Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth

1 Introduction

This article studies the Klein-Gordon equation coupled with the Born-Infeld (BI) theory with critical growth

where ω > 0 is a constant and λ > 0 is a positive parameter. The Klein-Gordon equation can be used to develop the theory of electrically charged fields [14] and study the interaction with an assigned electromagnetic field [12]. The BI electromagnetic theory [2,3] was originally proposed as a nonlinear correction of the Maxwell theory in order to overcome the problem of infiniteness in the classical electrodynamics of point particles (see [15]). The Klein-Gordon equation coupled with BI theory system has attracted many theoretic physicists. For more physical applications, please refer to references [20,30] and the references therein.

In the past decades, many people have studied this system by using variational methods and have also obtained the existence of nontrivial solution under different assumptions. Let us recall some previous results.

The first result is due to d’Avenia and Pisani, in which the existence of infinitely many radially symmetric solutions for the following form:

where 4 < p < 6 and ∣ ω ∣ < ∣ m 0 ∣ in [13]. Mugnai [20] obtained the same result when 2 < p ≤ 4 and 0 < ω < 1 2 p − 1 ∣ m ∣ . Afterward, Wang [26] used Pohožaev identity to improve [13,20] and obtained the solitary wave solution when one of the following conditions is satisfied,

1. 3 < p < 6 and m > ω > 0 ,

2. 2 < p ≤ 3 and ( p − 2 ) ( 4 − p ) m 2 > ω 2 > 0 .

1. 4 < p < 6 and ∣ m ∣ > ω ,

2. 2 < p ≤ 4 and 1 2 p − 1 ∣ m ∣ > ω .

Under some appropriate assumptions on V ( x ) , λ , k ( x ) , and g ( x ) , they obtained the existence of multiple nontrivial solutions when 1 < q < 2 < p < 6 . Recently, for general potential V ( x ) and ∣ u ∣ p − 2 u by a continuous nonlinearity f ( x , u ) with polynomial growth, Wen et al. [27] obtained infinitely many solutions and least energy solution. Che and Chen [7] used the genus theory to obtain the nontrivial solution.

We know that many papers about the nonlinearity are on subcritical growth. When the nonlinearity term is accompanied by critical growth, it is one of the most dramatic cases of loss compactness. To our best knowledge, there are only two works about the Klein-Gordon-Born-Infeld system with critical growth. Teng and Zhang [24] investigated the following system:

They obtained (1.4) has at least a nontrivial solution when 4 < p < 6 and m < ω . Later, the authors of [16] proved that there is a ground state solution for 2 < p < 4 . For some other equations where nonlinearity is critical growth, such as Choquard-Kirchhoff equation, Schrödinger equation, Schrödinger-Poisson system, and Schrödinger-Choquard-Kirchhoff equation, there are many mathematicians have obtained many interesting results. We just list a few, for example, see [10,18,19,21] and the references therein.

Motivated by the aforementioned works, in this article, we will use some new tricks to generalize [16,24] under the following conditions:

1. V ∈ C 1 ( R 3 , R ) and there is a V 0 > 0 such that V ( x ) ≥ V 0 for all x ∈ R 3 .

2. V ( x ) → ∞ as ∣ x ∣ → ∞ .

3. f ∈ C ( R ) , lim u → 0 f ( u ) u = 0 .

4. lim ∣ u ∣ → ∞ f ( u ) u = + ∞ .

Our first result is as follows.

To prove the existence of the nontrivial solution, we adapt a similar argument as in [17,29]. Here, we briefly explain the process. First, we make a suitable cut-off function to replace f ( u ) in problem (1.1), so we can obtain a new system. Second, we prove the new system have a nontrivial solution. Finally, we use the Moser iteration to obtain the existence of a nontrivial solution to the original Klein-Gordon equation coupled with BI theory.

In the second part of this article, when β = 0 , problem (1.1) will become a Klein-Gordon-Maxwell system with critical growth, namely:

This system has been extensively studied by many authors. A pioneering work is due to Cassani [6]. He considered the following Klein-Gordon-Maxwell critical system:

where λ > 0 , 2 < p < 6 , and 0 < ω < m . When N = 3 , he obtained the existence of a radially symmetric solution for any λ > 0 if p ∈ ( 4 , 6 ) and for λ is sufficiently large if p = 4 . Afterward Carrião et al. [5] complement the result of [6] and extend it in higher dimensions. They obtained the existence nontrivial solution of the previous equation provided one of those conditions satisfies

1. N = 4 and N ≥ 6 for 2 < p < 2 ∗ and ∣ m ∣ > ω if λ > 0 ;

2. N = 5 and either 2 < p < 8 3 if λ > 0 or 8 3 ≤ p < 2 ∗ if λ is sufficiently large;

3. N = 3 and either 4 < p < 2 ∗ if λ > 0 or 2 < p ≤ 4 if λ is sufficiently large.

1. 4 < p < 6 , 0 < ω < m and λ > 0 ;

2. 3 < p ≤ 4 , 0 < ω < m and λ is sufficiently large;

3. 2 < p ≤ 3 , 0 < ω < ( p − 2 ) ( 4 − p ) m and λ is sufficiently large.

admits that a nontrivial solution has been proved by Zhang [31]. Instead of the expression “ λ sufficiently large” in the aforementioned existing works, Tang et al. [23] give a certain range λ ≥ λ 0 , which admits a ground state solution when V is positive and periodic.

Similar to the method of Theorem 1.1, we can also obtain a nontrivial solution. Compared with the hypothesis of subcritical perturbation in the aforementioned article, in this article, the perturbation term f ( u ) can be not only a subcritical perturbation but also a supercritical perturbation. What’s more the restriction on λ is no longer sufficiently larger or greater than a certain number, we can only require λ ∈ ( 0 , λ 2 ∗ ) , where λ 2 ∗ ≥ 0 . Our second result is as follows.

This article is organized as follows. In Section 2, we present some preliminary lemmas. In Section 3, we prove Theorems 1.1. In Section 4, we prove Theorem 1.4.

2 Preliminaries

In this section, we explain the notations and some auxiliary lemmas, which are useful later.

H 1 ( R 3 ) denotes the usual Sobolev space equipped with the standard norm.

L ℓ ( R 3 ) , ℓ ∈ [ 1 , + ∞ ) denotes the Lebesgue space with the norm ∣ u ∣ ℓ = ∫ R 3 ∣ u ∣ ℓ d x 1 ℓ .

Under ( V 1 ) and ( V 2 ) , we define the Hilbert space

with respect to the norm

Then, the embedding E ↪ H 1 ( R 3 ) is continuous. The embedding from E into L q ( R 3 ) is compact for q ∈ [ 2 , 6 ) , and its detailed proof process can be seen in Lemma 3.4 in [33].

We denote by D ( R 3 ) the completion of C 0 ∞ ( R 3 ) with respect to the norm

It is easy to know that D ( R 3 ) is continuously embedded in D 1 , 2 ( R 3 ) , where D 1 , 2 ( R 3 ) is the completion of C 0 ∞ ( R 3 ) with respect to the norm ∥ ϕ ∥ D 1 , 2 ( R 3 ) = ∫ R 3 ∣ ∇ ϕ ∣ 2 d x 1 2 . Moreover, D 1 , 2 ( R 3 ) is continuously embedded in L 6 ( R 3 ) by Sobolev inequality and D ( R 3 ) is continuously embedded in L ∞ ( R 3 ) .

C 1 , C 2 , … denote positive constant possibly different in different places.

Indeed, solutions of (1.1) are critical points of functional G λ : E ( R 3 ) × D ( R 3 ) → R , defined by

Due to the strong indefiniteness of functional (2.1), we use the reduction method that can reduce the study of G λ ( u , ϕ ) to study a new functional I λ ( u ) as in [1].

We state some properties of the second equation of problem (1.1).

From the second equation in (1.1) and Lemma 2.1, we obtain

Consider the functional I λ ( u ) : E → R defined by I λ ( u ) = G λ ( u , ϕ u ) , and by combining with (2.3), we obtain

Of course I λ ( u ) ∈ C 1 ( E , R ) and for any u , v ∈ E , we have

From ( f 2 ) , there exist T > 0 large enough such that f ( T ) > 0 . Let

where f satisfies ( f 1 ) , ( f 2 ) , and C T = f ( T ) T p − 1 ( 4 < p < 6 ) . h T is a continuous function and satisfies the following properties:

1. lim t → 0 + h T ( t ) t = 0 .

2. lim t → + ∞ H T ( t ) t 4 = + ∞ , where H T ( t ) = ∫ 0 t h T ( s ) d s .

3. ∣ h T ( t ) ∣ ≤ C T ∗ ∣ t ∣ + C T ∣ t ∣ p − 1 , where C T ∗ = max t ∈ [ 0 , T ] ∣ f ( t ) ∣ t .

4. There exists μ = μ ( T ) > 0 such that t h T ( t ) − 4 H T ( t ) ≥ − μ t 2 for all t ≥ 0 .

We will study the critical points for the functional

From direct calculation, we know that the function h T is continuous, and we have J λ , T ∈ C 1 ( E , R ) , and for any u , v ∈ E ,

The next lemma shows that functional J λ , T ( u ) satisfies the mountain pass geometry.

From Lemma 2.3, we can obtain a ( P S ) sequence, namely, there exists a sequence { u n } ⊂ E satisfying

where