In this paper, we are concerned with the system of the Schrödinger-Maxwell equations where λ, b > 0 are constants, and 3 < p < 6. Under appropriate assumptions on V and K, we prove the existence of positive solutions in the case 3 < p < 4 via the truncation technique. Moreover, suppose that V may change sign, we also obtain the multiplicity of solutions for the case 4 < p < 6.

1. Introduction and Main Results

Consider the following system of Schrödinger-Maxwell equations:

(1)

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 ∈ C(R³, R) and V is bounded below.

(V₂) there exists a constant c > 0 such that the set {x ∈ R³ : V(x) ≤ c} is nonempty and meas{x ∈ R³ : V(x) ≤ c} < +∞, where meas denote the Lebesgue measure in R³.

(V₃)Ω = intV⁻¹(0) is nonempty and has smooth boundary and .

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 V(x) 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 ∈ C(R³, R), where a₀ > 0 is a constant. Moreover, for any M > 0, meas{x ∈ R³ : V(x) ≤ M} < +∞, 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 (V₄) by more general assumptions (V₁) and (V₂), the compactness of the embedding fails and this situation becomes more complicated. Recently, [16, 17] considered this case. The authors studied the following problem

(2)

where λ > 0 is a parameter, the potential V may change sign and f is either the superlinear or sublinear in u as |u|⟶∞. Very recently, Liu and Mosconi [18] considered the following system with a coercive sign-changing potential and a 3-sublinear nonlinearity:

(3)

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 [20–23] and the reference therein.

Here, we should point out that for the power-type nonlinearity f(u) = |u|^p−2u, in order to get the boundedness of a (PS) sequence, the methods heavily rely on the restriction p ∈ (4, 6). Meanwile, the condition (V₁) − (V₃) cannot guarantee the compactness of the embedding of H¹(R³) into the Lebesgue spaces L^s(R³), s ∈ [2, 6). This prevents from using the variational techniques in a standard way. Motivated by the works mentioned above and [24–28], 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:

Theorem 1. Suppose that V ≥ 0, (V₁)- (V₃), , K(x) ≥ 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 ∈ (3, 4). 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 (V₄) and (K₁) 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 ∈ L³(R³) ∪ L^∞(R³) makes the less strong influence of the nonlocal term K(x)ϕu (The conclusions remain valid if K ∈ L²(R³)). In this sense, Theorem 1 can be viewed as an improvement of Theorem 1.3 in Zhao et al. [23].

Theorem 3. Suppose that (V₁), (V₂), K ∈ L³(R³) ∪ L^∞(R³), K(x) ≥ 0, and 4 < p < 6 hold. Then, system (1) possesses infinitely many distinct pairs of nontrivial solutions whenever λ > 0 is sufficiently large.

Remark 4. In Theorem 3, V is allowed to be sign-changing for p ∈ (4, 6). We obtain the multiplicity of solutions for (1).

2. Preliminaries

Let

(4)

is the usual Sobolev space with the standard inner product and norm

(5)

In our problem, we work in the space defined by

(6)

with the inner product and the norm

(7)

where V^±(x) = max{±V(x), 0}, V(x) = V⁺(x) − V⁻(x). Obviously, it follows from (V₁), (V₂) that the embedding E_λ↪H¹(R³) is continuous.

As in [26], let

(8)

and denote the orthogonal complement of F_λ in E_λ by

. Consider the eigenvalue problem

(9)

In view of (V₁), (V₂), the quadratic form is weakly continuous. We have the following proposition.

Proposition 5 ([26], Lemma 8). Suppose (V₁), (V₂), and V ≠ 0. Then, for each fixed j,

(i)
μ_j(λ)⟶0 as λ⟶+∞
(ii)
μ_j(λ) is a nonincreasing continuous function of λ, where is sequence of positive eigenvalues of problem (P) satisfying μ₁(λ) ≤ μ₂(λ) ≤ ⋯≤μ_j(λ)⟶∞ as j⟶∞ and the corresponding eigenfunctions .

Let

(10)

Then,

(11)

Moreover dim for every fixed λ > 0.

In the sequel, we denote the usual L^p-norm by ‖·‖_p, and C stands for different positive constants. For any R > 0, B_R(x) denotes the open ball of radius R centered at x. Since the continuity of the following embedding

(12)

there are constants a_s > 0 and a > 0 such that

(13)

It is well known that system (1) is the Euler-Lagrange equation of the functional J : E_λ × D^1,2(R³)⟶R defined by

(14)

Evidently, the action functional J belongs to C¹(E_λ × D^1,2(R³), R) 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 L_u defined in D^1,2(R³) by

(15)

If K ∈ L^∞(R³), the Hölder inequality and the Sobolev inequality imply

(16)

while for K ∈ L³(R³), we have

(17)

Hence, by the Lax-Milgram theorem, there exists a unique ϕ_u ∈ D^1,2(R³) such that

(18)

Moreover, we can write an integral expression for ϕ_u in the form:

(19)

So, we can consider the functional I_λ : E_λ⟶R defined by I_λ(u) = J(u, ϕ_u). Then,

(20)

It follows from (16), (17), (18), and the Sobolev inequality that

(21)

(22)

Thus, I is a well-defined C¹ functional with derivative given by

(23)

By the proposition 2.3 in [12], we know that (u, ϕ) ∈ E_λ × D^1,2(R³) 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.

Theorem 6 (see [30].)Let X be a real Banach space with its dual space X^∗, and suppose that J ∈ C¹(X, R) satisfies

(24)

for some μ < η, ρ > 0 and e ∈ X with ‖e‖_X > ρ. Let c ≥ η be characterized by

(25)

where Γ = {γ ∈ C([0, 1], X): γ(0) = 0, γ(1) = e} is the set of continuous paths joining 0 and e. Then, there exists a sequence {u_n} ⊂ X such that

(26)

Theorem 7 see ([31], Theorem 9.12). Let E be an infinite dimensional Banach space and let I ∈ C¹(E, R) be even, satisfy (PS), and I(0) = 0. If E = V ⊕ X, where V is finite dimensional and I satisfies.

(I₁) there are constants ρ, α > 0 such that , and

(I₂) for each finite dimensional subspace , there is an such that I ≤ 0 on ,

then, I possesses an unbounded sequence of critical values.

Lemma 8 ([17], Lemma 2.3). The function ϕ_u possess the following properties:

(1)
The mapping u⟶ϕ_u maps bounded sets of E_λ into bounded sets of D^1,2(R³);
(2)
If u_n⇀u in E_λ, then in D^1,2(R³);
(3)
The mapping u⟶ϕ_u : E_λ⟶D^1,2(R³) is continuous.

3. Proof Of Theorem 1

In this section, we give the proof of Theorem 2 by using Theorem 6. Let η ∈ C^′([0, ∞), R) be a cut-off function satisfying 0 ≤ η ≤ 1, η(t) = 1 if 0 ≤ t ≤ 1, η(t) = 0 if t ≥ 2, max_t>0|η^′(t)| ≤ 2 and η^′(t) ≤ 0 for each t > 0. For every T > 0 we consider the truncated functional J : E_λ⟶R defined by

(27)

It is easy to see that J is of C¹. Moreover, for each u, v ∈ E_λ, we have

(28)

Lemma 9. Suppose that 3 < p < 4 and (V₁) − (V₃),V(x) ≥ 0 and , K(x) ≥ 0 hold. Then,

(i)
there exists a v ∈ E_λ\{0},with J(v) < 0
(ii)
there exists M > 0 independent of T, λ and b such that , where , Γ = {γ ∈ C([0, 1], E_λ): γ(0) = 0, γ(1) = v}

Proof. For any t > 0, by (19), we have

(29)

Let x₀ ∈ Ω and

for some r₀ > 0. Let

be such that

. Set v_t = t²u(t(x − x₀)), then

for t > 1. Hence, it is easy to see that

(30)

Since p > 3, we have that J(v_t)⟶−∞ as t⟶+∞. Taking v = v_t for t large enough, we have J(v) < 0.

(iii)
since there exists a constant M > 0 (independent of T, λ and b) such that .

Lemma 10. Suppose that 3 < p < 4 and (V₁), (V₂) hold. Then, there exists α > 0 such that .

Proof. For any u ∈ E_λ, we have

(31)

Since p > 3, we conclude that there exists ρ > 0 such that J(u) ≥ α > 0 for all u ∈ E_λ with

From Lemmas 9 and 10 and Theorem 6, we thus deduce that there exist a Cerami sequence {u_n} ⊂ E_λ such that

(32)

Lemma 11. Let 3 < p < 4, V(x) ≥ 0, (V₁) − (V₂), K(x) ≥ 0, and K ∈ L³(R³) ∪ L^∞(R³) be satisfied. Then, there exists Λ > 1 such that for each λ ∈ (Λ, ∞), if {u_n} ⊂ E_λ is sequence satisfying (32), then {u_n} has a convergent subsequence in E_λ for b small enough.

Proof. Let {u_n} be a Cerami sequence satisfying (32). Let , we show that . We first prove that . Suppose by contradiction that there exist a subsequence of {u_n}, still denoted by {u_n}, such that , we obtain

(33)

which is a contradiction by Lemma 9. Suppose that there exists no subsequence of {u_n} which is uniformly bounded by T. Then, we deduce that . We handle the case of K ∈ L³(R³) (The conclusions remain valid if K ∈ L^∞(R³)). By (22), we obtain

(34)

This is contradiction by choosing b sufficiently small. Since , 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, u_n⇀u inE_λ implies.

(35)

Furthermore, for any φ ∈ E_λ, we have

(36)

In fact, if K ∈ L^∞(R³), we just need to show that

(37)

Note that

(38)

The first limit on the right is 0 by the fact

inL²(R³) and so is the second limit because

inL⁶(R³) and uφ ∈ L^6/5(R³). While for K ∈ L³(R³), on the one hand, since Ku² ∈ L^6/5(R³), using (35) (c), we have

(39)

as n⟶∞. On the other hand, since K ∈ L³(R³), for any ε₁ > 0 there exists ρ₁ = ρ₁(ε₁) > 0 such that

(40)

Moreover, in view of the Sobolev embedding theorem, u_n⇀u implies that

(41)

Hence, for large n, we obtain

(42)

Consequently,

(43)

From above inequality and (39), one has

(44)

Then, J^′(u_n)⟶0 implies that

(45)

Let v_n≔u_n − u. It follows from (V₁), (V₂) that

(46)

Moreover, let 0 < α < min{(6 − p/2), 1}, 2 < p < 6. Then, 2 < 2(p − α)/2 − α < 6. By the Sobolev inequalities and Hölder inequality, one has

(47)

We know

(48)

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 {u_n} is also a Cerami sequence of I_λ satisfying , the conclusion follows from Lemmas 9, 10, and 11 and Theorem 6.

4. 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.

Lemma 13. Let (V₁), (V₂), K(x) ≥ 0, and K ∈ L³(R³) ∪ L^∞(R³) be satisfied. Then, there exist α, ρ > 0 such that I_λ(u) ≥ α for all with .

Proof. By proposition 5, for each fixed λ > Λ, there exists a positive integer n_λ such that μ_j(λ) ≤ 1 for j < n_λ and μ_j(λ) > 1 for j ≥ n_λ. Thus, for any , we have

(49)

for all , where . Since p > 2, the conclusion follows by choosing ρ sufficiently small.

Lemma 14. Let (V₁), (V₂), K(x) ≥ 0, and K ∈ L³(R³) ∪ L^∞(R³) be satisfied. Then, for any finite dimensional subspace , there is a large r > 0 such that I_λ(u) < 0 on .

Proof. Since all norms are equivalent in a finite dimensional space, there is a constant b₁ > 0 such that

(50)

By (22), there is a constant C_K > 0 such that

(51)

Hence, for all ,

(52)

Since p > 4, there is a large r > 0 such that I(u) < 0 on .

Lemma 15. Let (V₁), (V₂), K(x) ≥ 0, and K ∈ L³(R³) ∪ L^∞(R³) be satisfied. Then, there exists Λ > 1 such that, for each λ ∈ (Λ, ∞), I_λ satisfies the (PS)_c condition.

Proof. Let {u_n} be a (PS)_c sequence, that is I_λ(u_n)⟶c and . If {u_n} is unbounded in E_λ, up to a subsequence, we can assume that

(53)

as n⟶∞, after passing to a subsequence. Set

, we can assume that w_n⇀w in E_λ and w_n(x)⟶w(x) a.e.x ∈ R³.

If w = 0, since is weakly continuous, we have

(54)

a contradiction. If w ≠ 0, then the set Ω = {x ∈ R³ : ω(x) ≠ 0} has the positive Lebesgue measure. For x ∈ Ω, one has|u_n(x)|⟶∞ as n⟶∞, Fatou’s lemma shows that as n⟶∞. Thus, by (22), we obtain

(55)

This is a contradiction. This implies {u_n} 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.

Proof of Theorem 3. Obviously, I_λ(0) = 0. Furthermore, I_λ is even. The conclusion follows from Lemmas 13, 14, and 15, and Theorem 7.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgment

This work is supported partially by the Foundation of Baoshan University (BYPY202016).

Open Research

Data Availability

No data were used to support this study.

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

Abstract

1. Introduction and Main Results

2. Preliminaries

3. Proof Of Theorem 1

4. Proof Of Theorem 3

Conflicts of Interest

Acknowledgment

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References

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

Abstract

1. Introduction and Main Results

2. Preliminaries

3. Proof Of Theorem 1

4. Proof Of Theorem 3

Conflicts of Interest

Acknowledgment

Open Research

Data Availability

References

Citing Literature

References

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