Multiple solutions for critical Choquard-Kirchho type equations

where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, 2μ = (2N − μ)/(N − 2) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(RN), with r = 2*/(2* − q) if 1 < q < 2* and r = ∞ if q ≥ 2*. According to the di erent range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.


Introduction and main results
In this paper, we consider the following Kirchho -type equation with Hardy-Littlewood-Sobolev critical nonlinearity in R N : where a > , b ≥ , < µ < N, N ≥ , α and β are positive real parameters, * µ = ( N − µ)/(N − ) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, k ∈ L r (R N ), with r = * /( * − q) if < q < * and r = ∞ if q ≥ * . The paper was motivated by some works appeared in recent years. On one hand, the following Choquard or nonlinear Schrödinger-Newton equation −∆u + V(x)u = (Kµ * u )u + λf (x, u) in R N , (1.2) was studied by Pekar [41] in the framework of quantum mechanics. Subsequently, it was adopted as an approximation of the Hartree-Fock theory in [27]. Recently, Penrose [38] settled it as a model of the selfgravitational collapse of a quantum mechanical wave function. The rst existence and symmetry results of solutions to (1.2) go back to the works of Lieb [27] and Lions [30]. Equations of type (1.2) have been extensively studied, see e.g. [3,15,16,18,20,27,[34][35][36]43] for the study of Choquard-type equations. In the fractional Laplacian framework, we refer to the recent papers [32,40,45].
On the other hand, existence of solutions for Kirchho -type problems involving the critical Sobolev exponent has been considered by many authors. In [10], Chen, Kuo and Wu studied the following Kirchho -type problem −M( ∇u L )∆u = λf (x)|u| q− u + g(x)|u| p− u in Ω, u = on ∂Ω, where M(t) = a + bt, a, b > and f and g are continuous real valued sign changing functions. In [10] the authors prove existence and multiplicity of solutions by using the classical Nehari manifold method. The literature on Kirchho -type problems and related elliptic problems is very interesting and quite large, here we just list a few, for example, see [2, 12, 13, 24-26, 33, 37, 39, 47, 48] for the recent existence results.
Motivated by the above works, especially by the ideas of [11,19,21], in this paper we study the multiplicity of solutions for the Kirchho -type equations (1.1), with Hardy-Littlewood-Sobolev critical nonlinearities. There is no doubt that we encounter serious di culties because of the lack of compactness. To overcome the challenge we use the second concentration compactness principle and the concentration compactness principle at in nity in order to prove the (PS)c condition at special levels c.
The equation (1.1) is variational, so that the (weak) solutions of (1.1) are just the critical points of the underlying functional J α,β in D , (R N ). The rst two multiplicity results cover the cases < q < and q = .
Theorem 1.2. Let < µ < , q = and β = . Then, there exists a positive constant a * such that for each a > a * and α ∈ ( , aS k − r ) equation (1.1) has at least n pairs of nontrivial solutions.
In [45] Wang and Xiang obtain, in the fractional setting, the existence of at least two nontrivial solutions, when < q < * , N > µ ≥ . For the Laplacian counterpart of Theorem 1.1 in [45] their result can be stated as follows.
then there exists α * such that equation (1.1) admits at least two nontrivial solutions in D , (R N ) for all α > α * .
In the following, we are interested in looking for more solutions in the case < q < * . To this end, we shall employ the genus theory to obtain multiplicity of solutions. Regrettably, we have to restrict ourselves to the special case N = and < q < * := . More precisely, we obtain the following result.  1) is not covered in this paper, when either < q ≤ and N = , , or * ≤ q ≤ . However, the approaches used in this paper do not seem to be applicable in the above cases. Thus, these missing values will be studied in future work.
The paper is organized as follows. In Section 2, we recall some preliminaries and set up the underlying functional J α,β associated to (1.1). In Section 3, we prove the Palais-Smale condition at some special energy levels.
In Section 4, we introduce a truncation argument for the functional J α,β and prove Theorem 1.1 by using the Kajikiya new version of the symmetric mountain pass theorem. In Section 5, existence and multiplicity of nontrivial solutions for (1.1) is proved when q = . Section 6 deals with the existence of two nontrivial solutions for (1.1) when < q < * and β = , that is with the proof of Theorem 1.3. Finally, Section 7 is devoted to the proof of Theorem 1.4, that is to the proof of existence and multiplicity of solutions for (1.1) when N = , < q < and α = β.

Preliminaries
Here and in what follows, · p denotes the canonical L p (R N ) norm for any exponent p > . First, let us recall the Hardy-Littlewood-Sobolev inequality, see [28,Theorem 4.3].
for some A ∈ C, ≠ γ ∈ R and x ∈ R N .
Let us introduce D , (R N ) as the completion of C ∞ (R N ) with respect to the norm u = ( R N |∇u| dx) / . Then, the best constant for the embedding of Obviously, S > , see [44]. By the Hardy-Littlewood-Sobolev inequality, the integral for p > such that ( /p) + (µ/N) = , that is p = N/( N − µ). Hence, in D , (R N ) we must have The exponent * µ is called the (upper) critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. In particular, for all u ∈ D , (R N ). Hence, we set and clearly S H,L > . For more details on S H,L , we refer to the following result.
The energy functional associated to (1.1) is J α,β : D , (R N ) → R de ned by The Hardy-Littlewood-Sobolev inequality (2.1) gives . Consequently, the functional J α,β is of class C (D , (R N )). Moreover, where δx is the Dirac function of mass concentrated at x ∈ R N . Finally, for all i ∈ I However, roughly speaking, the second concentration compactness principle, stated in Lemma 2.3, is only concerned with a possible concentration of a weakly convergent sequence at nite points and it does not provide any information about the loss of mass of a sequence at in nity. The next concentrationcompactness principle at in nity was developed by Chabrowski [8], Bianchi, Chabrowski, Szulkin [6], Ben-Naoum, Troestler, Willem [5] and provides some quantitative information about the loss of mass of a sequence at in nity. Then Sζ * ∞ ≤ ω∞ and The next result is the concentration compactness principle at in nity for the critical Choquard equation, as proved by Gao et al. in [17].
and un u a.e. in R N . Let ω, ζ , and ν be the bounded nonnegative Radon measures, while let ω∞ and ζ∞ be the numbers given as in Lemmas . and . . Assume that Then there exists a nonnegative number ν∞ satisfying the relations

The Palais-Smale condition
In this section, we use the second concentration compactness principle and concentration compactness principle at in nity to prove that the (PS)c condition holds, when c < and < q < . We recall in passing that throughout the paper α and β in (1.1) are positive real parameters, without further mentioning.
Proof. Let (un)n be a sequence in D , (R N ) such that as n → ∞ Using the Hölder inequality and the Sobolev embedding theorem, we get for all u ∈ D , (R N ) This implies at once that (un)n is bounded in D , (R N ), since < µ < gives · * µ > and since < q < .
Proof. Let c < and let (un)n be a (PS)c sequence of for all n and all R > , with p ∈ [ , * ). Furthermore, by Proposition 1.202 of [14] there exist bounded nonnegative Radon measures ω, ζ and ν such that as n → ∞ in the sense of measure. Hence, by Lemma 2.3, there exist a at most countable set I, a sequence of points {z i } i∈I ⊂ R N and families of nonnegative numbers Therefore, as ε → we nally get On the other hand, the Hölder inequality yields Therefore, aω i ≤ βν i . Combining this with Lemma 2.3, we obtain that either We claim that the rst case can never occur. Otherwise, there exists i ∈ I such that , the Hölder inequality, the Sobolev embedding and the Young inequality imply that (3.5) According to this fact, we have Thus, for any β > , we choose α > so small that for every α ∈ ( , α ) the right-hand side of (3.6) is greater than zero, which is an obvious contradiction. Similarly, if α > is given, we take β > so small that for every β ∈ ( , β ) again the right-hand side of (3.6) is greater than zero. This gives the required contradiction. Consequently, ω i = for all i ∈ I in (3.4).
To obtain the possible concentration of mass at in nity, similarly, we de ne a cut o function ψ R in C ∞ (R N ) such that ψ R = in B R ( ), ψ R = in R N \ B R+ ( ), and |∇ψ R | ≤ /R in R N . On the one hand, the Hardy-Littlewood-Sobolev and the Hölder inequalities give On the other hand, the fact that J α,β (un), un ψ R → implies that Therefore aω∞ ≤Ĉβζ * µ * ∞ . Combining this with the Lemma 2.4, we obtain that either Therefore, as in (3.5) and (3.6), we have (3.8) Thus, for any β > , we choose α > so small that for every α ∈ ( , α ) the right-hand side of (3.8) is greater than zero, which is a contradiction. Similarly, if α > is given, we select β > so small that for every β ∈ ( , β ) the right-hand side of (3.8) is greater than zero. This gives the required contradiction. Therefore, ω∞ = in (3.7).
Then, for any c < and β > we have ω i = for all i ∈ I and ω∞ = for all α ∈ ( , Λ).
Similarly, for any c < and α > we again have ω i = for all i ∈ I and ω∞ = for any β ∈ ( , Λ).
Since ( un )n is bounded and J α,β (u) = , the weak lower semicontinuity of the norm and the Brézis-Lieb lemma yield as n → ∞ Thus (un)n strongly converges to u in D , (R N ). This completes the proof.

Proof of Theorem 1.1
In this section, we prove the existence of in nitely many solutions of (1.1) which tend to zero and we assume, without further mentioning, that all the assumptions of Theorem 1.1 hold. To this aim, we apply a new version of the symmetric mountain pass lemma, due to Kajikiya in [21, Theorem 1].

Lemma 4.1. Let E be an in nite-dimensional Banach space and J ∈ C (E). Suppose that the following properties hold. (J ) J is even, bounded from below in E, J( ) = and J satis es the local Palais-Smale condition.
(J ) For each n ∈ N there exists An ∈ Σn such that sup Then J admits a sequence of critical points (un)n such that J(un) ≤ , un ≠ for each n and (un)n converges to zero as n → ∞.
To obtain in nitely many solutions of (1.1), we need some technical lemmas. Let J α,β be the functional de ned in (2.
Clearly, h(t ) = = h(t ) and h(t * ) = = h(t * ). Following the same idea as in [19], we consider the truncated functional J α,β of J α,β , de ned for all u ∈ D , (R N ) by where ψ(u) = τ( u ) and τ : R + → [ , ] is a non-increasing C ∞ function such that τ(t) = if t ∈ [ , t ] and τ(t) = if t ≥ t . It is clear that J α,β ∈ C D , (R N ) and J α,β is bounded from below in D , (R N ). From the above arguments, recalling that all the assumptions of Theorem 1.1 hold, we have the next result.
Thus, for any u ∈ En, with u = ρ, we have where C and C are some positive constants, since all the norms are equivalent in the nite dimensional space En. Hence, J α,β (u) < provided that ρ > is su ciently small, being < q < . Therefore, As proved in the book [9] by Chang γ ({u ∈ En : u = ρ}) = n.
Choosing An = {u ∈ En : J α,β (u) < }, we have An ∈ n and sup u∈An J α,β (u) < . Therefore, all the assumptions of Lemma 4.1 are satis ed, since D , (R N ) is a real in nite Hilbert space. Thus, there exists a sequence (un)n in D , (R N ) such that J α,β (un) ≤ , un ≠ , J α,β (un) = for each n and un → as n → ∞.
Combining with Lemma 4.2 and taking n so large that un ≤ ρ is small enough, then these in nitely many nontrivial functions un are solutions of (1.1).

Proof of Theorem 1.2
In this section we study (1.1), when q = , < µ < and β = , and shall apply the mountain pass theorem for even functionals, in order to obtain a multiplicity result for (1.1). Actually, here (1.1) reduces to Clearly, the associated functional Jα to (5.1) is Then (un)n contains a strongly convergent subsequence.
Proof. The Hölder inequality and the Sobolev embedding theorem imply that for each u ∈ D , (R N ). Fix a (PS)c sequence (un)n for Jα in D , (R N ) at level c < c * . By the facts that α ∈ , aS k − L r , < µ < and by (5.2), proceeding as in proof of Lemma 3.2, in place of (3.6) we get which is impossible. Therefore, the compactness of the Palais-Smale sequence follows as in the proof of Lemma 3.2.
Now, let us recall a version of the mountain pass theorem for even functionals, which is the main tool for proving Theorem 1.2. For its proof readers are referred to [42]. From now on we assume that all the assumptions of Theorem 1.2 hold, without further mentioning. Proof. First, the fact that α ∈ ( , aS k − r ), the de nitions of S and S H,L yield Since < · * µ , there exists ϱ > such that Jα(u) ≥ ϱ for all u ∈ D , (R N ), with u = ρ, where ρ is chosen su ciently small. Thus, Jα satis es (I ).
Since α ∈ ( , aS k − r ), a direct consequence of Lemma 5.1 implies that Jα satis es (I ), with Let E be a nite dimensional subspace of D , (R N ). Thus, for any u ∈ E, with u large enough, by Lemma 2.2, we have for some positive constants c , c > , since all the norms on nite dimensional space are equivalent. Since < · * µ , we conclude that Jα(u) < for all u ∈ E, with u ≥ R, where R is chosen large enough. Consequently, Jα veri es (I ), as stated. Proof. The proof is similar to that presented in [46,Lemma 5]. From the de nition of c α n and the fact that k ≥ , k ≢ in R N , we deduce that Thus, for all α ∈ ( , aS k − r ) and a > a * , we get < c α ≤ c α ≤ · · · ≤ c α n < Mn < c * .
An application of Proposition 5.1 guarantees that the levels c α ≤ c α ≤ · · · ≤ c α n are critical values of Jα. Thus, if c α < c α < · · · < c α n , then the functional Jα has at least n critical points. Now, if c α j = c α j+ for some j = , , · · · , k − , again Proposition 5.1 implies that K c α j is an in nite set, see [42,Chapter 7], and so in this case, (5.1) has in nitely many solutions. Consequently, (5.1) has at least n pairs of solutions in D , (R N ), as stated.

Proof of Theorem 1.3
In this section we require that all the assumptions of Theorem 1.3 are satis ed. Thus, (1.1) becomes This case was investigated in [45,Theorem 1.1] in the fractional Laplacian context. For the convenience of the reader, we present a concise treatment. The aim of this section is to obtain two nontrivial solutions of (6.1). The rst is a least energy solution and the latter is a mountain pass solution. To begin with, let us introduce the functional Iα associated to (6.1) for all u ∈ D , (R N ). Since < q < * , ≤ µ < N and k ∈ L r (R N ), with r = * /( * − q), the Hardy-Littlehood-Sobolev inequality and the Sobolev inequality, show that Iα is well-de ned and of class C D , (R N ) . Next, we give a compactness result, which is crucial to prove Theorem 1.3.
Moreover, we easily deduce that (6.2) Put wn = un − u for all n. Without loss of generality, we assume that limn→∞ wn = . Theorem 2.3 of [40] in the subcase s = and p = , see also [16], yields Since (un)n is a (PS)c sequence, by the boundedness of (un)n, we have thanks to (6.2) In (6.3) we have used the weak convergence of (un)n in D , (R N ), which implies that Now, (6.3) yields as n → ∞ Thus, as n → ∞ Let us now recall the following well-known inequality, see [22]: for any p ≥ there holds for all s, t ∈ R. From the inequality (6.4) and the de nition of S H,L , we get as n → ∞ Letting n → ∞, we have When µ = and S − H,L < b, it follows from (6.5) that = , since · * µ = . Thus, un → u in D , (R N ). When µ > , it follows from (6.5) and the Young inequality that where b * is given in (1.3). Therefore, (b − b * ) ≤ . Hence, assumption (1.3) implies that = . In conclusion, un → u in D , (R N ) in both cases, as required.
Proof of Theorem 1.3. First, we show that (6.1) has a nontrivial least energy solution. Clearly, is well-de ned. Now we claim that there exists α * > such that m < for all α > α * . Indeed, x a function v ∈ D , (R N ), with v = and v k,q > , which is possible since k ≥ and k ≢ in R N . Then, for all α > α * , with α * = q a + b v q k,q . This proves the claim. Hence, by Lemma 6.1 and [31,Theorem 4.4], there exists u ∈ D , (R N ) such that Iα(u ) = m and I α (u ) = . Therefore, u is a nontrivial least energy solution of (6.1), with Iα(u ) < . Now we prove that (6.1) has a mountain pass solution. We deduce from (2.2) that u for all u ∈ D , (R N ). Since < q < * , there exists ρ > small enough and ϱ > such that Iα(u) > ϱ for all u ∈ D , (R N ), with u = ρ. De ne Iα(ξ (t)), Then c > . Lemma 6.1 yields that Iα satis es the assumptions of the mountain pass lemma, see [1, Theorem 2.1]. Hence, there exists u ∈ D , (R N ) such that Iα(u ) = c > and I α (u ) = . Thus, u is a nontrivial solution of (6.1), independent of u .