Existence result for the critical Klein-Gordon-Maxwell system involving steep potential well

: The Klein-Gordon-Maxwell system has received great attention in the community of mathematical physics. Under a special superlinear condition on the nonlinear term, the existence of solution for the critical Klein-Gordon-Maxwell system with a steep potential well has been solved. In this paper, under two general superlinear conditions, we obtain the existence of ground state solution for the critical Klein-Gordon-Maxwell system with a steep potential well. The general superlinear conditions bring challenge in proving the boundedness of Cerami sequence, which is a key step in the proof of the existence. To solve this, we construct a Pohoˇzaev identity and adopt some analytical techniques. Our results extend the previous results in the literature.


Introduction
The Klein-Gordon-Maxwell (KGM) system [1,2] describes the solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field.It is widely employed in many mathematical physics contexts, such as quantum electrodynamics, semiconductor theory, nonlinear optics and plasma physics.In this paper, we will investigate the existence of solution for two cases of critical Klein-Gordon-Maxwell system with steep potential well.
To review the existing work, we start with the following KGM system: where m, ω > 0 are constants, standing for the particle's mass and the phase, respectively; ϕ : R 3 → R and f : R 3 × R → R defines the electric potential and the nonlinear term of the particle's field u, respectively.The nonlinear term f describes the interaction between unknown particles or external nonlinear perturbations.If f does not explicitly depend on x, but only on u, we say f is autonomous.When f (x, u) = |u| q−2 u, Benci and Fortunato [1] proved that (1.1) has infinitely many radially symmetric solutions if 4 < q < 6 and |m| > |ω|; D'Aprile and Mugnai [3] proved that (1.1) has no solution if q ≥ 6 or q ≤ 2; further, Azzollini and Pomponio [4] studied the existence of a ground state solution for (1.1) when one of the following conditions holds: (i) 4 ≤ q < 6 and m > ω; (ii) 2 < q < 4 and m q − 2 > ω 6 − q.
Zhang et al. [10] extended the range of ω for which the ground state solution of (1.4) exists.
Comparing the conditions (f1)-(f2) (used in Zhang [11]) with (F1)-(F3) (used in Tang [7]), we find that, (f1) is essentially the same as (F1) for the positive ground state solutions; while the inequality f (t)t − 4F(t) ≥ 0 in (f2) is a special case of the inequality f (t)t ≥ θF(t) with θ = 4 in (F2), and the exponentially increasing property in (f2) is stronger than that in (F3).Therefore, we apply the conditions (F1)-(F3) rather than (f1)-(f2) to the nonlinear function f in (1.5) and its following extension: and give results about the existence of ground state solution.Note that, in (1.6), we use a potential K, instead of the constant µ in (1.5).Assume K : R 3 → R satisfies the following assumptions: For A, instead of (A3), we apply the following weaker condition: (A3') ⟨∇A(x), x⟩ ≥ 0 for all x ∈ R 3 .Our first result is as follows.
Theorem 1.1.Assume that A satisfies (A1), (A2) and (A3'), f satisfies (F1)-(F3), then problem (1.5) has a ground state solution when one of the following conditions holds: where µ 0 is a positive constant determined by A, α and s.Theorem 1.2.Assume that A satisfies (A1), (A2) and (A3'), f satisfies (F1)-(F3) and K satisfies (K1)-(K2), then problem (1.6) has a ground state solution when one of the following conditions holds: 3. After we replaced (f2) with (F2) and (F3), the following function still satisfies (F2) and (F3) but not (f2): It is easy to verify that K satisfies (K1) and (K2).There seems no results dealing with the nonlinearity which is combined with K and f in the existing results by using Pohožaev identity since it is difficult to prove the compactness of functional associated with problem (1.6).
The paper is organized as follows.In Section 2, some preliminary results are presented.In Section 3, we give the proof of Theorem 1.1.At last, the proof of Theorem 1.2 is given in Section 4.

Preliminaries
In this section, we will introduce some notations and lemmas which will be used in the proof of our theorems.
. Define E as the variational space equipped with the norm where A satisfies (A1) and (A2).By (A1), (A2) and the Poincaré inequality, E → H 1 (R 3 ) is continuous for any s ∈ [2,6], and there exists γ s > 0 such that Moreover, the map

Proof of Theorem 1.1
Similar to the argument in [3], we define the functional J λ (u) : E → R associated with (1.5) by By Lemmas 2.1 and 2.2, J λ ∈ C 1 (E, R), we have Let M := {u ∈ H 1 (R 3 )\{0} : J ′ λ (u) = 0} be the collection of the critical points of J λ .Any critical point u of J λ satisfies the following Pohožaev identity [3]: where Proof.The proof of Lemma 3.1 is similar to [19, Theorem 2.2], so we omit it here.□ In the following, we first estimate the upper bound of critical value c λ and prove the mountain pass geometry of energy function J λ by using Brézis-Nirenberg techique [19].Then we give the proof of Theorem 1.1.Lemma 3.2.Assume that (F1)-(F3) and (A1)-(A2) hold.If one of the following conditions holds: Then, we have c λ < 1 3 S 3/2 , where S is the best Sobolev constant for the embedding D 1,2 (R 3 ) → L 6 (R 3 ) and µ 0 is a positive constant given in (3.28).
Then, from Lemma 3.3 and c λ > 0, we assume that there exists a constant l > 0 such that This is a contradiction.Then there exists κ > 0 such that lim n→∞ sup From (A2) and Lemma 3.3, there exists a constant C 4 > 0 such that uniformly in n.Combining Hölder and Sobolev inequality, we obtain where s ∈ (2,6].Since meas(D R ) → 0 as R → 0 by (A2), for any δ > 0 and taking δ < κ 4 , there exists R * such that R > R * , and we obtain uniformly in n.From u n → 0 in L s loc with s ∈ (2, 6), (3.42) and (3.43), we have This is a contradiction.Hence, there exists u ∈ E\{0} such that By (3.44) and (3.45), we have J λ (ȗ) = m = inf M J λ (u).That is to say ȗ is a ground state solution for (1.5).□

Proof of Theorem 1.2
The proof is easy for case (i).For the proof of cases (ii) and (iii), due to the non-constant potential K in the nonlinearity of (1.6), it is challenging in obtaining the boundness of the Cerami sequence through the Pohožaev identity.We tackle this by applying the condition (K2) and some analytical techniques.
The proof of rest part is similar to that in Lemma 3.3.So, {u n } is bounded in E when θ ∈ (2, 4).□ The following lemma is used to prove that the sequence obtained in Lemma 4.2 is non-vanishing.Lemma 4.4.Under assumptions of Theorem 1.2, problem (1.6) admits a nontrivial solution.
Proof.Since K(x) > 0 and it is bounded, by using the deduction in the proof of Theorem 1.1, we can easily prove this lemma.So we omit here.□ Now, Theorem 1.2 can be proved by using Lemmas 4.1-4.4.

Conclusions
In this paper, we investigate a ground state solution for the critical KGM system with a steep potential well and its extension using general conditions and Pohožaev identity.Obviously, the techniques we use have been successfully applied to find the solution of the critical KGM system, and hope that these results can be widely used in other systems.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.