Ground states for a class of asymptotically periodic Schrödinger – Poisson systems with critical growth

The purpose of this paper is to study the existence of ground state solution for the Schrödinger–Poisson systems: { −∆u + V(x)u + K(x)φu = Q(x)|u|4u + f (x, u), x ∈ R3, −∆φ = K(x)u2, x ∈ R3, where V(x), K(x), Q(x) and f (x, u) are asymptotically periodic functions in x.


Introduction
For past decades, much attention has been paid to the nonlinear Schrödinger-Poisson system where h is the Planck constant.Equation (1.1) derived from quantum mechanics.For this equation, the existence of stationary wave solutions is often sought, that is, the following form of solution Ψ(x, t) = e it u(x), x ∈ R 3 , t ∈ R.
As far as we know, in [4], Azzollini and Pomponio firstly obtained the ground state solution to the Schrödinger-Poisson system (1.2).They obtained that system (1.2) has a ground state solution when V is a positive constant and 2 < q < 5, or V is non-constant, possibly unbounded below and 3 < q < 5. Since it's great physical interests, many scholars pay attention to study ground state solutions to the Schrödinger-Poisson system (1.2) and similar problems [1,8,11,12,14,15,20,37,38,45,46].
In [45], Zhang, Xu and Zhang considered existence of positive ground state solution for the following non-autonomous Schrödinger-Poisson system (1.4) In some weaken asymptotically periodic sense compare with that of in [1], they obtained the positive ground state solution to system (1.4) when V, K and f are all asymptotically periodic in x.
More recently, Zhang, Xu, Zhang and Du [46] completed the results obtained in [45] to Schrödinger-Poisson system with critical growth x ∈ R 3 . (1.5) In [46], V, K, Q satisfy: On the other hand, when K = 0 the Schrödinger-Poisson system (1.4) becomes the standard Schrödinger equation (replace The Schrödinger equation (1.6) has been widely investigated by many authors in the last decades, see [2, 6, 19, 24, 25, 29-31, 40, 41, 43] and reference therein.Especially, in [19,24,25,29,40,41], they studied the nontrivial solution or ground state solution for problem (1.6) with subcritical growth or critical growth in which V, f satisfy the asymptotically periodic condition.Other context about asymptotically periodic condition, we refer the reader to [18,21,35,36] and reference therein.Motivated by above results, in this paper, we will study ground state solutions to system (1.5) under reformative condition about asymptotically periodic case of V, K, Q and f at infinity.
The next theorem is the main result of the present paper.
(i) Functional sets A 0 in V, Q, K and A in ( f 5 ) were introduced by [24,25] in which Liu, Liao and Tang studied positive ground state solution to Schrödinger equation (1.6) with subcritical growth or critical growth.
(ii) Since F ⊂ A 0 , our assumptions on V, Q and K are weaker than [46].Furthermore, V(x) ≥ 0 in our paper but in [46] they assumed V(x) > 0.
(iii) In [46], to obtained the ground state to system (1.5), they firstly consider the periodic system −∆u Then a solution of system (1.5) was obtained by applying inequality between the energy of periodic system (1.7) and that of system (1.5).In this paper, we do not use methods of [46] and prove Theorem 1.1 directly.

The variational framework and preliminaries
To fix some notations, the letter C and C i will be repeatedly used to denote various positive constants whose exact values are irrelevant.B R (z) denotes the ball centered at z with radius R.
We denote the standard norm of L p by |u| p = ( R 3 |u| p dx) The Sobolev space H 1 (R 3 ) endowed with the norm The space D 1,2 (R 3 ) endowed with the standard norm ) and R 3 V(x)u 2 dx < ∞} be the Sobolev space endowed with the norm Lemma 2.1 ([24]).Suppose (V) holds.Then there exists two positive constants C 1 and C 2 such that H for all u ∈ E.Moreover, E → L p (R 3 ) for any p ∈ [2, 6] is continuous.The system (1.5) can be transformed into a Schrödinger equation with a nonlocal term.In fact, for all u ∈ E (then u ∈ H 1 (R 3 )), considering the linear functional L u defined in D 1,2 (R 3 ) by By the Hölder inequality, we have (2.1) Therefor, the Lax-Milgram theorem implies that there exists a unique Namely, φ u is the unique solution of −∆φ = K(x)u 2 .Moreover, φ u can be expressed as Substituting φ u into the systems (1.5), we obtain By (2.1), we get Then, we have (2.3) So the energy functional I : E → R corresponding to Eq. (2.2) is defined by where F(x, s) = s 0 f (x, t)dt.Moreover, under our conditions, I belongs to C 1 , so the Fréchet derivative of I is ) is unique solution of the following equation Moreover, φ u can be expressed as Let where F p (x, s) = s 0 f p (x, t)dt.Then I p is the energy functional corresponding to the following equation ) is a solution of periodic system (1.7) if and only if u ∈ E is a critical point of I p and φ = φ u .Lemma 2.2.Suppose (K) holds.Then, Proof.The proof is similar to that of in [27], so we omitted here.
Proof.Set h(x) := K(x) − K p (x).By (K), we have h(x) ∈ A 0 .Then for any ε > 0, there exists We cover R 3 by balls B 1 (y i ), i ∈ N. In such a way that each point of R 3 is contained in at most N + 1 balls.Without any loss of generality, we suppose that Like the argument of [45], we define By the Hölder inequality and the Sobolev embeddings, we have Let ε → 0, we obtain Q 11 → 0. By the condition u n 0, one has Let ε → 0, we have Q 2 → 0.Then, we get E 1 → 0. In the same way, we can prove E 2 → 0 and Then N is a Nehari type associate to I, and set c := inf u∈N I.
Lemma 2.8.Suppose that (V), (K), (Q) and ( f 1 )-( f 3 ) hold.For any u ∈ F, there is a unique t u > 0 such that t u u ∈ N .Moreover, the maximum of I(tu) for t ≥ 0 is achieved.
In fact, by ( f 1 ) and ( f 2 ), ∀δ > 0 there exists a C δ > 0 such that So, we get that Hence, g(t) > 0 for t small.On the other hand, let Θ = {x ∈ R 3 : u(x) > 0}, we have that Hence, it is easy to see that g(t) → −∞ as t → +∞.
Therefore, there exists a t u such that I(t u u) = max t>0 I(tu) and t u u ∈ N .Suppose that there exist t 1 > t 2 > 0 such that t 1 u, t 2 u ∈ N .Then, we have that Therefore, one has that which is absurd according to ( f 3 ) and t 1 > t 2 > 0.
Remark 2.9.As in [31,43], we have Lemma 2.10.Suppose that (V), (K), (Q) and ( f 1 )-( f 3 ) hold.Then there exists a bounded sequence Proof.From the proof of Lemma 2.8, it is easy to see that I satisfies the mountain pass geometry.By [33], there exists an {u n } such that I(u n ) → c and (1 . By ( f 3 ), we can obtain Then, we have that Therefor, {u n } is bounded and the proof is finished.
The proof of next lemma similar to that of [24,26].For easy reading, we give the proof.
Remark 2.12.For any u ∈ F, by Lemma 2.8, there exists t u > 0 such that t u u ∈ N and then I(t u u) ≥ c.Using V(x) ≤ V p (x), Q(x) ≥ Q p (x) and F(x, s) ≥ F p (x, s), we have c ≤ I(t u u) ≤ I p (t u u) ≤ max t>0 I p (tu).Then we obtain c ≤ c p .

Estimates
In this section, we will estimate the least energy c, and the method comes from the celebrated paper [7].Let In fact, S is the best constant for the Sobolev embedding D 1,2 (R 3 ) → L 6 (R 3 ).
Without loss of generality, we assume that x 0 = 0.For ε > 0, the function w ε : R 3 → R defined by is a family of functions on which S is attained.
Thus for ε small enough, one has {x:|x|<ε  Combining with F(x, s) ≥ 0 and the arbitrariness of R, we can obtain the claim.By (2.3) and (3.5), we get ≤ C 2 ε.

The proof of main result
The proof of Theorem 1.1.From Lemma 2.10, there exists a bounded sequence {u n } ∈ E satisfying I(u n ) → c and I (u n ) E −1 → 0. Then there exists u ∈ E such that, up to a subsequence, Using Remark 2.12, I p (w) = c p = c.By the properties of c and N , there exits t w > 0 such that t w w ∈ N .Thus, we obtain c ≤ I(t w w) ≤ I p (t w w) ≤ I p (w) = c.So c is achieved by t w w.By Lemma 2.11, we have I (t w w) = 0.
In a word, we obtain that Eq. (2.2) has a nonnegative ground state solution u ∈ E.

F
(x, t ε v ε )dx ≥ CR Since we are looking for a nonnegative solution, we may assume that f 1 p and |u| ∞ = ess sup x∈R 3 |u|.