈ L ∞ (R N ) and This completes the proof.
We can see that there exists v ∈ K ∩ L ∞ (R N ) such that |v| ∞ < 1. Since (f 1 ), there exists C > 0 independent of λ such that Combining (12), (13) and (14), we have Since 2 ≤ p, there exists λ 1 ≥ λ 0 such that |u| ∞ ≤ δ * for λ ≥ λ 1 . Therefore, from the definition of f , we can conclude that u is also a positive solutions of Eq. (P) for any λ ≥ λ 1 . This completes the proof of Theorem 1.1.
Under the assumption (V 3 ), the embedding E into L s (R N ) for any s ∈ [2, 2 * ) is compact from [4,29]. Combining with the proof of Theorem 1.1, we can directly prove the following result.