Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: −Δu+V(x)u=[|x|−μ∗|u|p]|u|p−2u,x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ \end{document} where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 3$\end{document}, 0<μ<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\mu <N$\end{document}, 2N−μN≤p<2N−μN−2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$\end{document}, ∗ represents the convolution between two functions. We assume that the potential function V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x)$\end{document} satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.


Introduction and main result
Consider the following Choquard equation: u + V (x)u = |x| -μ * |u| p |u| p-2 u, x ∈ R N , (1.1) where N ≥ 3, 0 < μ < N , 2N-μ N ≤ p < 2N-μ N-2 . Problem (1.1) arises from the study of the existence of standing wave solutions for the following equation: which appears naturally in optical systems with a thermal [21] and influences the propagation of electromagnetic waves in plasmas [2] and plays an important role in the theory of Bose-Einstein condensation [9]. Here ψ : R N × R → C represents the wave function of the state of an electron, and W is the external potential. In the present paper, we are mainly interested in studying the standing wave solutions of the form ψ(x, t) = u(x)e -iEt . This type of particle-like solution does not change its shape as it evolves in time, hence has a soliton-like behavior. Obviously, u(x) solves (1.1) iff ψ(x, t) solves the above equation with V (x) = W (x) -E. Additionally, it is easy to see that problem (1.1) has nonlocal characteristics in the nonlinearity due to the effect of the convolution part, which is different from a local problem.
Especially, when p = 2, N = 3, and μ = 1, problem (1.1) is called Choquard-Pekar equation u + V (x)u = |x| -1 * |u| 2 u, x ∈ R 3 , (1.2) which arises in the description of the quantum theory of a polaron at rest by Peak [28] in 1954 and in the modeling of an electron trapped in its own hole in 1976 in the work of Choquard, see [18]. Moreover, people also call this equation the Schrödinger-Newton equation, which was introduced by Penrose in his discussion on the self-gravitational collapse of a quantum mechanical wave function [29]. Up to reparametrization, Penrose suggested that the solutions to (1.2) are the basic stationary states that do not spontaneously collapse any further within a certain time scale. Hence, it is very interesting to study these basic solutions. Besides, equation (1.2) has many interesting applications in the quantum theory of large systems of nonrelativistic bosonic atoms and molecules, we refer readers to [18,28] for more physical backgrounds. Mathematical work on nonlinear Choquard equations like the above has been investigated in recent years, and the existence and multiplicity results for such type equations have been considered in many papers under some different assumptions on the potential and nonlinearity by using various variational arguments. For readers' convenience, next we briefly summarize the related study on the existence and multiplicity of nontrivial solutions to problem (1.1).
We pointed out that Lieb [18] and Lions [20] firstly studied the existence and symmetry of the solutions to (1.2). More precisely, up to translations, Lieb [18] obtained the existence and uniqueness of the ground state solutions with V being a positive constant. Lions [20] showed the existence of a sequence of radially symmetric solutions. Since then people began to pay attention to studying the existence of nontrivial solutions for nonlinear Choquard equations, not only from the mathematical curiosity. Such nonlocal problems are also widely used in optimization, finance, phase transitions, stratified materials, anomalous diffusion, and so on. Although Lieb [18] established the uniqueness of the ground state solutions, the classification of positive solutions has been an open problem for many years. The fundamental reason is that people cannot use the standard method of moving planes (based on the maximum principle) to obtain the radial symmetry of the solutions. Until 2010, inspired by the works of Chen et al. [4] and Li et al. [17], under the assumptions p ≥ 2 and Ma and Zhao [22] proved that all the positive solutions to equation (1.1) with V = Const and 2 ≤ p < 2N-μ N-2 are radially symmetry and monotone decreasing about some fixed point. And by using of the new method of moving planes introduced in [4] and Riesz and Bessel potentials, they deduced the problem into an elliptic system. At last, using the radial symmetry, up to translations, they proved that the positive solution to (1.2) (not only the ground state) is unique, which solved the open problem in [18]. After that, in the spirit of [22], Cingolani, Clapp, and Secchi [5] considered the following nonlinear Choquard equation with magnetic field: and obtained the existence of multiple complex-valued solutions that satisfy the symmetry condition u(gx) = τ (g)u(x) for all g ∈ G, where τ : G → S 1 is a given group homomorphism into the unit complex numbers, A is a real-valued C 1 -vector potential, V is a real-valued bounded continuous scalar potential with inf V > 0, N ≥ 3, 0 < μ < N , and In [24], Moroz and Van Schaftingen eliminated this restriction (1.3) to establish regularity and positivity of the groundstates for equation (1.1). They also proved the radial symmetry and monotonic decay of positive groundstates. We pointed out that they had many interesting works about Choquard equation, see [25][26][27] and the references therein. Recently, the strongly indefinite Choquard equations with critical exponent in the whole space were also studied in [14,31,32,39] where the existence and multiplicity were obtained by using a linking theorem. Furthermore, by using the minimax procedure and perturbation technique, Gao et al. [15] showed the existence of infinitely many solutions for a class of critical Choquard equations with zero mass.
Very recently, for other related topics involving the singularly perturbed problem, there have been some works devoted to the study of a concentration phenomenon of semiclassical states. For instance, by using a Lyapunov-Schmidt type reduction, Wei and Winter [37] constructed families of solutions for the following equation: with potential inf V > 0 and characterized the concentration behavior around the global minimum points of V . Moreover, they also showed that the groundstate to (1.2) is up to translations a nondegenerate critical point. Not long after that the existence of a family of solutions having multiple concentration regions located around the minimum points of the potential was obtained in [6]. With the help of the mountain pass lemma and the genus theory, Ding et al. [10] obtained the existence and multiplicity of semiclassical states to the Choquard equation They also constructed the multiplicity of high energy semiclassical states by using the Ljusternik-Schnirelmann theory. In [42], the authors proved the existence and concentration of semiclassical solutions under Berestycki-Lions type conditions. Furthermore, the other type of Choquard equations has also attracted great interest. For example, instead of the classical Laplacian operator, many scholars considered the following type of Choquard equation: which is called fractional Choquard equation that is used to model the dynamics of pseudo-relativistic boson stars. About the study of fractional Choquard equations, please see [8,23,34] and their references therein. More related results about ground state solutions and infinitely many solutions for other problems, we refer the readers to [1,3,7,13,16,30,33,41,43,44] and the references therein.
Motivated by the results mentioned above, in this paper we consider the case that the potential V is a general periodic function and prove the existence of a ground state solution and infinitely many pairs of geometrically distinct solutions for problem (1.1) by using the method of the Nehari manifold developed by Szulkin and Weth [36]. To the best of our knowledge, it seems that there is no work that considered this problem in the literature before. In order to state the main result, we list the assumption as follows: Then, according to condition (V ), the norm · is equivalent to the norm · H 1 , where According to the periodicity condition (V ), if u 0 is a solution of (1.1), then k * u 0 is also a solution of (1.1) for all k ∈ Z N . Set which means the orbit of u 0 with respect to the action of Z N . If u 0 is a critical point of the Our main result of this paper is the following:

Proof of theorem
Throughout this paper, we denote by · s the usual norm of the space L s , 1 ≤ s ≤ ∞, and c or c i (i = 1, 2, . . . ) denotes the different positive constants. Firstly, in order to overcome the nonlocality of problem (1.1) and study the property of the energy functional, we will use the classical Hardy-Littlewood-Sobolev inequality frequently. So, we give the Hardy-Littlewood-Sobolev inequality due to [19]. Proposition 2.1 (Hardy-Littlewood-Sobolev inequality, [19]) Let s, t > 1 and 0 < μ < N with 1 s . Then there exists a sharp constant C s,N,μ,t , independent of g, h, such that Remark 2.2 Obviously, Hardy-Littlewood-Sobolev inequality implies that the integral Since we will work with u ∈ E, in order to make the integral well defined, tq must fall in the interval [2, 2 * ], i.e., That is why the exponent 2N-μ N is called the lower critical exponent and the exponent 2N-μ N-2 is called the upper critical exponent. We need to point out that Hardy-Littlewood-Sobolev inequality plays an important role for nonlocal problems.
From a viewpoint of variational methods, it is clear that equation (1.1) is the Euler-Lagrange equation associated with the energy functional J : E → R given by Since p ∈ [ 2N-μ N , 2N-μ N-2 ), then by Hardy-Littlewood-Sobolev inequality and Sobolev embedding theorem, it is easy to prove that J is well defined on E and belongs to C 1 (E, R). Moreover, there holds In order to seek for the ground state solutions of problem (1.1), we consider the following Nehari set: Proof According to Hardy-Littlewood-Sobolev inequality and Sobolev embedding theorem, we have as t → +∞. Hence f has a positive maximum and there exists t v > 0 such that from where it follows that f (t) > 0 for 0 < t < t v and f (t) < 0 for t v < t and t v is unique. The proof is completed.
For the sake of convenience, for r > 0, set B r := {v ∈ E : v ≤ r} and S r := {v ∈ E : v = r}.
Proof (i) Let v ∈ E, by the Hardy-Littlewood-Sobolev inequality and the Sobolev embedding theorem, we get Observe that 0 < μ < N , then p > 1. So, from the above fact, it is easy to see that there exist > 0 and α > 0 such that inf S J ≥ α > 0. On the other hand, for every v ∈ N , there exists t 0 > 0 such that t 0 v ∈ S . Then by Lemma 2.3 we get which implies that v ≥ √ 2c > 0 for all v ∈ N . The proof is completed. N , i.e., J(v) → +∞ as v ∈ N and v → ∞.

Lemma 2.5 J is coercive on
Proof Arguing by contradiction, we assume that there exist a sequence {v n } ⊂ N and a positive number m such that v n → ∞ and J(v n ) ≤ m. Set u n = v n v n . Then, passing to a subsequence, there exists u ∈ E such that u n u in E, u n → u in L q loc (R N ) for any 2 ≤ q < 2 * and u n (x) → u(x) a.e. on R N . By the fact that u n = 0, there exists a point y ∈ R N such that It is easy to prove that γ (z) is continuous on R N by the absolute continuity of integral. Take large R > 0 with Note that γ is continuous andB R+1 (0) is a compact set, there exists y n ∈B R+1 (0) such that γ (y n ) = sup z∈B R+1 (0) γ (z). Hence, According to the assumption of periodicity, we can assume that {y n } is bounded in R N . In the sequel, we prove using the vanishing lemma [38, Lemma 1.21], we have u n → 0 in L s (R N ) for 2 < s < 2 * . Hence, for any t > 0, there holds this is impossible if t is large enough. Therefore, we know that u = 0. Set = {x ∈ R N : u(x) = 0}, then meas( ) > 0. In the light of Fatou's lemma we deduce that Then it follows that as n → ∞, which yields a contradiction. The proof is completed.
Proof Without loss of generality, we may assume that v = 1 for every v ∈ W. Suppose that there exist v n ∈ W and w n = t n v n such that J(w n ) ≥ 0 and t n → ∞ as n → ∞. Passing to a subsequence, we may assume that v n → v ∈ S 1 in W ⊂ E\{0}. Consequently, using Fatou's lemma, we obtain as n → ∞, a contradiction. This completes the proof. Since E is a Hilbert space, by Lemmas 2.3, 2.4, and 2.6 we can verify that the hypotheses A 2 and A 3 in [36] hold. Hence, we have Lemmas 2.7-2.8, the details of proofs can be found in [36]. (iv) If J is even, then so is ψ.

Lemma 2.9
The mapping m -1 defined in Lemma 2.7 is Lipschitz continuous.
Proof For any v, w ∈ N , by Lemma 2.4-(ii) we have v ≥ √ 2c and w ≥ √ 2c. Moreover, using Lemma 2.7, we obtain From the above fact, it is easy to see that the mapping m -1 is Lipschitz continuous. This completes the proof.
For any a ∈ R, set ψ a := {v ∈ S : ψ(v) ≤ a}. We have the following lemma about the discreteness property of the Palais-Smale sequence.  Consequently, by the Hardy-Littlewood-Sobolev inequality, Hölder's inequality, and the boundedness of {u 1 n } and {u 2 n } in E, we know that there exists a constant c 2 > 0 such that In the following we divide into two cases to discuss.
2N-μ → 0 as n → ∞. It follows from (2.1) that u 1 nu 2 n → 0. Moreover, from Lemma 2.9 we have v 1 2N-μ 0 as n → ∞. By the boundedness of {u 1 n } and {u 2 n } in E and the vanishing lemma [38, Lemma 1.21], we can deduce that Thereby, up to a subsequence, there exists ε 0 > 0 such that Following the proof of Lemma 2.5, there exists y n ∈ R N such that Similar to [12], with the help of Lemma 2.4 and Lemma 2.5, we can prove that To complete the proof of Theorem 1.1, we take the following two lemmas.

Lemma 2.11
Problem (1.1) has at least a ground state solution.
Making use of Proposition 9 in [36], we know thatψ : E \ {0} → R is a class of C 1 . Moreover, it is easy to see that According to Corollary 3.4 in [11], there exists a sequence {w n } ⊂ S such that ψ(w n ) → c and there exists α n ∈ R such that ψ (w n )α n ϕ (w n ) E * → 0. Hence, Set v n = m(w n ) ∈ N for all n ∈ N. In view of Lemma 2.8-(ii) we have Moreover, {v n } is bounded in E by Lemma 2.5. Consequently, up to a subsequence, there In what follows, we claim that J (v) = 0. In fact, for any φ ∈ C ∞ 0 (R N ), there holds According to the fact that v n v in E, we get Consequently, according to the above facts, we can obtain Let n → ∞, by the Hardy-Littlewood-Sobolev inequality, Hölder's inequality, and the Sobolev embedding theorem, we get which yields that J (v) = 0. Notice that As in the proof of Lemma 2.10, we know that there is a constant c 3 > 0 such that Consequently, up to a subsequence, there exists ε 0 > 0 such that The proof of Lemma 2.5 indicates that there exists y n ∈ R N such that By the assumption of periodicity, we can assume that {y n } is bounded in R N . Therefore, there exists a bounded domain 1 ⊂ R N such that However, since 1 is bounded, the embedding theorem implies that Clearly, this yields a contraction. Hence v nv 2Np 2N-μ → 0 as n → ∞. Then, by (2.2), we see that v n → v in E and J(v) = c > 0. Therefore, v ∈ N is a ground state solution of problem (1.1). The proof is completed.
Next we are devoted to looking for infinitely many geometrically distinct solutions for problem (1.1). Observe that, by Lemma 2.8-(iv), we know that ψ is even. Set For A ∈ , we denote the Krasnoselskii genus of A by i(A). Define for all k ∈ N. It is not difficult to prove that c ≤ c k and c k ≤ c k+1 .

Lemma 2.12 c k is a critical value of ψ.
Proof If the conclusion is false, then for any w ∈ S one has ψ(w) = c k or ψ (w) = 0. It implies that there exists δ > 0 such that N c k ,δ := w ∈ S : ψ(w)c k < δ, ψ (w) E * < δ = ∅.