Multiple positive solutions of a class of Schrödinger–Poisson equation involving indefinite nonlinearity in R3

Applied Mathematics Letters 93 (2019) 111–116

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Applied Mathematics Letters www.elsevier.com/locate/aml

Multiple positive solutions of a class of Schrödinger–Poisson equation involving indefinite nonlinearity in R3 Wenbin Gan ∗, Shibo Liu 1 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

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Article history: Received 20 November 2018 Received in revised form 24 January 2019 Accepted 24 January 2019 Available online 5 February 2019

abstract In this paper, we study the existence of multiple positive solutions of Schrödinger– Poisson type equations with indefinite nonlinearity. Our main tool is the mountain pass theorem. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Poisson type equation Positive solution Mountain pass theorem

1. Introduction In this paper, we consider the following Schr¨ odinger–Poisson equation { −∆u + u + µϕu = a(x) | u |p−2 u + λk(x)u, u ∈ H 1 (R3 ), −∆ϕ = u2 , ϕ ∈ D1,2 (R3 ),

(1.1)

where λ, µ > 0, 2 < p < 4. k(x) is a positive function in R3 , a(x) ∈ C(R3 ) and a(x) changes sign, that is why we call it indefinite nonlinearity in the title. According to a classical model, the interaction of a charge particle in an electromagnetic field can be described by coupling the nonlinear Schr¨ odinger and the Poisson’s equations. In recent years, there are a lot of works dealing with the following Schr¨ odinger–Poisson systems { −∆u + V (x)u + l(x)ϕu = f (x, u), u ∈ H 1 (R3 ), (1.2) −∆ϕ = l(x)u2 , ϕ ∈ D1,2 (R3 ), p−1

It was first introduced by Benci and Fortunato in [1]. Under V (x) = 1 and f (x, u) = |u| u, 1 < p < 5, Eq. (1.2) has been widely studied, e.g. [2–5]. For 3 ≤ p < 5, Coclite [4] studied the existence of positive radial solution for Eq. (1.2) by the mountain pass theorem. ∗ 1

Corresponding author. E-mail addresses: [email protected] (W. Gan), [email protected] (S. Liu). Supported by National Natural Science Foundation of China (No.11671331).

https://doi.org/10.1016/j.aml.2019.01.032 0893-9659/© 2019 Elsevier Ltd. All rights reserved.

W. Gan and S. Liu / Applied Mathematics Letters 93 (2019) 111–116

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The non-radial solution was considered in [2]. Ruiz [3] obtained some general results about existence and nonexistence of ground state solutions with 1 < p < 5, while Ambrosetti and Ruiz [5] obtained existence of multiple bound state solutions. Furthermore, Huang, Rocha and Chen [6] considered the case V (x) = 1 and f (x, u) = a(x) | u |p−2 u + λk(x)u, p > 4, for Eq. (1.2), they obtained the same result as our Theorem 1.1, where l(x) is a square integrable function. Recently, Chen [7] obtained the same results as in [6] for the case of l(x) = 1 by analyzing delicately the behavior of the non-local term. [7] and [3] pointed out that the case of 2 < p ≤ 4 is much more delicate, the method of [6] and [7] cannot be applied to Eq. (1.1) with 2 < p ≤ 4, it would be an interesting problem. Shen and Han [8] considered Eq. (1.1) for the case p = 4. They obtained the same results by the Nehari manifold and the concentration compactness principle, which can be regarded as the complementary work of Huang, Rocha and Chen [6]. But the condition (∫ )2 ∫ 2 4 a(x)e1 dx − b |∇e1 | dx < 0 R3

R3

is required, where e1 is the first eigenfunction in (1.3). Through the above analysis, as for Eq. (1.1), the case that p ≥ 4 has already been considered. But the case that 2 < p < 4 has not yet been considered up to now. The main difficulty is to get the boundedness of (PS) sequence, the methods in [7] are no longer applicable. In view of the above main difficulties, fortunately, we may use the method in [9] to obtain the boundedness of (PS) sequences. Therefore, our manuscript can be regarded as complementary work of [6–8]. The purpose of the present paper is to prove that Eq. (1.1) possesses multiple positive solutions for 2 < p < 4. To obtain our main results, we assume that the following conditions hold for the functions a(x) and k(x). (A1 ) a(x) ∈ C(R3 ) and lim|x|→∞ a(x) = a∞ < 0. (A2 ) a(x) is sign changing, i.e., |Ω ± | = ̸ 0, where Ω + = {x ∈ R3 : a(x) > 0} and Ω − = {x ∈ R3 : a(x) < 0}. Moreover, |Ω 0 | = 0, where Ω 0 = {x ∈ R3 : a(x) = 0}. 3 (K1 ) k(x) ∈ L 2 (R3 ) and k(x) > 0 for x ∈ R3 . As Lemma 3.1 in [7], let λi (i = 1, 2 · · · ) be the ith of eigenvalues of the eigenvalue problem − ∆u + u = λk(x)u, u ∈ H 1 (R3 ),

(1.3)

the associated normalized eigenfunctions are denoted by e1 , e2 , . . . with ∥ej ∥ = 1, j = 1, 2, . . .. Our main results are described as follows: Theorem 1.1.

Let (A1 ), (A2 ), (K1 ) hold and 2 < p < 4. Furthermore, if µ > 0 small enough, then

(1) for 0 < λ ≤ λ1 , Eq. (1.1) has at least one positive solution in H 1 (R3 ) × D1,2 (R3 ); (2) there exists δ > 0 such that for λ1 < λ < λ1 + δ, Eq. (1.1) has at least one positive solutions in H 1 (R3 ) × D1,2 (R3 ) with negative critical value. 2. Preliminaries Let X be the usual Sobolev space H 1 (R3 ) with the inner product ∫ (u, v) = (∇u∇v + uv)dx R3 2

and the norm ∥u∥ = (u, u).

W. Gan and S. Liu / Applied Mathematics Letters 93 (2019) 111–116

A function u ∈ X is called a weak solution of Eq. (1.1) if ∫ ∫ ∫ ∫ p−2 (∇u∇v + uv)dx + µ ϕu uvdx = a(x)|u| uvdx + λ R3

R3

R3

for all v ∈ X, where 1 ϕu (x) = 4π

113

k(x)uvdx,

R3

u2 (y) dy |x − y|

∫ R3

is the unique solution of −∆ϕ = u2 in X, we refer to [10]. For u ∈ X, we define ∫ ∫ ∫ µ 1 λ 1 p 2 2 2 ϕu u dx − a(x)|u| dx − k(x)|u| dx. Iµ (u) = ∥u∥ + 2 4 R3 p R3 2 R3 Then I ∈ C 1 (X, R1 ) and ∫ ∫ ⟨Iµ′ (u), v⟩ = (∇u∇v + uv)dx + µ R3

∫ ϕu uvdx −

R3

a(x)|u|

p−2

∫ uvdx − λ

R3

k(x)uvdx R3

for any v ∈ X. 3. The Palais–Smale condition Lemma 3.1 (Lemma 2.13 in [11]). Assume that (K1 ) holds. Then the function Φk : u ∈ H 1 (R3 ) ↦→ ∫ ∫ ˜ k : u ∈ H 1 (R3 ) ↦→ 3 k(x)uvdx is weakly continuous for any k(x)u2 dx is weakly continuous and Φ R3 R v ∈ H 1 (R3 ). Lemma 3.2 (Lemma 2.2 in [7]). Suppose that (A1 ), (A2 ), (K1 ) hold and 2 < p < 4. If any (P S)c sequence is bounded in X for any c ∈ R, then the functional Iµ satisfies (P S)c condition, that is, any (P S)c sequence ( ) Iµ (un ) → c, Iµ′ (un ) → 0 in X −1 possesses a convergent subsequence. Proof . We can refer to Lemma 2.2 in [7], for the reader’s convenience, we sketch the proof briefly. Let {un } ⊂ X be a bounded (P S)c sequence. Up to a subsequence, we may assume that there is u ∈ X such that ⎧ un ⇀ u, in X, ⎪ ⎪ ⎨ un → u, a.e. x ∈ R3 , (3.4) ∇un ⇀ ∇u, in L2 (R3 ), ⎪ ⎪ ⎩ 2 3 un ⇀ u, in L (R ). Noting that, for any ψ ∈ X, by (3.4), we can obtain ∫ ∫ p−2 a(x)|un | un ψdx → R3

and

p−2

a(x)|u|

uψdx

R3

∫

∫ (∇un ∇ψ + un ψ)dx →

(∇u∇ψ + uψ)dx

R3

R3

as n → ∞. Moreover, by Lemma 3.1, we also have ∫ ∫ k(x)un ψdx → R3

k(x)uψdx.

R3

Denote u ˜n := un − u, combining with Brezis–Lieb lemma, Iµ′ (un ) → 0 and the above results, we deduce that for n large enough, ∫ ∫ ∫ p o(1)∥un ∥ = ∥˜ un ∥2 + ϕun u2n dx − a(x)|˜ un | dx + o(1) + o(1)∥u∥ − ϕun un udx, R3

R3

R3

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which implies that for n large enough, ∫ ∫ p 2 ∥˜ un ∥ = a(x)|˜ un | dx + o(1)∥un ∥ − o(1)∥u∥ − o(1) + R3

∫ ϕun un udx −

R3

R3

ϕun u2n dx.

Moreover, by (2.25) in [7], we know that for n large enough, ∫ ∫ ϕun un udx − ϕun u2n dx ≤ 0. R3

R3

From (A1 ) we may assume without loss of generality, there is R0 > 0 such that a(x) < −1 for |x| > R0 . We obtain from the Sobolev embedding theorem that ∫ p a(x)|un − u| dx → 0, |x|≤R0

as n → ∞. From the above three formulas we obtain ∫ ∫ p 0 ≤ lim sup ∥un − u∥2 ≤ lim inf a(x)|un − u| dx ≤ lim n→∞

n→∞

R3

n→∞

p

a(x)|un − u| dx = 0.

|x|≤R0

This proves that un → u in X as n → ∞. □ To get a bounded (P S)c sequence, we introduce a cut-off function η ∈ C ∞ (R+ , R+ ) satisfying η(t) = 1, for t ∈ [0, 1]; 0 ≤ η(t) ≤ 1, for t ∈ (1, 2); η(t) = 0, for t ∈ [2, +∞); |η ′ (t)|∞ ≤ 2. We then define a modified functional ∫ ∫ ∫ µ 1 λ 1 p 2 ϕu u2 dx − a(x)|u| dx − k(x)|u| dx, Iµ,M (u) = ∥u∥2 + ψM (u) 2 4 p R3 2 R3 R3 2

where ψM (u) := η( ∥u∥ ) for M > 0, and M2 ∫ ∫ ∫ p−2 ′ ⟨Iµ,M (u), φ⟩ = (∇u∇φ + uφ)dx + µψM (u) ϕu uφdx − a(x)|u| uφdx 3 3 3 R R R ∫ ∫ ∫ 2 1 2 (∇u∇φ + uφ)dx, − λ R3 k(x)uφdx + µ2 η ′ ( ∥u∥ ϕ u dx · ) u 3 3 2 2 R R M M

(3.5)

for any φ ∈ X. Lemma 3.3.

Assume that (A1 ), (A2 ), (K1 ) hold and 2 < p < 4.

(i) If 0 < λ < λ1 , then u = 0 is a local minimum of Iµ,M ; (ii) there is u ∈ X with ∥u∥ > ρ such that Iµ,M (u) < 0 for any λ > 0. Proof . (i). (A1 ) and (A2 ) imply that a(x) is bounded in R3 , by the characterization of λ1 and the continuity of the Sobolev embedding of X in Lp (R3 ), we deduce that ∫ ∫ ∫ 1 µ 1 λ p 2 2 2 Iµ,M (u) = ∥u∥ + ψM (u) ϕu u dx − a(x)|u| dx − k(x)|u| dx 2 4 p R3 2 R3 R3 ( ) 1 λ 1 λ ≥ ∥u∥2 − C∥u∥p − ∥u∥2 = ∥u∥2 − − C∥u∥p−2 . (3.6) 2 2λ1 2 2λ1 ( ) Choosing ρ = ∥u∥ small enough, we obtain that Iµ,M (u) ≥ 41 1 − λλ1 ρ2 . Therefore the conclusion (i) is proved. (ii). We choose the function v ∗ ∈ X such that supp v ∗ ⊂ Ω + such that v ∗ (x) ≥ 0, ∀x ∈ Ω + and ∗ v = t0 e1 + v with t0 ̸= 0, where v ∈ {span{e1√ }}⊥ . We can assume ∥v ∗ ∥ = 1(otherwise we replace v ∗ by v∗ ∗ u = ∥v∗ ∥ ). For each M > 0, there exists tM ≥ 2M > 0 large enough such that ∫ ∫ tpM 1 2 λ 2 2 ∗ ∗ p Iµ,M (tM v ) = tM − a(x)|v | dx − tM k(x)|v ∗ | dx < 0. 2 p Ω+ 2 3 R Hence, (ii) holds by taking u = tM v ∗ .

□

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Define Cµ,M =

inf

max Iµ,M (γ(t)),

γ∈Γµ,M t∈[0,1]

where Γµ,M := {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = u}. Then, by Lemma 3.3, we have Cµ,M ≥ α > 0, ∀µ, M > 0. Furthermore, by the mountain pass theorem there exists {unµ,M } ⊂ X such that ′ Iµ,M (unµ,M ) → Cµ,M and (1 + ∥unµ,M ∥)∥Iµ,M (unµ,M )∥X −1 → 0,

(3.7)

where X −1 denotes the dual space of X. Lemma 3.4. Under the conditions of Lemma 3.3, let {unµ,M } be given by (3.7), then there exists M0 > 0 such that lim supn→∞ ∥unµ,M0 ∥ ≤ M20 for all µ ∈ (0, M0−3 ). Proof . Motivated by [9], we prove the lemma by contradiction. If the Lemma is not true, then for every M > 0, there exists µ∗ ∈ (0, M −3 ) such that lim sup ∥unµ∗ ,M ∥ > n→∞

M . 2

For simplicity, we denote unµ∗ ,M by un , and up to a subsequence, we get ∥un ∥ ≥

(3.8) M 2

for all n ∈ N . Because

1 Iµ∗ ,M (un ) − ⟨Iµ′ ∗ ,M (un ), un ⟩ p ( ) ( ) ∫ 1 1 1 1 2 = − ∥un ∥ + − µ∗ ψM (un )ϕun u2n dx 2 p( 4 ∫p R3 ) ( )∫ µ∗ ′ ∥un ∥2 ∥un ∥2 1 1 2 2 − η ϕu u dx − λ − k(x)|un | dx, 2p M2 M 2 R3 n n 2 p 3 R

(3.9)

) 1 1 1 − ∥un ∥2 − ∥Iµ′ ∗ ,M (un )∥X −1 ∥un ∥ p (2 p ) 1 1 1 2 ≤ − ∥un ∥ + ⟨Iµ′ ∗ ,M (un ), un ⟩ 2 p p ∫ 4−p ∗ ∗ = Iµ ,M (un ) + µ ψM (un )ϕun u2n dx 4p R3∫ ( ) )∫ ( µ∗ ′ ∥un ∥2 ∥un ∥2 1 1 2 2 + η ϕu u dx + λ − k(x)|un | dx. 2p M2 M 2 R3 n n 2 p 3 R

(3.10)

we deduce, (

Since 0 < λ < λ1 , there exists positive constant θ such that ( )[ ] ∫ 1 1 2 2 2 θ∥un ∥ ≤ − ∥un ∥ − λ k(x)|un | dx 2 p R3 1 ≤ ∥Iµ′ ∗ ,M (un )∥X −1 ∥un ∥ + Iµ∗ ,M (un ) p ( ) ∫ ∫ 4−p ∗ µ∗ ′ ∥un ∥2 ∥un ∥2 µ η + ψM (un )ϕun u2n dx + ϕun u2n dx 2 2 4p 2p M M 3 3 R R = I1 + I2 + I3 + I4 .

(3.11)

By (3.7), we have Iµ∗ ,M (un ) = Cµ∗ ,M + o(1). In the following, we estimate the above four terms. According to the proof of (ii) in Lemma 3.3, it follows that there exists M1 > 0 large enough such that Iµ∗ ,M (2M v ∗ ) < 0, ∀M ≥ M1 .

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By the definition of Cµ∗ ,M , we have Cµ∗ ,M ≤ max Iµ∗ ,M (2M v ∗ t) t∈[0,1]{ } ∫ ∫ 2p µ∗ p ≤ max 2(M t)2 − a(x)|v ∗ | dx · (M t)p + (2M )4 ϕv∗ (v ∗ )2 dx p Ω+ 4 t∈[0,1] 3 R ˜1 µ∗ M 4 . ≤ D1 + C

(3.12)

Hence, ˜1 µ∗ M 4 . I2 ≤ D1 + C Since

∫

R3

2

(3.13)

4

˜ ϕu u dx ≤ C∥u∥ , for any u ∈ X (see [3]), it is easy to obtain that ) ( ˜2 µ∗ M 4 , ψM (un ) = 0, as ∥un ∥2 ≥ 2M 2 . I3 ≤ C

(3.14)

˜3 µ∗ M 4 . From |η ′ (x)|∞ ≤ 2 and the definition of η(x), we have I4 ≤ C By (3.7), we know that I1 is bounded. Combining the above estimates of I1 , I2 , I3 and I4 , for all M ≥ M1 , we have ˜4 µ∗ M 4 + D4 , where C ˜4 , D4 are positive constants. θ∥un ∥2 ≤ C (3.15) Since µ∗ ≤ M −3 and ∥un ∥ ≥ conclusion. □

M 2 ,

(3.15) is impossible for M > 0 large enough. So we achieve the

Remark 3.5. If 0 < λ < λ1 + δ, then Lemma 3.3 implies that the functional Iµ,M0 has the mountain pass geometry, there is (P S) sequence {unµ,M0 } for Iµ,M0 and it follows from Lemmas 3.2 and 3.4 that the functional Iµ,M0 = Iµ satisfies (P S) condition for all 0 < µ < M0−3 . Proof of Theorem 1.1. By the analysis of Remark 3.5, and since Lemmas 3.2 and 3.4 imply unµ,M0 → u1 , for all 0 < µ < M0−3 . Hence, u1 is a nontrivial solution of Eq. (1.1) with Iµ (u1 ) > 0. Similar to the proof of [7], we can also obtain that u1 is a positive solution of Eq. (1.1). From Proposition 3.4 in [7], if λ1 < λ < λ1 + δ, we can also obtain the second positive solution u2 with Iµ (u2 ) < 0. Thus, the desired conclusion directly follows from the above argument and Proposition 3.4 in [7]. Hence we finish the proof of Theorem 1.1. □ References [1] V. Benci, D. Fortunato, An eigenvalue problem for the Schr¨ odinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998) 283–293. [2] T. d’Aprile, Non-radially symmetric solution of the nonlinear Schr¨ odinger equation coupled with Maxwell equations, Adv. Nonlinear Stud. 2 (2002) 177–192. [3] D. Ruiz, The Schr¨ odinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006) 655–674. [4] G.M. Coclite, A multiplicity result for the nonlinear Schr¨ odinger-Maxwell equations, Commun. Appl. Anal. 7 (2003) 417–423. [5] A. Ambrosrtti, D. Ruiz, Multiple bound states for the Schr¨ odinger-Poisson problem, Commun. Contemp. Math. 10 (2008) 391–404. [6] L.R. Huang, E.M. Rocha, J.J. Chen, Two positive solutions of a class Schr¨ odinger-Poisson system with indefinite nonlinearity, J. Differential Equations 255 (2013) 2463–2483. [7] J.Q. Chen, Multiple positive solutions of a class of non autonomous Schr¨ odinger-Poisson systems, Nolinear Anal. RWA. 21 (2015) 13–26. [8] Z.P. Shen, Z.Q. Han, Multiple solutions for a class of Schr¨ odinger-Poisson system with indefinite nonlinearity, J. Math. Anal. Anal. 426 (2015) 839–854. [9] Y.S. Jiang, Z.P. Wang, H.S. Zhou, Multiple solutions for a nonhomogeneous Schr¨ odinger-Maxwell system in R3 , Nonlinear Anal. TMA. 83 (2013) 50–57. [10] V. Benci, D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys. 14 (2002) 409–420. [11] M. Willem, Minimax Theorems, Birkh¨ auser, Boston, 1996.