Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent

: In this article, we consider the multiplicity of positive solutions for a static Schrödinger-Poisson-Slater equation of the type

( ), and ).Using Ekeland's variational principle and a measure representation concentration-compactness of Lions, when g has one local maximum point, we obtain two positive solutions for > μ 0 small; while g has k strict local maximum points, we prove that the equation has at least + k 1 distinct positive solutions for > μ 0 small by the Nehari manifold.Moreover, we show that one of the solutions is a ground state solution.

Introduction
In this article, we consider the following static Schrödinger-Poisson-Slater equation: where > μ 0, < < p 1 2 (6 is the critical exponent), and → f g , : 3  is continuous function satisfying: and its stationary counterpart is , in .
The interest on this problem stems from the Slater approximation of the exchange term in the Hartree-Fock model, see [27].Slater introduced the local term , and μ is the so-called Slater constant (up to renormalization).For more information on these models and their deduction, see [5,11,22].Recently, there has been much research on equation (1.2) by using variational methods in [5], the readers are referred to [1,2,19,23,24,26,29] and many others for detailed results.
2), then the equation reduces to the following static case: , in , which can be called a zero mass problem, see [6].H 1 3  ( ) is not the right space for problem (1.3) due to the absence of a phase term.Ruiz [25] introduced the following space: where the double integral expression is the so-called Coulomb energy of the wave and E 3 ( ) is the space of functions in 1,2 3  ( ) such that the Coulomb energy of the charge is finite.It was shown in the study by Ruiz [25] that E is a uniformly convex separable Banach space and that ↪ E L q 3 ( ) continuously for ∈ q 3, 6 [ ]. Ianni and Ruiz in [10] studied both the existence of ground and bound states, for > p 3 and proved that the problem has a radial solution, for = p 3. Furthermore, Liu et al. [20] considered the following type of the Schrödinger-Poisson-Slater equation with critical growth: they proved the existence of positive solutions to equation (1.4) by using the novel perturbation approach, together with the well-known Mountain-Pass theorem, for ∈ p 3, 6 ( )and by using the truncation technique, for ∈ p , 3  [3,12,34].In this article, we will discuss the existence of positive ground state solutions and the multiplicity of positive solutions for equation (1.1).The multiplicity of positive solutions for equation (1.1) is inspired by Liao et al. [15], which studied the following concave-convex elliptic equation with critical exponent: where )is an open bounded domain with smooth boundary, > < < μ q 0, 1 2, and = − 2* N N 2 2 is the critical Sobolev exponent, and the coefficient functions f and g satisfy the following conditions: (g 1 ) g is continuous on Ω and > g 0.
(g 2 ) There exist k points a a a , ,…, k . By the Nehari method, they proved that equation (1.5) has at least + k 1 positive solutions for > μ 0 small.For more problems on related results, see [7,9,13,14,17,21].In particular, Cao and Chabrowski [7] considered the multiplicity of solutions of this type for the critical problem for the first time.They studied the multiplicity of positive solutions for the following semilinear elliptic equation with critical exponent: where )is an open bounded domain with smooth boundary, ∈ f L Ω 2 ( ) is nonzero and nonne- gative, and ∈ g C Ω ( ) is positive which satisfies the following condition: x a i uniformly i.They obtained that equation (1.6) has at least k positive solutions for > μ 0 small.Our main results are the following.
inspired by [20,33], we adapt a measure representation concentration-compactness principle of Lions [18] to overcome the difficulties, and by exploring the parameter μ, we show that the associated energy functional satisfies, in general, the Palais-Smale condition at some level for > μ 0 small enough under our assumptions.Because the associated energy functional is not bounded below, we apply Ekeland's variational principle to obtain that equation (1.1) has at least + k 1 positive solutions by the Nehari manifold. Notations: ) denotes the open ball with center x and radius R in 3 .
The article is organized as follows: in Section 2, we give preliminary results; in Section 3, we give the proof of Theorem 1.1; and in Section 4, we prove Theorem 1.2.

Preliminary
Noting that the function is an extremal function for the minimum problem (2.1).For each > ε 0, is a positive solution of the critical problem Then, we have the following properties.
Proposition 2.1.[10,20,25] ) is a uniformly convex Banach space.Moreover, ∞ C 0 3 , then ∈ u E if and only if both u and ϕ u belong to 1,2 3 ( ).In such case, equation (1.1) can be rewritten as a system in the following form: then T is a continuous map, linear in each variable, and we have the following technical results in E, see [10,20,25].
For the sake of brevity, let us define → M E : as Similar to [10,20], we have the fact that for any ∈ u E, The corresponding energy functional of equation (1.1) is defined as follows: for all ∈ u E.Moreover, J μ is well-defined and And the critical points of J μ are the weak solutions of equation (1.1).Since J μ is unbounded below on E, we consider the functional on the Nehari manifold Adopting the method used in [30], we split μ into three parts: Now, we give some conclusions of the energy functional J μ on μ .
Multiple positive solutions for a Schrodinger-Poisson-Slater equation with critical exponent  5 Lemma 2.1.J μ is coercive and bounded from below on μ .
Proof.For ∈ u μ , by the Young and Hölder inequalities, we deduce from Proposition 2.3 that , which implies that J μ is coercive and bounded from below on μ due to < < p 1 2.This completes the proof of Lemma 2.1. □ Next, we define Then, we have the following lemma.
), then there exist unique + t and − t with ), i.e., m t ( ) achieves its maximum at t max and t max is unique.Let Then, by the Sobolev embedding theorem, it holds that and by simple computations, one has , then by the Sobolev embedding theorem and equation (2.10), for all and thus, there exist unique + t and − t with Similar to Case i, we can conclude that Then, the proof of Lemma 2.2 is complete.
Proof.(i) Assuming the contrary, there exist , for all 0, , This contradicts equation (2.11).Thus, Then, we have the following conclusion.
(ii) There exists a constant = > c c p S f μ , , , 0 Thus, there exists a constant = > c c p S f μ , , , 0 , for all ∈ μ μ 0, 0 ( ) ͠ .This completes the proof of Lemma 2.4.□ 3 Proof of Theorem 1.1 In this section, we give the proof of Theorem 1.1.We need to show that J μ satisfies the Palais-Smale condition at some level, for ∈ μ μ 0, * ( ) with some > μ * 0. To get compactness of the bounded PS ( )-sequence in E, we recall the well-known concentration-compactness principle of Lions [18]., where δ xj is the Dirac measure at point x j .
Multiple positive solutions for a Schrodinger-Poisson-Slater equation with critical exponent  9 To study the concentration at infinity of the sequence, we recall the following quantities: Lemma 3.2.[18] Let u n { } be a sequence weakly converging to u in 1,2 3 ( ) and define Then, the quantities ∞ ν and ∞ μ ˜are well defined and satisfy , where μ ˜and ν are defined in Lemma 3.1.
For detailed proofs of Lemmas 3.1 and 3.
Applying Lemmas 3.1 and 3.2, the following proof is almost identical to that of Lemma 2.6 of [33] and is omitted here for brevity.Then, the proof of Lemma 3.3 is complete.□ By Ekeland's variational principle [31] and using the same argument as in [8], we have the following lemma.
(ii) There exists a ), then there exists which is impossible.We obtain that ∈ + u μ μ and then ( ) , and it follows that (ii) We need to show that ).Since J μ is even, we can assume that ≥ u 0 μ , applying strong maximum principle, u μ is a positive ground state solution of equation (1.1).
(iii) By the Sobolev embedding theorem and Hölder inequality, it holds that ( ) be a radially symmetric function with , and where ρ 0 is a positive constant, when ), for all > ε 0, where U ε is given in equation (2.2).From the argument in [20,28], one has uniformly in i, as → + ε 0 , the last inequality holds true because the classic sharp Hardy-Littlewood-Sobolev inequality in [16].Then, we have the following results.Lemma 3.5.
Thus, one has For the arbitrariness of η, combining with equation (3.2), one obtains This completes the proof of Lemma 3.5.□ Lemma 3.6.There exists > ε 0 0 small enough such that for < < ε ε 0 0 , we have where u μ and μ * are given in Proposition 3.1 and Lemma 3.3, respectively.
Proof.By equation (2.4) and the Hölder's inequality, one has be a constant, then using the following two elementary inequalities: , for , 0,1 2; , for 2,0 , 1, and M 0 are two positive constants.By basic calculation, for each ≥ t 0, it holds that ( ) are bounded for all > ε 0 as follows: , as > t 0 small enough, uniformly for all ε and i.Thus, there exists > T 0 We have the following two cases.
, as → ε 0, and we can conclude that for all 0 small enough .
The proof of Lemma 3.5 is finished.□ Lemma 3.7.There exists , where ε 0 is given in Lemma 3.6.
Hence, there is a ( ) , so there exists Then, the proof of Lemma In this section, we prove Theorem 1.2.We define { } be a barycenter map defined as follows: For each ≤ ≤ i k 1 , we define , where ε ˜0 is given in Lemma 3.7.
and each 1 .
This completes the proof of Lemma 4.1.
and we have the following lemma.
. By the definition of S, we have ≥ ≥ l l Sl .Then, we have ( ) .
On the other hand, similar to Lemma 2.2, there exists Moreover, we also conclude that t ε i ,0 is uniformly bounded as the proof of Lemma 3.6.Applying equations (3.1), (3.2), and (3.4) and Lemma 3.5, one has One deduces from Lemma 2.1 that there exists a constant > C 0

Lemma 3 . 1 . [ 18 ] 2 | | weakly converges in 3 ( ) to a nonnegative measure μ ˜, (ii) u n 6 | | weakly converges in 3 (
Let u n { } be a sequence weakly converging to u in1,2 3  ( ).Then, up to subsequences, (i) ∇u n ) to a nonnegative measure ν, and there exist an at most countable index set I , a family ∈ x j I : j { }of distinct points of 3 , and families ∈ ν j I : j { } of positive numbers such that
Multiple positive solutions for a Schrodinger-Poisson-Slater equation with critical exponent  19 Proof of Theorem 1.2.By Lemma 4.7, there exists a PS β μ . We deduce from equation (4.3) and Lemma 3.3 that J μ has at least k distinct nontrivial critical points in − μ .If we consider