Existence Solutions for a Class of Schrödinger-Maxwell Systems with Steep Well Potential

where λ, b > 0 are constants, and 3 < p < 6. Such a system is also called the Schrödinger-Poission equation which is obtained while looking for the existence of standing waves for nonlinear Schrödinger equations interacting with an unknown electrostatic field. For more details on the physical aspects, we refer the reader to [1] and the references therein. On the potential V , we make the following assumptions: (V1) V ∈ CðR3, RÞ and V is bounded below. (V2) there exists a constant c > 0 such that the set fx ∈ R3 : VðxÞ ≤ cg is nonempty and measfx ∈ R3 : VðxÞ ≤ cg < +∞, where meas denote the Lebesgue measure in R3. (V3)Ω = int V−1ð0Þ is nonempty and has smooth boundary and Ω =V−1ð0Þ. In recent years, system (1) has been widely studied under various conditions on V and K . The greatest part of the literature focuses on the study of the system for V and K being constants or radially symmetric functions. We refer the reader to [2–9]. When the potential VðxÞ is neither a constant nor radially symmetric, in [10, 11], the existence of ground state solutions is proved for 3 < p < 6. In [12–14], the existence of nontrival solution is obtained via the variational techniques in a standard way under the following condition: (V4) V ∈ CðR3, RÞ, inf x∈R3 VðxÞ ≥ a0 > 0, where a0 > 0 is a constant. Moreover, for any M > 0, measfx ∈ R3 : VðxÞ ≤ Mg < +∞, where meas denote the Lebesgue measure in R3. It is worth mentioning that conditions (V4) were first introduced by Bartsch and Wang [15] to guarantee the compact embedding of the functional space. If replacing ðV4Þ by more general assumptions ðV1Þ and ðV2Þ, the compactness of the embedding fails and this situation becomes more complicated. Recently, [16, 17] considered this case. The authors studied the following problem


Introduction and Main Results
Consider the following system of Schrödinger-Maxwell equations: where λ, b > 0 are constants, and 3 < p < 6. Such a system is also called the Schrödinger-Poission equation which is obtained while looking for the existence of standing waves for nonlinear Schrödinger equations interacting with an unknown electrostatic field. For more details on the physical aspects, we refer the reader to [1] and the references therein.
On the potential V, we make the following assumptions: (V 1 ) V ∈ CðR 3 , RÞ and V is bounded below. (V 2 ) there exists a constant c > 0 such that the set fx ∈ R 3 : VðxÞ ≤ cg is nonempty and measfx ∈ R 3 : VðxÞ ≤ cg < +∞, where meas denote the Lebesgue measure in R 3 .
(V 3 )Ω = int V −1 ð0Þ is nonempty and has smooth boundary and Ω = V −1 ð0Þ. In recent years, system (1) has been widely studied under various conditions on V and K. The greatest part of the literature focuses on the study of the system for V and K being constants or radially symmetric functions. We refer the reader to [2][3][4][5][6][7][8][9].
It is worth mentioning that conditions (V 4 ) were first introduced by Bartsch and Wang [15] to guarantee the compact embedding of the functional space. If replacing ðV 4 Þ by more general assumptions ðV 1 Þ and ðV 2 Þ, the compactness of the embedding fails and this situation becomes more complicated. Recently, [16,17] considered this case. The authors studied the following problem where λ > 0 is a parameter, the potential V may change sign and f is either the superlinear or sublinear in u as juj ⟶ ∞.
Very recently, Liu and Mosconi [18] considered the following system with a coercive sign-changing potential and a 3sublinear nonlinearity: By using a linking theorem, the authors obtained the existence of nontrivial solutions. Nextly, Gu, Jin, and Zhang [19] investigated the existence of sign-changing solutions for system (3). By using the method of invariant sets of descending flow, the multiple radial sign-changing solutions are obtained in the subquadratic case as λ small. For more results about the Schrödinger-Poisson systems, we refer the reader to [20][21][22][23] and the reference therein.
Here, we should point out that for the power-type nonlinearity f ðuÞ = juj p−2 u, in order to get the boundedness of a (PS) sequence, the methods heavily rely on the restriction p ∈ ð4, 6Þ. Meanwile, the condition ðV 1 Þ − ðV 3 Þ cannot guarantee the compactness of the embedding of H 1 ðR 3 Þ into the Lebesgue spaces L s ðR 3 Þ, s ∈ ½2, 6Þ. This prevents from using the variational techniques in a standard way. Motivated by the works mentioned above and [24][25][26][27][28], in the present paper, we are mostly interested in sign-changing potentials and consider system (1) with more general potential V, K, and the range of p. Our main results are as follows: KðxÞ ≥ 0, and 3 < p < 4 hold. Then, system (1) possesses at least a nontrivial solution for b small and λ large. Remark 2. It is known that it is difficult to get the boundedness of a (PS) sequence when dealing with the case p ∈ ð3, 4Þ. To overcome the difficulty, motivated by [24,25], we use the truncation technique to obtain a bounded Cerami sequence for b small. In this case, the conditions ðV 4 Þ and ðK 1 Þ of Theorem 1.3 in [23] cannot be used. Moreover, in the process of proving the convergence of a bounded Cerami sequence, we use the observation that the condition K ∈ L 3 ðR 3 Þ ∪ L ∞ ðR 3 Þ makes the less strong influence of the nonlocal term KðxÞϕu (The conclusions remain valid if K ∈ L 2 ðR 3 Þ ). In this sense, Theorem 1 can be viewed as an improvement of Theorem 1.3 in Zhao et al. [23].

Preliminaries
Let is the usual Sobolev space with the standard inner product and norm In our problem, we work in the space defined by with the inner product and the norm where V ± ðxÞ = max f±VðxÞ, 0g, VðxÞ = V + ðxÞ − V − ðxÞ.
Obviously, it follows from ðV 1 Þ, ðV 2 Þ that the embedding E λ ↪H 1 ðR 3 Þ is continuous. As in [26], let and denote the orthogonal complement of F λ in E λ by F ⊥ λ . Consider the eigenvalue problem In view of ðV 1 Þ, ðV 2 Þ, the quadratic form u ↦ Ð R 3 V − ðx Þu 2 dx is weakly continuous. We have the following proposition.
In the sequel, we denote the usual L p -norm by k·k p , and C stands for different positive constants. For any R > 0, B R ð xÞ denotes the open ball of radius R centered at x. Since the continuity of the following embedding there are constants a s > 0 and a > 0 such that It is well known that system (1) is the Euler-Lagrange equation of the functional J : Evidently, the action functional J belongs to C 1 ðE λ × D 1,2 ðR 3 Þ, RÞ and its critical points are the solutions of system (1). It is easy to know that J exhibits a strong indefiniteness, namely, it is unbounded both from below and from above on infinitely dimensional subspaces. This indefiniteness can be removed using the reduction method described in [29], by which we are led to study a one variable functional that does not present such a strongly indefiniteness.
Actually, considering for all u ∈ E λ , the linear functional If K ∈ L ∞ ðR 3 Þ, the Hölder inequality and the Sobolev inequality imply while for K ∈ L 3 ðR 3 Þ, we have Hence, by the Lax-Milgram theorem, there exists a Moreover, we can write an integral expression for ϕ u in the form: So, we can consider the functional I λ : E λ ⟶ R defined by I λ ðuÞ = Jðu, ϕ u Þ. Then, It follows from (16), (17), (18), and the Sobolev inequality that Thus, I is a well-defined C 1 functional with derivative given by By the proposition 2.3 in [12], we know that ðu, ϕÞ ∈ E λ × D 1,2 ðR 3 Þ is a critical point of J if and only if u is a critical point of I and ϕ = ϕ u .
To complete the proof of our theorem, we need the following results.
Taking v = v t for t large enough, we have JðvÞ < 0: Proof. For any u ∈ E λ , we have Since p > 3, we conclude that there exists ρ > 0 such that JðuÞ ≥ α > 0 for all u ∈ E λ with kuk E λ = ρ: From Lemmas 9 and 10 and Theorem 6, we thus deduce that there exist a Cerami sequence fu n g ⊂ E λ such that Proof. Let fu n g be a Cerami sequence satisfying (32). Let T = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2pðM + 1Þ/p − 2 p , we show that ku n k E λ ≤ T. We first prove that ku n k E λ ≤ ffiffi ffi 2 p T. Suppose by contradiction that there exist a subsequence of fu n g, still denoted by fu n g, such that ku n k E λ > ffiffi ffi 2 p T, we obtain which is a contradiction by Lemma 9. Suppose that there exists no subsequence of fu n g which is uniformly bounded by T. Then, we deduce that Tku n k E λ ≤ ffiffi ffi 2 p T. We handle the case of K ∈ L 3 ðR 3 Þ (The conclusions remain valid if K ∈ L ∞ ðR 3 Þ). By (22), we obtain This is contradiction by choosing b sufficiently small. Since ku n k E λ ≤ T, passing to a subsequence if necessary, we can assume that there exists u ∈ E λ such that u n ⇀ u, inE λ . In view of the Sobolev embedding theorems and Lemma 8, Furthermore, for any φ ∈ E λ , we have In fact, if K ∈ L ∞ ðR 3 Þ, we just need to show that Note that The first limit on the right is 0 by the fact ϕ u n ðu n − uÞ ⇀ 0 inL 2 ðR 3 Þ and so is the second limit because ϕ u n − ϕ u ⇀ 0 inL 6 ðR 3 Þ and uφ ∈ L 6/5 ðR 3 Þ. While for K ∈ L 3 ðR 3 Þ, on the one hand, since Ku 2 ∈ L 6/5 ðR 3 Þ, using (35) (c), we have as n ⟶ ∞. On the other hand, since K ∈ L 3 ðR 3 Þ, for any ε 1 > 0 there exists ρ 1 = ρ 1 ðε 1 Þ > 0 such that Moreover, in view of the Sobolev embedding theorem, u n ⇀ u implies that Hence, for large n, we obtain Consequently, From above inequality and (39), one has

Journal of Function Spaces
Then, J ′ ðu n Þ ⟶ 0 implies that Let v n ≔ u n − u. It follows from ðV 1 Þ, Moreover, let 0 < α < min fð6 − p/2Þ, 1g, 2 < p < 6. Then, 2 < 2ðp − αÞ/2 − α < 6. By the Sobolev inequalities and Hölder inequality, one has We know Letting Λ > 0 be so large that the term in the brackets above is positive when λ ≥ Λ, we get v n ⟶ 0 in E λ . Since v n = u n − u and v n ⟶ 0, it follows that u n ⟶ u in E λ . This completes the proof.
Proof of Theorem 2. Note that fu n g is also a Cerami sequence of I λ satisfying ku n k E λ ≤ T, the conclusion follows from Lemmas 9, 10, and 11 and Theorem 6.

Proof Of Theorem 3
In this section, while V is sign-changing, we study the existence of solutions of (1) for the case 4 < p < 6 and give the proof of Theorem 3. Without loss of generality, we assume that b = 1.
Proof. Since all norms are equivalent in a finite dimensional space, there is a constant b 1 > 0 such that By (22), there is a constant C K > 0 such that Hence, for all u ∈Ẽ, Since p > 4, there is a large r > 0 such that IðuÞ < 0 oñ E \ B r ð0Þ.
This is a contradiction. This implies fu n g is bounded in E λ . Going if necessary to a subsequence, we can assume that u n ⇀ u in E λ . The following proof is similar to the proof of Lemma 11. We omit details of this.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.