MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHR¨ODINGER-POISSON SYSTEM

. This article concerns the existence of solutions to the Schr¨odinger-Poisson system


Introduction and statement of main results
This article concerns the existence of solutions to the Schrödinger-Poisson system where 4/3 < p < 12/5, p < q < p * = 3p 3−p , ∆ p u = div(|∇u| p−2 ∇u), λ > 0, and h = 0. The system (1.1) can be viewed as a perturbation of the system −∆ p u + |u| p−2 u + λφu = |u| q−2 u in R 3 , −∆φ = u 2 in R 3 . (1.2) This system was first considered by Du, Su, and Wang in [11] where the variational framework was built and the existence of nontrivial solutions was established via the mountain pass theorem. For p = 2, the system (1.2) reduces to the following classical Schrödinger-Poisson system where λ > 0 and q ∈ (2, 6). Such a system, also known as the nonlinear Schrödinger-Maxwell equation, has an interesting physical context. According to a classical model, the interaction of a charged particle with an electromagnetic field can be described by coupling a nonlinear Schrödinger equation and a Poisson equation. For more details on the physical aspects of the system we refer to the pioneering works of Benci and Fortunado [5,6] and the references therein. In the past decades, the existence of solutions to the system (1.3) has been discussed in [4] for q ∈ (3, 6), in [9,10] for q ∈ [4,6), and in [2,3,21,25,31] for q ∈ (2, 6) or general nonlinearity. For p = 2, the system (1.1) reduces to the nonhomogeneous Schrödinger-Poisson system −∆u + u + λφu = |u| q−2 u + h(x) in R 3 , where λ > 0, q ∈ (2, 6) and h(x) ≡ 0. In [22], Salvatore obtained multiple radial solutions to the system (1.4) for q ∈ (4, 6) and h ∈ L 2 (R 3 ) being radial with small L 2 -norm. In [16], Jiang, Wang and Zhou considered the system (1.4) with h ∈ C 1 (R 3 ) ∩ L 2 (R 3 ) being a nonnegative radial function and satisfying (x, ∇h) ∈ L 2 (R 3 ). Applying the Ekeland's variational principle and the mountain pass theorem, it was proved in [16] that the system (1.4) admitted two radial solutions for q ∈ (2, 6) with small L 2 -norm |h| L 2 (R 3 ) of h and for q ∈ (2, 3] with λ > 0 also small. For other works related to the system (1.4) or to similar systems involving certain potentials, we refer to [8,13,17,20,26,27,30,32,33] and the references therein.
After an accurate bibliographic review, we see that it is open question the existence of multiple solutions to the quasilinesr system (1.1) with 4/3 < p < 12/5 and h = 0. Inspired by this fact, We aim to establish the existence of multiple solutions to system (1.1). We use τ = τ τ −1 to denote the Hölder conjugate of τ > 1. We impose on h the following assumption.
(H1) h is a nonzero radial function and for (p * ) ≤ s ≤ p , where the gradient ∇h is in the weak sense. We will prove the following theorems. Theorem 1.1. Assume that (H1) holds and 6p p+2 < q < p * . Then there exists Λ > 0 such that for |h| L s (R 3 ) < Λ the system (1.1) admits two solutions for any λ > 0. Theorem 1.2. Assume that (H1)(i) holds and p < q ≤ 6p p+2 . There exist Λ > 0 and λ * > 0 such that for |h| L s (R 3 ) < Λ, system (1.1) admits two solutions for any λ ∈ (0, λ * ). Remark 1.3. The first attempt of the study on the Schrödinger-Poisson system (1.2) with p-Laplacian were made in [11]. Now the results in Theorems 1.1 and 1.2 extend the results in [16,22] from p = 2 to the quasilinear case 4/3 < p < 12/5. This range of p was first determined in [11]. We observe a phenomenon that the solvability of the system (1.1) can be considered for a large class of radial functions h satisfying (H1). In this sense the existence results in [16] may be extended to the case that h and (x, ∇h) belonging to L s (R 3 ) with 6/5 ≤ s ≤ 2.
Notice that for p = 2, it is difficult to prove the Pohožaev identity which is essential to establish the boundedness of Palais-Smale sequences for q ∈ (3, 6) in [16]. To overcome this difficulty, for 6p/(p + 2) < q < p * we introduce an auxiliary functional and use an indirect method to do that: see our proof of Lemma 4.3. It also should be pointed out that our method is more applicable. As far as we know, this article is the first attempt to study the nonhomogeneous Schrödinger-Poisson system with p-Laplacian.
The proofs of the main results will be obtained by exploiting suitable variational methods. In Section 2, we give some preliminary results concerning the variational structure for the system (1.1). In Section 3, with the aid of the Ekeland's variational principle [12], we obtain by Theorem 3.3 a solution of (1.1) with negative energy for p < q < p * . In Section 4 we obtain a solution of (1.1) with positive energy and discuss with two cases of 6p p+2 < q < p * and p < q ≤ 6p p+2 . In Subsection 4.1, we use the scaling technique beginning in [14] and developing in [11] to obtain the boundedness of a Palais-Smale sequence for 6p p+2 < q < p * and find a positive energy solution by using the mountain pass theorem [1], see Theorem 4.1. In Subsection 4.2, by using the cut-off technique as in [15] and combining some delicate analysis, we prove a positive energy solution of (1.1) with p < q ≤ 6p p+2 and λ > 0 small, see

Preliminaries
In this section we give some preliminary results related to the variational structure of system (1.1). We will use the following function spaces.
• W 1,p (R 3 ), the Sobolev space with the norm u = R 3 |∇u| p + |u| p dx 1/p , and It is a Hilbert space with the inner product v, w = R 3 ∇v∇w dx. It follows from the classical Sobolev embedding theorems that Restricted to the radial case, it holds that the embedding W 1,p We will use C to denote various positive constants. We will use the following elementary inequality (see [24, p240]) in later arguments: There exists c p > 0 such that for all ξ, η ∈ R 3 , we have (2.1) For each fixed u ∈ W 1,p (R 3 ), we define a linear functional K : By the Hölder and Sobolev inequalities, we have Therefore K is continuous on D 1,2 (R 3 ). By the Lax-Milgram theorem, there exists a unique φ u ∈ D 1,2 (R 3 ) satisfying the equation −∆φ u = u 2 . According to [18,Theorem 6.21], φ u has the explicit expression φ u (x) = 1 We note here that the third conclusion comes from a fact that the convolution of two radial functions is still radial. Now we are ready to establish the variational framework of (1.1). For h ∈ L s (R 3 ) with (p * ) ≤ s ≤ p , arguing as in [5,6], by Proposition 2.1 and the implicit function theorem, the functional is a solution of the system (1.1). Then we will prove Theorems 1.1 and 1.2 by looking for critical points of I λ .
The following result is crucial and can be proved by applying some ideas from Boccardo and Murat [7]. We include the proof for completeness.

A solution with negative energy
In this section we find a solution of (1.1) with negative energy for p < q < p * , and h satisfying (H1)(i) and small |h| L s (R 3 ) .
Proof. For u ∈ W 1,p (R 3 ) and λ > 0, since φ u ≥ 0, it follows from Hölder and Sobolev inequalities that where S denotes the embedding constant of Since q > p, there exists a unique ρ > 0 such that the function . Then by (3.1) we have that when |h| L s (R 3 ) < Λ, I λ (u) ≥ α for any u = ρ.
Next we work on the Sobolev space W 1,p r (R 3 ) of radial functions. Lemma 3.2. Assume that h satisfies (H1)(i). Then each bounded sequence {u n } ⊂ W 1,p r (R 3 ) satisfying I λ (u n ) → 0 has a strongly convergent subsequence.
Proof. We first find a function w ∈ W 1,p and h is radial. Therefore, there exists n 0 ∈ N such that By Hölder's inequality, we conclude that It follows that : u ≤ ρ} and ρ is given by Lemma 3.1. Applying the Ekeland variational principle [12], we obtain a sequence {u n } ⊂B ρ satisfying It must be that u n < ρ for all n ∈ N large. Otherwise, we may assume that u n = ρ, up to a subsequence. By Lemma 3.1, we see that I λ (u n ) ≥ α > 0. Then there is a contradiction by taking the limit in (3.9) as n → ∞. We can assume that u n < ρ for all n ∈ N. Now we show that I λ (u n ) → 0. For any z ∈ W 1,p r (R 3 ) with z = 1, we choose sufficiently small δ > 0 such that u n + tz < ρ for all |t| < δ. By (3.10), we have Letting t → 0, we obtain I λ (u n ), z ≥ −1/n. Similarly, replacing z with −z in the above arguments, we obtain I λ (u n ), z ≤ 1/n. Then, we deduce that, for any z ∈ W 1,p r (R 3 ) with z = 1, I λ (u n ), z → 0 as n → ∞. Thus {u n } is a bounded (PS) c * sequence of I λ . Finally, by Lemma 3.2, there exists u * ∈ W 1,p r (R 3 ) such that I λ (u * ) = c * < 0 and I λ (u * ) = 0.

A solution with positive energy
In this section we find a solution of (1.1) with positive energy. In Subsection 4.1 we consider the case 6p p+2 < q < p * and in Subsection 4.2 we consider the case p < q ≤ 6p p+2 . We still work on W 1,p r (R 3 ).
4.1. Case 6p p+2 < q < p * . In this subsection we will prove the following theorem.
Taking u n = Φ(σ n , v n ), we have Now we prove that {u n } is bounded in W 1,p r (R 3 ). By (4.4), for n large enough, It follows that

.5) EJDE-2023/28
It is easy to see that R 3 hu n dx ≤ C u n . We deduce from (H1), the Hölder and Sobolev inequalities that Therefore by (4.5) We adopt some techniques from [15] to do the proof. We introduce a smooth function χ ∈ C ∞ (R + , [0, 1]) which satisfies We define a penalized functional I λ,M : where M > 0 and L M (u) = χ u p M p . It is standard to prove that I λ,M belongs to C 1 , and for all u, v ∈ W 1,p r (R 3 ), where From the definition one sees that if u is a critical point of I λ,M and u ≤ M/2, then u is a critical point of I λ . We first verify that the penalized functional I λ,M possesses a mountain pass geometry for each M > 0.
(ii) Arguing as in the proof of Theorem 3.3, we can choose a function ω 1 ∈ W 1,p r (R 3 ) such that ω 1 = 1 and R 3 h(x)ω 1 (x) dx > 0. For each M > 0 and t ≥ M , it follows from the definition of χ that L M (tω 1 ) = 0. Thus h(x)ω 1 dx.
Since p < q, we take ω = t M ω 1 and t M > M large, so that ω > ρ and I λ,M (ω) < 0. This completes the proof. Proof. Let {u n } be bounded in W 1,p r (R 3 ). Up to a subsequence, there exists u ∈ W 1,p r (R 3 ) such that u n u in W 1,p r (R 3 ), u n → u in L q (R 3 ) for all p < q < p * and u n (x) → u(x) a.e. in R 3 . Therefore   It can be shown in a same way that |a λ,M (u)| is bounded. By Lemma 2.2, we have that ∇u n (x) → ∇u(x) a.e. in R 3 . Combing with u n (x) → u(x) a.e. in R 3 , we deduce by [29,Proposition 5.4.7] that [u, u n − u] = o(1). (4.14) It follows from (4.11) and (4.14) that Arguing as in the proof of Lemma 3.2 we obtain that u n − u → 0 as n → ∞.