Infinitely many small solutions for a sublinear Schrödinger–Poisson system with sign-changing potential

Computers and Mathematics with Applications 71 (2016) 2082–2088

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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa

Infinitely many small solutions for a sublinear Schrödinger–Poisson system with sign-changing potential Gui Bao School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong 264025, PR China

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Article history: Received 18 November 2015 Received in revised form 23 March 2016 Accepted 2 April 2016 Available online 23 April 2016 Keywords: Sublinear Schrödinger–Poisson system Dual approach Small solutions

In this paper, we study the existence of infinitely many small solutions for a class of sublinear Schrödinger–Poisson system with sign-changing potential. By using a dual approach, we prove that the problem has infinitely many small solutions. As a main novelty with respect to some previous results, we do not require any growth condition on the nonlinear term. © 2016 Published by Elsevier Ltd.

1. Introduction This paper is concerned with the existence of infinitely many solutions for a quasilinear Schrödinger equation of the form

 −1u + V (x)u + φ u = f (x, u), −1φ = u2 , lim φ(x) = 0, |x|→∞

x ∈ R3 , x ∈ R3 .

(1.1)

In recent years, system (1.1) or similar system has been widely studied under various conditions on f . By using variational methods, some interesting results were obtained; see, for example, [1–17]. In [9], Ruiz considered the following problem

 −1u + u + λφ u = |u|p−2 u, −1φ = u2 , lim φ(x) = 0, |x|→∞

in R3 , in R3 ,

(1.2)

and gave existence and nonexistence results for 2 < p < 6 and λ > 0. Azzollini and Pomponio [1] proved the existence of a ground state solution of (1.1) when f (u) = |u|p−2 u and p ∈ (3, 6). Lv [6] obtained the existence and multiplicity of solutions to (1.1) for a sublinear term f , by using the minimizing theorem and the dual fountain theorem, respectively. In [11], Sun proved the multiplicity of solutions to (1.1) for a sublinear term f . See also [7,12] for a sublinear term f . We should point out that in the above mentioned papers, some kind of coercive conditions at infinity are needed. In the present paper, without imposing any growth condition on the nonlinear term, we prove that (1.1) has infinitely many small solutions in R3 . Moreover, unlike [6,11], the potential V (x) in this paper is allowed to be sign-changing. We introduce the following hypotheses for the problem (1.1).

(V) V ∈ C (R3 , R), infx∈R3 V (x) > −∞ and for each M > 0, meas{x ∈ R3 , V (x) ≤ M } < +∞, where meas denotes the

Lebesgue measure in R3 . (F1 ) limt →0 f (xt,t ) = +∞ uniformly for x ∈ R3 . E-mail address: [email protected]. http://dx.doi.org/10.1016/j.camwa.2016.04.006 0898-1221/© 2016 Published by Elsevier Ltd.

G. Bao / Computers and Mathematics with Applications 71 (2016) 2082–2088

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(F2 ) There exist constants δ1 > 0 and 1 < r1 < 2 such that f ∈ C (R3 × [−δ1 , δ1 ], R) and |f (x, t )| ≤ a(x)|t |r1 −1 ,

|t | ≤ δ1 , ∀x ∈ R3 ,

2

where a(x) ∈ L 2−r1 (R3 ) is a positive continuous function. (F3 ) There exists a constant δ2 > 0 such that f (x, −t ) = −f (x, t ) for all |t | ≤ δ2 and all x ∈ R3 . Denote F (x, u) =

u 0

f (x, s)ds. Our main result for (1.1) is the following theorem:

Theorem 1.1. Suppose that (V) and (F1 )–(F3 ) hold. Then the problem (1.1) has infinitely many nontrivial solutions (uk , φk ) satisfying 1 2

 R3

(|∇ uk |2 + V (x)u2k )dx −

1



4

|∇φk |2 dx + R3

1



2

R3

φk u2k dx −

 R3

F (x, uk )dx ≤ 0

and uk → 0 as k → ∞. Throughout this paper, C > 0 denote various positive constants which are not essential to our problem and may change from line to line. 2. Preliminaries and useful lemmas From (V), we know that there exist constants V0 > 0 and σ > 0 such that V˜ (x) := V (x) + V0 > σ for all x ∈ R3 . Let f˜ (x, u) = f (x, u) + V0 u and consider the following new equation

 −1u + V˜ (x)u + φ u = f˜ (x, u), −1 φ = u 2 , lim φ(x) = 0, |x|→∞

x ∈ R3 , x ∈ R3 .

(2.1)

Then problem (1.1) is equivalent to the problem (2.1). In the following, we just need to study the equivalent problem (2.1). Let Ω ⊆ R3 and Lp (Ω ), 1 ≤ p ≤ +∞ be a Lebesgue space, the norm in Lp (Ω ) is denoted by | · |p,Ω . Let H01 (Ω ), Ω ⊂ R3 , and H 1 (R3 ) denote the usual Sobolev spaces. Set



 (|∇ u|2 + V˜ (x)u2 )dx < +∞ , Ω    E = E (R3 ) = u ∈ H 1 (R3 ) : (|∇ u|2 + V˜ (x)u2 )dx < +∞

E (Ω ) =

u ∈ H01 (Ω ) :



R3

with the norms

 ∥u∥Ω = ∥u∥E ,Ω =

Ω

 ∥u∥ = ∥u∥E ,R3 =

 12 (|∇ u|2 + V˜ (x)u2 )dx ,

 21 2 ˜ (|∇ u| + V (x)u )dx , 2

R3

respectively. Define the function space D1,2 (R3 ) := u ∈ L2 : ∇ u ∈ (L2 (R3 ))3 ∗



with the norm ∥u∥D1,2 =



R3



 21 |∇ u|2 dx .

By the continuity of the embeddings E (Ω ) ↩→ Lr (Ω ), r ∈ [2, 2∗ ], Ω ⊆ R3 and D1,2 (R3 ) ↩→ L2 (R3 ), there exist constants τr > 0, 2 ≤ r ≤ 2∗ and τ∗ such that |u|r ,Ω ≤ τr ∥u∥E ,Ω , ∀u ∈ E (Ω ), and |u|2∗ ,R3 ≤ τ∗ ∥u∥D1,2 , ∀u ∈ E. Moreover, similar to Lemma 3.4 in [18], we may prove that under the assumption (V), the embedding E (Ω ) ↩→ Lr (Ω ) is compact for 2 ≤ r < 2∗ , Ω ⊆ R3 . Let 0 < l ≤ 12 min{δ1 , δ2 , 1}. We define an even function h ∈ C 1 (R, R+ ) such that 0 ≤ h(t ) ≤ 1, h(t ) = 1 for |t | ≤ l; ∗

h(t ) = 0 for |t | ≥ 2l and h is decreasing in [l, 2l]. Let fh (x, u) = f˜ (x, u)h(u). Consider the cutoff functional Ih : Ih (u, φ) =

1 2

∥ u∥ 2 −

1 4



|∇φ|2 dx + R3

1 2

 R3

φ u2 dx −

 R3

Fh (x, u)dx.

The critical points of Ih are weak solutions of the problem

 −1u + V˜ (x)u + φ u = fh (x, u), −1φ = u2 , lim φ(x) = 0, |x|→∞

x ∈ R3 , x ∈ R3 .

(2.2)

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G. Bao / Computers and Mathematics with Applications 71 (2016) 2082–2088

Then u ∈ E, satisfies |u| ≤ l, is a critical point of the functional Ih , u is a weak solution of (1.1). It is easy to know that Ih exhibits a strong indefiniteness. This indefiniteness can be removed by using the reduction method described in [19]. Now we recall this method. For any u ∈ E, consider the linear functional Tu : D1,2 (R3 ) → R defined as Tu (v) =



u2 v dx.

R3

By the Hölder’s inequality, we have

 R3

u2 v dx ≤ |u2 | 6 ,R3 |v|6,R3 ≤ |u|212 5

5

1 ,2

So, Tu is continuous on D

µ(u, v) =



,R3

|v|6,R3 ≤ τ 212 τ∗ ∥u∥2 ∥v∥D1,2 . 5

(R ). Set 3

∇ u · ∇v dx,

R3

for all u, v ∈ D1,2 (R3 ). Obviously, µ(u, v) is bilinear, bounded and coercive. Hence, the Lax–Milgram theorem implies that for every u ∈ E, there exists a unique φu ∈ D1,2 (R3 ) such that Tu (v) = µ(φu , v) for any v ∈ D1,2 (R3 ), that is

 R3



u2 v dx =

R3

∇φu · ∇v dx.

Using integration by parts, we have



 ∇φu · ∇v dx = − R3

R3

v 1φu dx,

for any v ∈ D1,2 (R3 ), therefore

− 1φu = u2

(2.3)

in a weak sense. We can write an integral expression for φu in the form

φu =

u2 (y)



1 4π

|x − y|

R3

dy

for any u ∈ E. Clearly, φu ≥ 0 and φ−u = φu for all u ∈ E. It follows from (2.3) that

 R3

φu u2 dx =

 R3

φu (−1φu )dx =

 R3

|∇φu |2 dx,

(2.4)

and by the Hölder’s inequality, we have 2 u D1,2

∥φ ∥



φu u dx ≤ 2

= R3

 R3

φ

6 u dx

 61 

 56 12 |u| 5 dx

R3

≤ τ∗ ∥φu ∥D1,2 |u| 12 ,R3 2

5

and it follows that

∥φu ∥D1,2 ≤ τ∗ |u|212 ,R3 .

(2.5)

5

Hence,

 R3

φu u2 dx ≤ τ∗2 |u|412 ,R3 ≤ τ∗2 τ 412 ∥u∥4 := C ∥u∥4 . 5

(2.6)

5

So, we can consider the functional Φ : E → R defined by Φ (u) = I (u, φu ). By (2.4), the reduced functional takes the form

Φ ( u) =

1 2

∥u∥2 +

1 4

 R3

φu u2 dx −

By (F2 ), we have

|Fh (x, u)| ≤

a(x) r1

|u|r1 + V0 u2

 R3

Fh (x, u)dx.

G. Bao / Computers and Mathematics with Applications 71 (2016) 2082–2088

for all x ∈ R3 and |u| ≤ 2l, where r1 ∈ (1, 2) and a(x) ∈ L

 R3

Fh (x, u)dx ≤

a(x)



r1

|x|<2l

≤ ≤

1 r1 1 r1 1 r1

|u|r1 dx +

(R3 ). Then we obtain

V0 u2 dx

|a(x)|

2 2−r1

 2−2r1 

2

|u| dx

dx

|x|<2l

|a(x)|

2085

|x|<2l



≤



2 2−r1

 r21

 + V0

|x|<2l

2 2−r1

r1

+ V0 |u|22,R3

,R3 |u|2,R3

r

τ21 |a(x)|

2 2−r1

,R3 ∥u∥

u2 dx

|x|<2l

r1

+ V0 τ22 ∥u∥2 .

Combining this with (2.6), we get that Φ is well defined. Furthermore, it is well known that Φ is a C 1 functional with derivative given by

⟨Φ ′ (u), v⟩ =

   ∇ u · ∇v + V (x)uv + φu uv − fh (x, u)v dx. R3

It is easy to verify that (u, φu ) ∈ E × D1,2 (R3 ) is a solution of problem (1.1) if and only if u ∈ E is a critical point of the functional Φ . Let Γk denote the family of closed symmetric subsets A of E such that 0 ̸∈ A and the genus γ (A) ≥ k. The following critical point theorem was established in [20]. Lemma 2.1. Let E be an infinite dimensional Banach space and I ∈ C 1 (E , R) satisfy (A1 ) and (A2 ) below.

(A1 ) I (u) is even, bounded from below, I (0) = 0 and I (u) satisfies the Palais–Smale condition (PS). (A2 ) For each k ∈ N, there exists an Ak ∈ Γk such that supu∈Ak I (u) < 0. Then I (u) admits a sequence of critical points uk such that I (uk ) ≤ 0, uk ̸= 0 and limk→∞ uk = 0. 3. Proof of main results Lemma 3.1. Φ is bounded from below and satisfies the (PS) condition. Proof. By (V), (F2 ) and the definition of h, we have

|Fh (x, v)| ≤

a( x ) r1

|v|r1 + V0 v 2 ,

∀(x, v) ∈ (R3 , R).

(3.1)

For any given v ∈ E. Let Ω = {x ∈ R3 : |v| ≤ 1}. By (3.1) and Hölder’s inequality, one has

Φ (v) =

≥ ≥ ≥ ≥ ≥

1 2 1 2 1 2 1 2 1 2 1 2

∥v∥2 + ∥v∥2Ω + ∥v∥2Ω − ∥v∥2Ω − ∥v∥2Ω −



1

4 R3  1 4

R3

φv v 2 dx −

φv v 2 dx −

  a(x) Ω 

r1 a(x)

Ω

r1

1 r1



|a(x)|

R

3

Ω

Fh (x, v)dx Fh (x, v)dx

 |v|r1 + V0 |v|2 dx  |v|r1 + V0 |v|r1 dx

2 2−r1

r1

,Ω |v|2,Ω

r

− V0 |v|r11 ,Ω

r

∥v∥2Ω −

τ21 r1

|a(x)|

2 2−r1

r1

,R3 ∥v∥Ω

r

− V0 τrr11 ∥v∥Ω1 .

(3.2)

Note that r1 ∈ (1, 2), (3.2) implies that Φ is bounded from below. Next, we prove that Φ satisfies the (PS) condition. Let {vn } ⊂ E be any (PS) sequence of Φ , i.e., {Φ (vn )} is bounded and Φ ′ (vn ) → 0 in E ∗ . For each n ∈ N, set Ωn = {x ∈ R3 : |vn | ≤ 1}. Then by (3.2), we have C ≥ Φ (vn ) ≥

1 2

r

∥vn ∥2Ωn −

τ21 r1

|a(x)|

2 2−r1

r1

,R3 ∥vn ∥Ωn

r

− V0 τrr11 ∥vn ∥Ω1n .

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G. Bao / Computers and Mathematics with Applications 71 (2016) 2082–2088

This implies that ∥vn ∥Ωn ≤ C and C is independent of n. Thus, 1



2

Ωn

|∇vn |2 dx +

1



2

Ωn

V˜ (x)vn2 dx +

1



4

Ωn

φvn vn2 dx ≤ Φ (vn ) +

 Ωn

Fh (x, vn )dx

r

≤C+

τ2 1 r1

|a(x)|

2 2−r1

r1

,R3 ∥vn ∥Ωn

r

+ V0 τrr11 ∥vn ∥Ω1n

≤ C,

(3.3)

where C is independent of n. Similarly, 1

Φ (vn ) =

2 1

≥

2



1

2

|∇vn | dx +

2

R3

 R3 \Ωn

 R3

1

|∇vn |2 dx +

2

1 V˜ (x)vn2 dx + 4

 R3 \Ωn



φ v 

2 vn n dx

R3

V˜ (x)vn2 dx +

 − Ωn

φvn vn2 dx −



≤ Φ (vn ) +



1

4

Fh (x, vn )dx

R3 \Ωn

Ωn

Fh (x, vn )dx.

Therefore, 1



2

2

R3 \Ωn

|∇vn | dx +

1



2

R3 \Ωn

1 V˜ (x)vn2 dx +



4

R3 \Ωn

φ v

2 vn n dx

Ωn

Fh (x, vn )dx

r

≤C+

τ2 1 r1

|a(x)|

2 2−r1

r1

,R3 ∥vn ∥Ωn

r

+ V0 τrr11 ∥vn ∥Ω1n

≤ C,

(3.4)

where C is independent of n. Then from (3.3) and (3.4), we get that D2n :=

1 2



|∇vn |2 dx + R3

1



2

R3

V˜ (x)vn2 dx +

1



4

R3

φvn vn2 dx

is bounded independent of n.   From [21], we know that there exists constant C > 0 such that D2n ≥ 21 R3 |∇vn |2 dx + 12 R3 V˜ (x)vn2 dx ≥ C ∥vn ∥2 . Then {vn } is bounded in E by the above arguments. Thus, up to a subsequence, we have vn ⇀ v in E, vn → v in Lp (R3 ) for 2 ≤ p < 2∗ and vn → v a.e. on R3 . By (F2 ) and Hölder’s inequality, one has

         r1 r1  ≤C f ( x , v ) − f ( x , v) (v − v) dx | a ( x )| |v | + V |v | + | a ( x )| |v| + V |v| |(vn − v)|dx h n h n n 0 n 0  3  R R3  r ≤ C |a(x)| 2 ,R3 |vn |21,R3 + V0 |vn |2,R3 2−r1  r + |a(x)| 2 ,R3 |v|21,R3 + V0 |v|2,R3 |vn − v|2,R3 2−r1

= on (1).

(3.5)

On the other hand, by the Hölder’s inequality and (2.5), we get that

 R3

φvn vn (vn − v)dx ≤ |φvn vn |2,R3 |vn − v|2,R3 ≤ |φvn |6,R3 |vn |3,R3 |vn − v|2,R3 ≤ τ∗ ∥φvn ∥D1,2 |vn |3,R3 |vn − v|2,R3 ≤ τ∗2 |vn |212 ,R3 |vn |3,R3 |vn − v|2,R3 5

≤ τ∗2 τ 212 τ3 τ2 ∥vn ∥3 ∥vn − v∥ 5

≤ ∞. Then by Lebesgue’s convergence theorem, we have

 R3

φvn vn (vn − v)dx → 0,

n → ∞.

It is obvious that

 R3

φv v(vn − v)dx → 0,

n → ∞.

(3.6)

G. Bao / Computers and Mathematics with Applications 71 (2016) 2082–2088

2087

This together with (3.6) implies

 R3

(φvn vn − φv v)(vn − v)dx → 0,

n → ∞.

(3.7)

From (3.5) and (3.7), we have on (1) = ⟨Φ (vn ) − Φ (v), vn − v⟩ = ∥vn − v∥ + ′

′

2

−

 R3

(φvn vn − φv v)(vn − v)dx

  R3



fh (x, vn ) − fh (x, v) (vn − v)dx

= ∥vn − v∥2 + on (1). Hence vn → v in E. The proof is completed.



By a similar argument as in [22], we have Lemma 3.2. For any n ∈ N, there exists a closed symmetric subsets An ⊂ E such that the genus γ (An ) ≥ n and supv∈An Φ (v) < 0. Proof. Let En be any n-dimensional subspace of E. Since all norms are equivalent in a finite dimensional space, there is a constant α = α(En ) such that

∥v∥ ≤ α|v|2,R3 for all v ∈ En . Claim. There exists a constant κ > 0 such that 1



2



|v|2 dx ≥ R3

|v|2 dx

(3.8)

|v|>l

for all v ∈ En with ∥v∥ ≤ κ . Indeed, if (3.8) is false, there exists a sequence {vk } ⊂ En \ {0} such that vk → 0 in E and 1 2



|vk |2 dx <

R3



|vk |2 dx,

k ∈ N.

|vk |>l

v

Let uk = |v | k . Then k 2,R3 1 2

<



|uk |2 dx,

k ∈ N.

(3.9)

|vk |>l

On the other hand, we can assume that uk → u in E since En is finite dimensional. Hence uk → u in L2 (R3 ). Moreover, it can be deduced from vk → 0 in E that meas{x ∈ R3 : |vk | > l} → 0,

k → ∞.

Therefore,



|uk |2 dx ≤ 2



|uk − u|2 dx + 2



R3

|vk |>l

|u|2 dx → 0,

k → ∞,

|vk |>l

which contradicts (3.9) and hence (3.8) holds. By (F1 ), we can choose l small enough such that f (x, v) ≥ 8

1 2

 1 + τ∗2 τ 412 α 2 v 4

5

for all x ∈ R and 0 ≤ v ≤ 2l. This inequality implies that 3

1

Fh (x, v) = F (x, v) ≥ 4

2

 1 + τ∗2 τ 412 α 2 v 2 , 4

5

∀(x, v) ∈ R3 × [0, l].

The assumption (F3 ) implies Fh (x, v) is even in v . Thus, by (3.10), we have

 φv v 2 dx − Fh (x, v)dx 2 4 R3 R3  1 1 ≤ ∥v∥2 + τ∗2 τ 412 ∥v∥4 − Fh (x, |v|)dx

Φ (v) =

1

2

∥v∥2 +

1

4



5

|v|≤l

(3.10)

2088

G. Bao / Computers and Mathematics with Applications 71 (2016) 2082–2088

  1 + τ∗2 τ 412 α 2 |v|2 dx 2 4 2 4 5 5 |v|≤l   1 1   1 1 2 2 2 2 4 2 4 |v| dx − ≤ + τ∗ τ 12 ∥v∥ − 4 + τ∗ τ 12 α

≤

1

1

∥v∥2 + τ∗2 τ 412 ∥v∥2 − 4

2

4

1

2

5

4

5

R3

|v|2 dx



|v|>l

1 1  ≤− + τ∗2 τ 412 ∥v∥2 , 2

4

5

for all v ∈ En with ∥v∥ ≤ min{κ, 1}. Let 0 < ρ ≤ min{κ, 1} and An = {v ∈ En : ∥v∥ = ρ}. We conclude that γ (An ) ≥ n and supv∈An Φ (v) ≤

−



1 2

 + 14 τ∗2 τ 412 ρ 2 < 0. The proof is completed.



5

Proof of Theorem 1.1. By (F1 ) and (F3 ), we get that Φ is even and Φ (0) = 0. Then from Lemmas 2.1, 3.1 and 3.2 we obtain that Φ has a critical sequence {vn } converging to 0. By a similar argument as Lemma 3.3 in [22], we may get that there exists n1 such that |vn |∞,R3 ≤ l for n ≥ n1 . Then we get infinitely many small solutions of (1.1). The proof is completed.  Acknowledgments This work is supported by the Research Award Fund for the Natural Science Foundation of Shandong Province, Grant No. ZR2015PA009. The authors wish to thank the anonymous reviewer so very much for his/her valuable suggestions and comments. References [1] A. Azzollini, A. Pomponio, Ground state solutions for the nonlinear Schödinger–Maxwell equations, J. Math. Anal. Appl. 45 (1) (2008) 90–108. [2] A. Ambrosetti, D. Ruiz, Multiple bound states for the Schrödinger–Poisson problem, Commun. Contemp. Math. 10 (2008) 391–404. [3] D. Bonheure, C. Mercuri, Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials, J. Differential Equations 251 (2011) 1056–1085. [4] Y. Jiang, H. Zhou, Bound state for a stationary nonlinear Schrödinger–Poisson system with sign-changing potentional in R3 , Acta Math. Sci. 29B (4) (2009) 1095–1104. [5] F. Li, Q. Zhang, Existence of positive solutions to the Schrödinger–Poisson systems without compactness conditions, J. Math. Anal. Appl. 401 (2013) 754–762. [6] Y. Lv, Existence and multiplicity of solutions for a class of sublinear Schrödinger-Maxwell equations, Bound. Value Probl. 2013 (1) (2013) 1–22. [7] C. Liu, Z. Wang, H. Zhou, Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differential Equations 245 (2008) 201–222. [8] C. Mercuri, Positive solutions of nonlinear Schrödinger–Poisson systems with radial potentials vanishing at infinity, Rend. Lincei Sci. Fis. Nat. 19 (2008) 211–227. [9] D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2) (2006) 655–674. [10] J. Sun, H. Chen, J.J. Nieto, On ground state solutions for some non-autonomous Schröinger-Poisson systems, J. Differential Equations 252 (2012) 3365–3380. [11] J. Sun, Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations, J. Math. Anal. Appl. 390 (2) (2012) 514–522. [12] Y. Ye, C.L. Tang, Existence and multiplicity of solutions for Schrödinger–Poisson equations with sign-changing potential, Calc. Var. Partial Differential Equations 53 (1–2) (2015) 383–411. [13] L. Zhao, H. Liu, F. Zhao, Existence and concentration of solutions for the Schrödinger–Poisson equtations with steep well potential, J. Differential Equations 255 (2013) 1–23. [14] G. Cerami, G. Vaira, Positive solutions for some non-autonomous Schrödinger–Poisson systems, J. Differential Equations 248 (2010) 521–543. [15] I. Ianni, G. Vaira, On concentration of positive bound states for the Schrödinger–Poisson problem with potentials, Adv. Nonlinear Stud. 8 (2008) 573–595. [16] J. Sun, T.F. Wu, Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system, J. Differential Equations 260 (2016) 586–627. [17] J. Sun, T.F. Wu, On the nonlinear Schrödinger–Poisson systems with sign—changing potential, Z. Angew. Math. Phys. 66 (2015) 1649–1669. [18] W.M. Zou, M. Schechter, Critical Point Theory and Its Applications, Springer, NewYork, 2006. [19] V. Benci, D. Fortunato, An eigenvalue problem for the Schödinger-Maxwell equations, J. Juliusz Schauder Cent. 11 (1998) 283–293. [20] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005) 352–370. [21] X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations 256 (2014) 2619–2632. [22] F. Zhou, K. Wu, Infinitely many small solutions for a modified nonlinear Schrödinger equations, J. Math. Anal. Appl. 411 (2014) 953–959.