Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 72, pp. 1–14. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.72 NEW TYPE OF MULTI-BUMP SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS TAO WANG, XIAOYU TIAN, WENLING HE Abstract. In this article, we study the existence of non-radial positive solutions of the Schrödinger- Poisson system −∆u+ u+ V (|x|)Φ(x)u = Q(|x|)|u|p−1u, x ∈ R3, −∆Φ = V (|x|)u2, x ∈ R3, where 1 < p < 5 and V,Q are radial potential functions. By developing some refined estimates, via the Lyapunov-Schmidt reduction method, we construct infinitely many multi-bump solutions when V,Q have some suitable algebraical decay at infinity. The maximum points of those multi- bump solutions are located on the top and bottom circles of a cylinder. This result not only gives a new type of multi-bump solutions but also extends the existence of multi-bump solutions to a general class of potential functions with a relatively slow decay rate at infinity. 1. Introduction We are interested in the Schrödinger-Poisson system −∆u+ u+ V (|x|)Φ(x)u = Q(|x|)|u|p−1u, x ∈ R3, −∆Φ = V (|x|)u2, x ∈ R3, (1.1) where 1 < p < 5, V and Q are potential functions satisfying (A1) V (|x|) = a |x|m +O ( 1 |x|m+θ ) as |x| → +∞, (A2) Q(|x|) = Q0 + b |x|n +O ( 1 |x|n+κ ) as |x| → +∞, where Q0, θ, κ, a > 0, b ∈ R. This system has physical origins in quantum mechanics and semicon- ductor theory (see for example [5, 6, 21]). As we see, there are numerous results on the existence and qualitative properties of solutions for system (1.1), such as positive radial solutions, semiclas- sical states, nodal solutions and so on. When Q(x) ≡ 1, V (x) ≡ λ > 0, D’Aprile and Mugnai [9] proved that (1.1) with 3 ≤ p < 5 admits a radial positive solution by using the Mountain pass theorem, see also [8]. Ruiz [27] introduced a new manifold by using the Pohažev identity and showed that (1.1) has at least one positive radial solution for all λ > 0 and 2 < p < 5. Moreover, if 0 < λ < 1 4 , the author established the existence of two positive radial solutions when 1 < p < 2 and one positive radial solution when p = 2. Ambrosetti and Ruiz [2] studied that (1.1) has infinitely many pairs of radial solutions for 2 < p < 5 and λ > 0. For more related results, one can refer to [1, 14, 15, 16, 18, 23, 26, 28, 31, 32] and the references therein. In recent years, the existence of multi-bump solutions has attracted much attention from re- searchers. When V has a positive local minimum, Ruiz and Vaira [29] established the existence of infinitely many multi-bump solutions to (1.1) via the Lyapunov-Schmidt reduction method, whose bumps concentrate near the local minimum of V . When V and Q have a fast algebraic 2020 Mathematics Subject Classification. 35B09, 35J05, 35J15, 35J60, 35Q55. Key words and phrases. Schrödinger-Poisson system; multi-bump solutions; Lyapunov-Schmidt reduction method. ©2025. This work is licensed under a CC BY 4.0 license. Submitted April 17, 2025. Published July 14, 2025. 1 2 T. WANG, X. TIAN, W. HE EJDE-2025/72 decay, Li, Peng and Yan [20] and Ding, Li and Ye [11] constructed infinitely many non-radial positive multi-bump solutions of (1.1) concentrating near infinity on a plane, respectively. For more related results, one can refer to [13, 17, 25, 24, 22] and references therein. Motivated by the above work, this paper is devoted to constructing a new type of multi-bump solutions for (1.1) when the potentials V and Q just have a slower algebraic decay by using the Lyapunov-Schmidt reduction method and some delicate estimates. Our main result is as follows. Theorem 1.1. Let V,Q satisfy (A1), (A2). Then system (1.1) has infinitely many multi-bump solutions whose maximum points lie on the top and bottom circles of a cylinder, provided that either 2m,n ≥ p+1 2p when p ∈ (1, 2), or 2m,n > 1/2 when p ∈ [2, 5). Remark 1.2. Compared with [10, 20], this type of multi-bump solutions is new, which concen- trates not on a plane, but on a cylinder in R3, which is one novelty. Besides, the range of m,n is extended at least from 2m,n ≥ 1 to a slow decay 2m,n > 1/2 when p ∈ [2, 5). This is another novelty of this paper. 2. Notation and energy expansion Throughout this paper, we use the following notation. • A1 = p−1 p+1 ∫ R3 U p+1, A2 = 1 8π ∫ R3 ∫ R3 U2(x)U2(y) |x−y| , A3 = 2 p+1 ∫ R3 U p+1, B1 = ∫ R3 U pe−x1 ; • H1(R3) is the usual Sobolev space endowed with inner product (u, v) = ∫ R3(∇u∇v+uv)dx and norm ∥u∥2 = ∫ R3(|∇u|2 + u2)dx; • D1,2(R3) is the completion of C∞ 0 (R3) with respect to the norm ∥u∥2D1,2 = ∫ R3 |∇u|2dx; • Hk and Dk are symmetric Sobolev subspaces defined by Hk = { u ∈ H1(R3) : u(r cos θ, r sin θ, x3) = u ( r cos ( θ + 2jπ k ) , r sin ( θ + 2jπ k ) , x3 ) , j = 1, . . . , k, and u is even in x2 } , Dk = { Φ ∈ D1,2(R3) : u(r cos θ, r sin θ, x3) = u ( r cos ( θ + 2jπ k ) , r sin ( θ + 2jπ k ) , x3 ) , j = 1, . . . , k, and u is even in x2 } . By using the Lax-Milgram theorem, for every u ∈ H1(R3), system (1.1) is equivalent to the single equation −∆u+ u+ V (|x|)Φu(x)u = Q(|x|)up−1u, u > 0, x ∈ R3, with a convolution term Φu defined by Φu(x) = 1 4π ∫ R3 V (y)u2(y) |x−y| dy. By using Hölder inequality and Sobolev inequality, we obtain ∥Φu∥2D1,2 = ∫ R3 Φuu 2dx ≤ ∥Φu∥L6∥u∥2L12/5 ≤ C∥Φu∥D1,2∥u∥2L12/5 ,∫ R3 Φuu 2dx ≤ C∥u∥4L12/5 ≤ C∥u∥4. Moreover, if u ∈ Hk, then Φu ∈ Dk. To obtain positive solutions of (1.1), we shall consider the functional I : u ∈ H1(R3) → R I(u) = 1 2 ∫ R3 (|∇u|2 + u2)dx+ 1 4 ∫ R3 V (x)Φuu 2dx− 1 p+ 1 ∫ R3 Q(x)up+1 + dx. which is well defined and is a C1−functional with derivative ⟨I ′(u), v⟩ = ∫ R3 (∇u · ∇v + uv + V (x)Φuuv −Q(x)up+v)dx, ∀v ∈ H1(R3). Without confusion, we shall denote by u instead of u+ for simplicity. For j = 1, . . . , k, we set Ωj := { x = (x1, x2, x3) ∈ R3 : 〈 (x1, x2) |(x1, x2)| , ( cos 2(j − 1)π k , sin 2(j − 1)π k )〉 ≥ cos π k } , EJDE-2025/72 MULTI-BUMP SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS 3 Ω+ j := { x = (x1, x2, x3) ∈ Ωj , x3 ≥ 0 } , Ω− j := { x = (x1, x2, x3) ∈ Ωj , x3 < 0 } . Then Ωj = Ω+ j ∪ Ω− j . Let Dk := [(min{2m,n} 2π − α ) k ln k, (min{2m,n} 2π + α ) k ln k ] × [(min{2m,n}+ 2 2 − β ) ln k, (min{2m,n}+ 2 2 + β ) ln k ] , where α, β > 0 are small constants. For (r, h) ∈ Dk, we take 2k points P+ j = ( r cos 2(j − 1)π k , r sin 2(j − 1)π k , h ) , P− j = ( r cos 2(j − 1)π k , r sin 2(j − 1)π k , −h ) . Clearly, P± j ∈ Ω± j . Hereafter, we always assume (r, h) ∈ Dk. Let U be a ground state solution of −∆U + U = Up, U ∈ H1(R3), U(0) = max x∈R3 U(x) and U > 0. (2.1) As is proved in [12] and [19], U is radially symmetric, unique and satisfies lim |x|→∞ U(x)e|x||x| = C < +∞ and lim |x|→∞ U ′(x) U(x) = −1. We denote by UP+ j (x) = U(x− P+ j ), UP− j (x) = U(x− P− j ), and set an approximate solution of (1.1) as Wr,h (x) = k∑ j=1 ( UP+ j (x) + UP− j (x) ) . We firstly cite the following estimates. Lemma 2.1 ([4, Lemma 2.1]). For each η ∈ (0, 1], there is a constant C > 0 such that for each x ∈ Ω+ 1 , k∑ i=2 ( UP+ i (x) + UP− i (x) ) ≤ Ce−ηπ r k e−(1−η)|x−P+ 1 |, UP− 1 (x) ≤ Ce−ηhe−(1−η)|x−P+ 1 |. Lemma 2.2 ([4, Proposition 3.2]). There exists σ > 0 such that 1 2 ∫ R3 ( |∇Wr,h|2 + |Wr,h|2 ) − 1 p+ 1 ∫ R3 Q(|x|)|Wr,h|p+1 = k [ A1 − 2B1e −2π r k ( k 2πr ) −B1e −2h ( 1 2h ) − bA3 rn + nbA3 2 h2 rn+2 +Ok ( e−2(1+σ)π r k ) +Ok ( e−2(1+σ)h ) +Ok ( 1 rn+σ ) +Ok ( h2 rn+2+σ )] . (2.2) To obtain some estimates, we establish the following integral estimate, which will play an important role in our proof. It can be regarded as an extension of the well-known translation estimate [3, Proposition 1.2] to the convolution case. Lemma 2.3. Suppose that S, T,K : R3 → R are positive continuous radial functions satisfying |x|a1eb1|x|S(|x|) → c1, |x|a2eb2|x|T (|x|) → c2, |x|dK(|x|) → c3, as |x| → ∞, (2.3) where ai ∈ R and bi, ci, d > 0. Then there is τ > 0 such that∫ R3 ∫ R3 K(|x|)K(|y|) |x− y| Sξ(x)Tξ(y) dy dx = c23 |ξ|2d ∫ R3 ∫ R3 S(x)T (y) |x− y| dy dx+O ( 1 |ξ|2d+τ ) (2.4) as |ξ| → ∞, where Sξ(x) = S(x− ξ) and Tξ(y) = T (y − ξ). 4 T. WANG, X. TIAN, W. HE EJDE-2025/72 Proof. By rotation, we can assume ξ |ξ| = (1, 0, 0). Let 0 < λ < 1, B = Bλ|ξ|(0) and BC = R3\Bλ|ξ|(0). Then by (2.3) and Hardy-Littlewood-Sobolev inequality, there is some τ > 0 such that ∫ R3 ∫ R3 K(|x|)K(|y|) |x− y| Sξ(x)Tξ(y) dy dx = (∫ B ∫ B +2 ∫ B ∫ BC + ∫ BC ∫ BC )K(|x+ ξ|)K(|y + ξ|) |x− y| S(x)T (y) dy dx = c23 |ξ|2d ∫ R3 ∫ R3 S(x)T (y) |x− y| dy dx+O ( 1 |ξ|2d+τ ) +O (c3e−τλ|ξ| |ξ|d ∫ B ∫ BC c2 |x− y| S(|x|)|y|−a2e−(b2−τ)|y| dy dx ) +O ( e−2τλ|ξ| ∫ BC ∫ BC c1c2 |x− y| |x|−a1e−(b1−τ)|x||y|−a2e−(b2−τ)|y| dy dx ) = M |ξ|2d +O ( 1 |ξ|2d+τ ) +O (e−τλ|ξ| |ξ|d ) +O ( e−2τλ|ξ| ) = M |ξ|2d +O ( 1 |ξ|2d+τ ) , where M = c23 ∫ R3 ∫ R3 S(x)T (y) |x−y| dy dx. The proof is complete. □ 3. Reduction equation Let Z1 = ∂Wr,h ∂r , Z2 = ∂Wr,h ∂h and E = {v ∈ Hk : ⟨Z1, v⟩ = 0 and ⟨Z2, v⟩ = 0} . To apply the reduction method, we define a functional J : E → R by J(ϕ) = I(Wr,h + ϕ). Then by the Taylor’s expansion, we obtain J(ϕ) = J(0) + l(ϕ) + 1 2 ⟨Lϕ, ϕ⟩ −R(ϕ), (3.1) where J(0) = I(Wr,h), l(ϕ) = ∫ R3 ( k∑ i=1 ( Up P+ i + Up P− i ) −Q(|x|)W p r,h ) ϕ+ ∫ R3 V (|x|)ΦWr,h Wr,hϕ, ⟨Lv1, v2⟩ = ∫ R3 ( ∇v1∇v2 + v1v2 − pQ(|x|)W p−1 r,h v1v2 ) + ∫ R3 V (|x|)ΦWr,h v1v2 + 2 ∫ R3 V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hv1dy ) Wr,hv2dx, for all v1, v2 ∈ E, R(ϕ) = ∫ R3 V (|x|)ΦϕWr,hϕ+ 1 4 ∫ R3 V (|x|)Φϕϕ 2 − 1 p+ 1 ∫ R3 Q(|x|) ( |Wr,h + ϕ|p+1 −W p+1 r,h − (p+ 1)W p r,hϕ− 1 2 (p+ 1)pW p−1 r,h ϕ2 ) . In particular, ⟨Lϕ, ϕ⟩ = ∫ R3 ( |∇ϕ|2 + |ϕ|2 − pQ(|x|)W p−1 r,h ϕ2 ) + ∫ R3 V (|x|)ΦWr,h ϕ2 + 2 ∫ R3 V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hϕdy ) Wr,hϕdx In the following, we prove that L is invertible and bounded in E, whose proof is slightly different from [20], because of the presence of two variable quantities r, h of the points P± j . EJDE-2025/72 MULTI-BUMP SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS 5 Lemma 3.1. There exists a constant ρ > 0 independent of k such that for any (r, h) ∈ Dk, it holds ∥Lv∥ ≥ ρ∥v∥, v ∈ E. (3.2) Proof. We shall prove it by contradiction. Suppose on the contrary that there exist (rk, hk) ∈ Dk and vk ∈ E with ∥vk∥2 = k such that as k → +∞, ∥Lvk∥ = o(1)∥vk∥. (3.3) By using the symmetry, it holds that for any ψ ∈ E,∫ Ω1 ( ∇vk∇ψ + vkψ − pQ(|x|)W p−1 r,h vkψ ) + ∫ Ω1 ( V (|x|)ΦWr,h vkψ ) + 2 ∫ Ω1 V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hvkdy ) Wr,hψ = 1 k ⟨Lvk, ψ⟩ = o ( 1√ k ) ∥ψ∥. (3.4) Clearly, if ψ = vk, then we have ∫ Ω1 (|∇vk|2 + v2k) = 1 (3.5) and ∫ Ω1 ( |∇vk|2 + v2k − pQ(|x|)W p−1 r,h v2k ) + ∫ Ω1 ( V (|x|)ΦWr,h v2k ) + 2 ∫ Ω1 V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hvkdy ) Wr,hvk = o(1). (3.6) Let v̄k(x) = vk(x− P+ 1 ) and R1 > 0 satisfy BR1 (P+ 1 ) ⊂ Ω1. Then∫ BR1 (0) (|∇vk|2 + v2k) ≤ 1, and thus there is some v ∈ H1(R3) such that as k → +∞, up to a subsequence, v̄k ⇀ v weakly in H1(R3), v̄k → v strongly in L2 loc(R3). Since v̄k ∈ E is even in x2, v is also even in x2 and satisfies∫ R3 Up−1 ∂U ∂x1 v = 0 and ∫ R3 Up−1 ∂U ∂x3 v = 0. (3.7) For any R1 > 0 and ψ ∈ C∞ 0 (BR1 (0)) being even in x2, let ψk,j(x) := ψ(x− P+ j ) ∈ C∞ 0 (BR1 (P+ j )). Then k∑ j=1 ψk,j ∈ Hk. Moreover, there are bk,1, bk,2 ∈ R such that ψ̄k(x) = k∑ j=1 ψk,j − bk,1Z1 − bk,2Z2 ∈ E, (3.8) where bk,1 and bk,2 satisfy( ∥Z1∥2 ⟨Z1, Z2⟩ ⟨Z1, Z2⟩ ∥Z2∥2 )( bk,1 bk,2 ) = ( ⟨ ∑k j=1 ψk,j , Z1⟩ ⟨ ∑k j=1 ψk,j , Z2⟩ ) = k ( ⟨ψk,1, Z1⟩ ⟨ψk,1, Z2⟩ ) . Observe that ∥Z1∥2 = 2k ( ∥ ∂U ∂x1 ∥2 + o(1) ) , ∥Z2∥2 = 2k ( ∥ ∂U ∂x3 ∥2 + o(1) ) , ⟨Z1, Z2⟩ = o(k). Then a direct computation shows that there is some C > 0 independent of k such that max k {|bk,1|, |bk,2}| ≤ C. 6 T. WANG, X. TIAN, W. HE EJDE-2025/72 This together with (3.8), implies ∥ψ̄k∥2 ≤ Ck. Then by inserting ψ = ψ̄k into (3.4), we obtain∫ Ω1 ( ∇vk∇ψ̄k + vkψ̄k − pQ(|x|)W p−1 r,h vkψ̄k ) + ∫ Ω1 ( V (|x|)ΦWr,h vkψ̄k ) + 2 ∫ Ω1 V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hvkdy ) Wr,hψ̄k = 1 k ⟨Lvk, ψ̄k⟩ = o ( 1√ k ) ∥ψ̄k∥ = o(1). Thus, ⟨Lvk, ψk,1⟩ = 1 k ⟨Lvk, k∑ j=1 ψk,j⟩ = 1 k ⟨Lvk, ψ̄k⟩+ 1 k 2∑ i=1 bk,i⟨Lvk, Zi⟩ = 1 k ⟨Lvk, ψ̄k⟩+ 2∑ i=1 γk,i⟨ψk,1, Zi⟩, (3.9) where ( γk,1 γk,2 )T = ( ⟨Lvk, Z1⟩ ⟨Lvk, Z2⟩ )T ( ∥Z1∥2 ⟨Z1, Z2⟩ ⟨Z1, Z2⟩ ∥Z2∥2 )−1 . Let η ∈ C∞ 0 (BR1(P + 1,k)) be a cutoff function satisfying that η = 1 in BR1 2 (P+ 1,k), |∇η| ≤ CR1 −1, and |∇2η| ≤ CR−2 1 . Then by taking ψk,1 = ηZj in (3.9), we obtain that 2∑ i=1 γk,i⟨ηZj , Zi⟩ = ⟨Lvk, ηZj⟩+ o(1) = ⟨vk, L(ηZj)⟩+ o(1) = o(1). (3.10) Since ⟨ηZj , Zi⟩ = ⟨Zj , Zi⟩+ o(1), we see from (3.10) that( ∥Z1∥2 + o(1) o(1) o(1) ∥Z2∥2 + o(1) )( γk,1 γk,2 ) = ( o(1) o(1) ) , which shows that γk,i = o(1) for i = 1, 2. Then by (3.9), we have ⟨Lvk, ψk,1⟩ = o(1), namely,∫ R3 ( ∇vk∇ψk,1 + vkψk,1 − pQ(|x|)W p−1 r,h vkψk,1 ) + ∫ R3 ( V (|x|)ΦWr,h vkψk,1 ) + 2 ∫ R3 V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hvkdy ) Wr,hψk,1 = o(1). This implies that for any ψ ∈ C∞ 0 (BR1 (0)) being even in x2, ⟨Lv̄k, ψ⟩ = ∫ R3 ( ∇v̄k∇ψ + v̄kψ − pQ(|x|)W p−1 r,h v̄kψ ) + ∫ R3 ( V (|x|)ΦWr,h v̄kψ ) + 2 ∫ R3 V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hv̄kdy ) Wr,hψ = o(1). (3.11) Next, we assert that∫ R3 ( V (|x|)ΦWr,h v̄kψ + 2V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hv̄kdy ) Wr,hψ ) → 0 as k → +∞. (3.12) EJDE-2025/72 MULTI-BUMP SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS 7 In fact, by (A1), (A2), Lemma 2.1 and [3, Proposition 2.1], we have ∥ΦWr,h ∥2D1,2(Ω1) := ∫ Ω1 |∇ΦWr,h |2 = 2 ∫ Ω+ 1 V (|x|)ΦWr,h W 2 r,h ≤ C (∫ Ω+ 1 V (|x|)6/5 ( U2 P+ 1 + ( k∑ i=2 UP+ i + k∑ j=1 UP− j )2)6/5 dx )5/6 ∥ΦWr,h ∥L6(Ω+ 1 ) ≤ C [( ∫ Ω+ 1 V (|x|)6/5U12/5 P+ 1 )5/6 + ( e−ηπ r k + e−ηh )2(∫ Ω+ 1 V (|x|)6/5e− 12 5 (1−η)|x−P+ 1 |dx )5/6] ∥ΦWr,h ∥D1,2(Ω1) ≤ C ( 1 |P+ 1 |m + ( e−ηπ r k + e−ηh )2 1 |P+ 1 |m ) ∥ΦWr,h ∥D1,2(Ω1) ≤ C |P+ 1 |m ∥ΦWr,h ∥D1,2(Ω1), (3.13) that is, ∥ΦWr,h ∥D1,2(Ω1) ≤ C |P+ 1 |m . Then by (3.5) and [3, Proposition 1.2], we obtain that for any ψ ∈ C∞ 0 (BR1 (0)),∫ R3 V (|x|)ΦWr,h v̄kψ = ∫ BR1 (0) V (|x|)ΦWr,h v̄kψ ≤ C∥ΦWr,h ∥D1,2(BR1 (0)) (∫ BR1 (0) (V (|x|)v̄kψ)6/5 )5/6 ≤ C |P+ 1 |m (∫ BR1 (0) (V (|x|)v̄k)12/5 )5/12(∫ BR1 (0) ψ12/5 )5/12 ≤ C |P+ 1 |2m ∥ψ∥. (3.14) Now, we claim that ∫ R3 (V (|y|)Wr,h(y)v̄k) 6/5 ≤ C |P+ 1 | 65m ∥v6/5k ∥. (3.15) Indeed, note from [30, Proposition A.3] that for any ϑ > 0, k∑ i=2 e−ϑ|P+ 1 −P+ i | ≤ Ce−2πϑ r k and e−ϑ|P+ 1 −P− 1 | ≤ Ce−2ϑh, (3.16) and that∫ R3 (V (|y|)Wr,h(y)v̄k) 6/5 ≤ C k∑ i=1 ∫ R3 V 6/5(|y|)U6/5 P± i (y)v̄ 6/5 k (y) ≤ C (∫ R3 V 6/5(|y|)U6/5(y − P+ 1 )v 6/5 k (y − P+ 1 ) + k∑ i=2 ∫ R3 V 6/5(|y|)U6/5(y − P+ i )v 6/5 k (y − P+ 1 ) + ∫ R3 V 6/5(|y|)U6/5(y − P− 1 )v 6/5 k (y − P+ 1 ) ) =: C(V1 + V2 + V3), 8 T. WANG, X. TIAN, W. HE EJDE-2025/72 By using [7, Lemma 6.1.3], we see that V1 = ∫ R3 V 6/5(|y + P+ 1 |)U6/5v 6/5 k ≤ (∫ R3 V 12/5(|y + P+ 1 |)U12/5 ) )1/2 (∫ R3 v 12/5 k )1/2 ≤ C |P+ 1 | 65m ∥v6/5k ∥. (3.17) By using the symmetry, (3.16) and the exponential decay of U , we have V2 ≤ k∑ i=2 (∫ R3 V 12/5(|y|)U 12 5 −2τ (y − P+ 1 ) )1/2(∫ R3 U2τ (y − P+ i )v 12/5 k (y − P+ 1 ) )1/2 ≤ k∑ i=2 (∫ R3 V 12/5(|y + P+ 1 |)U 12 5 −2τ )1/2(∫ R3 U2τ (y + P+ 1 − P+ i )v 12/5 k )1/2 ≤ C |P+ 1 | 65m k∑ i=2 (∫ R3 e−2τ |y+P+ 1 −P+ i |v 12/5 k )1/2 ≤ C |P+ 1 | 65m k∑ i=2 (∫ R3 e−2τ |y|e2τ |P + 1 −P+ i |v 12/5 k )1/2 ≤ C |P+ 1 | 65m e2πτ r k (∫ R3 e−2τ |y|v 12/5 k )1/2 ≤ C |P+ 1 | 65m ∥v6/5k ∥. (3.18) By using a similar argument as (3.18), we can deduce that V3 ≤ C |P+ 1 | 65m e2τh (∫ R3 e−2τ |y|v 12/5 k )1/2 ≤ C |P+ 1 | 65m ∥v6/5k ∥. (3.19) Obviously, the claim (3.15) follows immediately from (3.17)-(3.19). Furthermore, by using the Hardy-Littlewood-Sobolev inequality and (3.15), we obtain∫ R3 V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hv̄kdy ) Wr,hψdx ≤ C (∫ BR1 (0) (V (|x|)Wr,h(x)ψ) 6/5 )5/6(∫ R3 (V (|y|)Wr,h(y)v̄k) 6/5 )5/6 ≤ C (∫ BR1 (0) ( V (|x+ P+ 1 |)Wr,h(x+ P+ 1 ) )12/5 )5/12(∫ BR1 (0) ψ12/5 )5/12 C |P+ 1 |m ∥vk∥ ≤ C |P+ 1 |2m ∥ψ∥ ∥vk∥. (3.20) This together with (3.14), gives∫ R3 ( V (|x|)ΦWr,h v̄kψ + 2V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hv̄kdy ) Wr,hψ ) ≤ C √ k |P+ 1 |2m ∥ψ∥. Thus the claim (3.12) follows immediately from 2m > 1/2. Now, by letting k → +∞ as in (3.11), we deduce that∫ R3 ∇v∇ψ + vψ − pUp−1vψ = 0 (3.21) for any ψ ∈ C∞ 0 (BR1 (0)) being even in x2. Moreover, for any ψ ∈ C∞ 0 (BR1 (0)), let φ(y) = ψ(y)+ψ(y1,−y2, y3) in (3.21). Clearly, φ is even in x2 and (3.21) holds for ψ = ϕ. Since v is even EJDE-2025/72 MULTI-BUMP SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS 9 in x2, (3.21) holds for ψ(y1,−y2, y3). So (3.21) holds for all ψ ∈ C∞ 0 (BR1 (0)). Namely, −△v + v − pUp−1v = 0 inR3. (3.22) Since v is even in x2, by the non-degeneracy property of U , we can get from (3.22) that there are c1, c2 ∈ R such that v = c1 ∂U ∂x1 + c2 ∂U ∂x3 . Inserting it into (3.7), we obtain easily that c1 = c2 = 0. Thus v ≡ 0. Therefore, for large k, ∫ BR1 (P+ 1 ) v2k = o(1). This implies∫ Ω1 ( |∇vk|2 + v2k − pQ(|x|)W p−1 r,h v2k ) ≥ 1 2 + o(1). (3.23) In view of (3.14) and (3.20), we can use a similar argument to deduce that∫ Ω1 ( V (|x|)ΦWr,h v2k + 2V (|x|) (∫ R3 V (|y|) 4π|x− y| Wr,hvkdy ) Wr,hvk ) ≤ C |P+ 1 |2m ∥vk∥ → 0, as k → +∞. (3.24) Inserting (3.23) and (3.24) into (3.6), we obtain a contradiction. So (3.2) holds and the proof is complete. □ Recall that ⟨lk, ϕ⟩ = ∫ R3 ( k∑ i=1 ( Up P+ i + Up P− i ) −Q(|x|)W p r,h ) ϕ+ ∫ R3 V (|x|)ΦWr,h Wr,hϕ. (3.25) Compared to [20], the following estimate is more acurate. Lemma 3.2. For (r, h) ∈ Dk, there exists some σ > 0 such that for k large enough, ∥lk∥ ≤  C0 k p p+1 (1−σ)min{2m,n}− 1 2 , if 1 < p < 2, C0 k(1−σ)min{2m,n}− 1 2 , if 2 ≤ p < 5, (3.26) where C0 > is independent of k. Proof. By a similar calculation as for (3.13), we can deduce that ∫ Ω1 (V (|x|)Wr,h) 3 ≤ C |P+ 1 |3m . Then by the symmetry, we obtain∫ R3 V (|x|)ΦWr,h Wr,hϕ ≤ k ∫ Ω1 V (|x|)ΦWr,h Wr,hϕ ≤ Ck∥ΦWr,h ∥L6(Ω1) (∫ Ω1 (V (|x|)Wr,h) 3 dx )1/3 ∥ϕ∥L2(Ω1) ≤ C √ k∥ΦWr,h ∥D1,2(Ω1) (∫ Ω1 (V (|x|)Wr,h) 3 dx )1/3 ∥ϕ∥L2(R3) ≤ C √ k |P+ 1 |2m ∥ϕ∥H1(R3) ≤ C k2m− 1 2 ∥ϕ∥H1(R3). (3.27) Recall from [4, (4.15), (4.16)] that there is a small number σ > 0, such that∫ R3 ( k∑ i=1 ( Up P+ i + Up P− i ) −W p r,h ) ϕ ≤  C k p p+1 (1−σ)min{2m,n}− 1 2 ∥ϕ∥H1(R3), if 1 < p < 2, C k(1−σ)min{2m,n}− 1 2 ∥ϕ∥H1(R3), if 2 ≤ p < 5 (3.28) and from [4, Lemma 4.2] that ∫ R3 Q(|x|)W p r,hϕ ≤ C kn− 1 2 ∥ϕ∥H1(R3). (3.29) Thus (3.26) follows immediately from (3.25) and (3.27)-(3.29). The proof is complete. □ 10 T. WANG, X. TIAN, W. HE EJDE-2025/72 In view of (3.1), Lemmas 3.1 and 3.2, to find a critical point of J is equivalent to solving ϕ = A(ϕ) := −L−1lk − L−1R′(ϕ). (3.30) A direct calculation as [20, Lemma 2.3] shows that there is some constant C > 0, independent of k, such that ∥Ri(ϕ)∥ ≤ C∥ϕ∥min(3,p+1)−i, i = 0, 1, 2, (3.31) where Ri denotes the derivative of i order for R. Here we omit the details of the proof. Then we have the following result. Proposition 3.3. There exists an integer k0 > 0 such that for each k ≥ k0, there is a C1 map ϕ : Dk → Hk, ϕ = ϕ(r, h) ∈ E satisfying (3.30). Moreover, there is a small σ > 0 such that ∥ϕ∥ ≤  C∗ k p p+1 (1−2σ)min{2m,n}− 1 2 if 1 < p < 2, C∗ k(1−2σ)min{2m,n}− 1 2 if 2 ≤ p < 5. Proof. Define B := { ϕ ∈ E : ∥ϕ∥ ≤  C∗ k p p+1 (1−2σ)min{2m,n}− 1 2 if 1 < p < 2, C∗ k(1−2σ)min{2m,n}− 1 2 if 2 ≤ p < 5. } where σ > 0 is a small number as (3.26). With the aid of Lemma 3.2 and (3.31), a standard argument can show that there is a sufficiently large number C∗ > C0 such that for k sufficiently large, A is a strict contraction map such that A(B) ⊂ B. Then by the contraction mapping theorem, there is a C1 map ϕ : Dk → B such that J ′ (ϕ(r, h)) = 0. The proof is complete. □ 4. Energy estimate With the help of Lemmas 2.2 and 2.3, we obtain the following energy expansion of approximate solution. We firstly recall from [10, Lemma A.6] that r k ln k k∑ i=2 1 |P+ 1 − P+ i | = 1 π + o(1). (4.1) Then by using a similar argument, we can prove that there is some C0 > 0 such that k∑ i=2 1 |P+ 1 − P− i | = C0 + o(1). (4.2) Proposition 4.1. For all (r, h) ∈ Dk, there is a small number σ > 0 such that I(Wr,h) = k [ A1 + a2A2 r2m − ma2h2A2 r2m+2 − bA3 rn + nbA3 2 h2 rn+2 − 2B1e −2π r k ( k 2πr ) −B1e −2h ( 1 2h ) + a2A2k ln k πr2m+1 + a2A2C0k ln k r2m+1 + ma2A2h 2k ln k πr2m+3 + ma2A2C0h 2k ln k r2m+3 + a2A2 2hr2m + ma2A2h 2r2m+2 +Ok ( 1 r2m+σ ) +Ok ( h2 r2m+2+σ ) +Ok ( 1 rn+σ ) +Ok ( h2 rn+2+σ ) +O ( k ln k r2m+1+σ ) +O ( h2k ln k r2m+3+σ ) +O ( 1 r2m+σh ) +O ( h r2m+2+σ ) +Ok ( e−2(1+σ)π r k ) +Ok ( e−2(1+σ)h )] , (4.3) where Ai, B1 are defined in section 2. EJDE-2025/72 MULTI-BUMP SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS 11 Proof. For fixed R > 0, by using Lemmas 2.1 and 2.3, there is some σ > 0 such that for k large enough, 1 8π k∑ i=2 ∫ BR(P+ 1 ) ∫ BR(P± i ) V (|x|)V (|y|) |x− y| W 2 r,h(x)W 2 r,h(y) dy dx = 1 8π k∑ i=2 ∫ BR(P+ 1 ) ∫ BR(P± i ) V (|x|)V (|y|) |x− y| (UP+ 1 + C(e−η πr k + e−ηh)e−(1−η)|x−P+ 1 |)2 × (UP± i + C(e−η πr k + e−ηh)e−(1−η)|y−P± i |)2 dy dx = 1 8π k∑ i=2 ∫ BR(P+ 1 ) ∫ BR(P± i ) V (|x|)V (|y|) |x− y| U2 P+ 1 U2 P± i +O( 1 |P+ 1 |2m+σ ) = 1 8π k∑ i=2 ∫ BR(0) ∫ BR(0) V (|x+ P+ 1 |)V (|y + P± i |) |x− y + P+ 1 − P± i | U2(x)U2(y) +O( 1 |P+ 1 |2m+σ ) = 1 8π|P+ 1 |2m k∑ i=2 ∫ BR(0) ∫ BR(0) a2 |x− y + P+ 1 − P± i | U2(x)U2(y) +O( 1 |P+ 1 |2m+σ ) ≤ a2A2 |P+ 1 |2m k∑ i=2 1 |P+ 1 − P± i | +O( 1 |P+ 1 |2m+σ ) = a2A2 |P+ 1 |2m k∑ i=2 1 |P+ 1 − P± i | +O( 1 |P+ 1 |2m+σ ). (4.4) Then by (4.1) and (4.2), it follows that k∑ i=2 ∫ BR(P+ 1 ) ∫ BR(P+ i ) V (|x|)V (|y|) |x− y| W 2 r,h(x)W 2 r,h(y) dy dx = a2A2 π|P+ 1 |2m k ln k r +O( 1 |P+ 1 |2m+σ ) k∑ i=2 ∫ BR(P+ 1 ) ∫ BR(P− i ) V (|x|)V (|y|) |x− y| W 2 r,h(x)W 2 r,h(y) dy dx = a2A2C0 |P+ 1 |2m k ln k r +O( 1 |P+ 1 |2m+σ ). (4.5) As for (4.4), we have 1 8π ∫ BR(P+ 1 ) ∫ BR(P− 1 ) V (|x|)V (|y|) |x− y| W 2 r,h(x)W 2 r,h(y) dy dx = a2A2 2h|P+ 1 |2m +O( 1 |P+ 1 |2m+σ ). (4.6) By the exponential decay of U , Lemmas 2.1 and 2.3, we have∫ Ω+ 1 \BR(P+ 1 ) ∫ R3 V (|x|)V (|y|) |x− y| W 2 r,h(x)W 2 r,h(y) dy dx ≤ C ∫ Ω+ 1 \BR(P+ 1 ) ∫ R3 V (|y|) |x− y| ( U2 P+ 1 (x) + ( k∑ i=2 UP+ i (x) + k∑ j=1 UP− j (x) )2) W 2 r,h(y) dy dx ≤ Ce−R ∫ Ω+ 1 \BR(P+ 1 ) ∫ R3 V (|y|) |x− y| e−|x−P+ 1 |W 2 r,h(y) dy dx + ( e−ηπ r k + e−ηh )2 ∫ Ω+ 1 \BR(P+ 1 ) ∫ R3 V (|y|) |x− y| e−2(1−η)|x−P+ 1 |W 2 r,h(y) dy dx ≤ C ( e−R + ( e−ηπ r k + e−ηh )2 ) ≤ C |P+ 1 |2m+σ . (4.7) 12 T. WANG, X. TIAN, W. HE EJDE-2025/72 Therefore, by Lemma 4.4 and (4.5)-(4.7), we obtain 1 4 ∫ R3 V (|x|)ΦWr,h W 2 r,h(x)dx = k 2 ∫ Ω+ 1 V (|x|) (( k∑ l=1 ∫ BR(P± l ) + ∫ R3\ (∑k l=1 BR(P± l ) ) ) 1 4π|x− y| V (y)W 2 r,h(y)dy ) W 2 r,h(x)dx = k 8π ∫ Ω+ 1 ∩BR(P+ 1 ) ∫ BR(P+ 1 ) V (|x|)V (|y|) |x− y| W 2 r,h(x)W 2 r,h(y) dy dx+ kOk ( 1 |P+ 1 |2m+σ ) + k 8π (∫ Ω+ 1 ∩BR(P+ 1 ) ∫ BR(P− 1 ) + k∑ l=2 ∫ Ω+ 1 ∩BR(P+ 1 ) ∫ BR(P± l ) )V (|x|)V (|y|) |x− y| W 2 r,h(x)W 2 r,h(y) dy dx = k 8π ∫ Ω+ 1 ∩BR(P+ 1 ) ∫ BR(P+ 1 ) V (|x|)V (|y|) |x− y| [ U2 P+ 1 (x) +Ok ( UP+ 1 (x) ( k∑ i=2 UP+ i (x) + k∑ j=1 UP+ 1 (x) ))] × [ U2 P+ 1 (y) +Ok ( UP+ 1 (y) ( k∑ i=2 UP+ i (y) + k∑ j=1 UP− j (y) ))] dy dx+ kOk ( 1 |P+ 1 |2m+σ ) + k 8π ( a2A2 π|P+ 1 |2m k ln k r + a2A2C0 |P+ 1 |2m k ln k r ) + k 8π ( a2A2 2h|P+ 1 |2m ) = k 8π ∫ Ω+ 1 ∩BR(0) ∫ BR(0) V (|x+ P+ 1 |)V (|y + P+ 1 |) |x− y| U2(x)U2(y) dy dx+ kOk ( 1 |P+ 1 |2m+σ ) + kOk (∫ Ω+ 1 ∩BR(0) ∫ BR(0) V (|x+ P+ 1 |)V (|y + P+ 1 |) |x− y| U2(x)U(y)e−(1−η)|y| ( e−ηπ r k + e−ηh ) dy dx ) + k ( a2A2 π|P+ 1 |2m k ln k r + a2A2C0 |P+ 1 |2m k ln k r ) + k ( a2A2 2h|P+ 1 |2m ) = k a2A2 |P+ 1 |2m + kOk ( 1 |P+ 1 |2m+σ ) + k ( a2A2 π|P+ 1 |2m k ln k r + a2A2C0 |P+ 1 |2m k ln k r ) + k ( a2A2 2h|P+ 1 |2m ) = k [a2A2 r2m − ma2h2A2 r2m+2 +Ok ( 1 r2m+σ ) +Ok ( h2 r2m+2+σ ) + a2A2k ln k πr2m+1 + a2A2C0k ln k r2m+1 + ma2A2h 2k ln k πr2m+3 + ma2A2C0h 2k ln k r2m+3 + a2A2 2hr2m + ma2A2h 2r2m+2 +O( k ln k r2m+1+σ ) +O( h2k ln k r2m+3+σ ) +O( 1 r2m+σh ) +O( h r2m+2+σ ) ] . (4.8) Note that I(Wr,h) = 1 2 ∫ R3 ( |∇Wr,h|2 + |Wr,h|2 ) − 1 p+ 1 ∫ R3 Q(|x|)|Wr,h|p+1 + 1 4 ∫ R3 V (|x|)ΦWr,h W 2 r,h. Then (4.3) follows from (4.8) and (2.2) immediately. The proof is complete. □ 5. Proof of Theorem 1.1 Let ϕr,h = ϕ(r, h) be the map obtained in Proposition 3.3. We define a function F : Dk → R by F (r, h) = I(Wr,h + ϕr,h). With the same argument in [4, Proposition 5.3], we can easily check that if (r, h) ∈ Dk is a critical point of F (r, h), then Wr,h + ϕr,h is a solution of (1.1). Proof of Theorem 1.1. In view of Lemma 3.2, Proposition 4.1 and (3.31), by using the Taylor expansion, we can deduce that F (r, h) = I(Wr,h) +Ok(∥lk∥ ∥ϕ∥+ ∥ϕ∥2) EJDE-2025/72 MULTI-BUMP SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS 13 = k [ A1 + a2A2 r2m − ma2h2A2 r2m+2 − bA3 rn + nbA3 2 h2 rn+2 − 2B1e −2π r k ( k 2πr ) −B1e −2h ( 1 2h ) +Ok ( 1 r2m+σ ) +Ok ( h2 r2m+2+σ ) +Ok ( 1 rn+σ ) +Ok ( h2 rn+2+σ ) +Ok ( e−2(1+σ)π r k ) +Ok ( e−2(1+σ)h )] . 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Tao Wang College of Mathematics and Computing Science, Hunan University of Science and Technology, Xiang- tan, Hunan 411201, China Email address: wt 61003@163.com Xiaoyu Tian College of Mathematics and Computing Science, Hunan University of Science and Technology, Xiang- tan, Hunan 411201, China Email address: Tianxymath@163.com Wenling He College of Mathematics and Computing Science, Hunan University of Science and Technology, Xiang- tan, Hunan 411201, China Email address: hwl202112@163.com 1. Introduction 2. Notation and energy expansion 3. Reduction equation 4. Energy estimate 5. Proof of Theorem 1.1 Acknowledgements References