Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth

1 Introduction

This study is concerned with the following Klein-Gordon-Maxwell system:

where ω > 0 is a constant, u , ϕ : R 2 → R , V : R 2 → R , and f : R → R is of critical exponential growth in the sense of Trudinger-Moser inequality which is stated in the later part of this study. It is well known that this system arises from describing the interaction between the nonlinear Klein-Gordon field and the electrostatic field. The unknowns of system (1.1) u and ϕ are the fields associated with the particle and electric potential, respectively. For more details and physical backgrounds, we refer the readers to [5,6].

In recent years, the majority of the literature focuses on the study of the special problem

and the first result is of the study of Benci and Fortunat [6] by using variational methods. When 0 < ω < m o and 4 < q < 6 , they showed the existence of infinitely many radially symmetric solutions for Problem (1.2). D’Aprile and Mugnai [21] complement the results in the case 0 < ω < ( q − 2 ) ⁄ 2 m 0 and 2 < q ≤ 4 . In another paper [22], they also proved that (1.2) only has a trivial solution when 0 < q ≤ 2 or q ≥ 6 by a Pohozaev-type argument. Based on the work of [6,21], Azzollini and Pomponio in [2] proved that Problem (1.2) has nontrivial solution if one of the following conditions holds:

1. 4 ≤ q < 6 and 0 < ω < m 0 ;

2. 2 < q < 4 and 0 < ω < ( q − 2 ) ⁄ ( 6 − q ) m 0 .

Furthermore, in (1.2), if m 0 2 − ω 2 is replaced by a potential V ( x ) and ∣ u ∣ q − 2 u by a continuous nonlinearity f ( x , u ) , it reduces to the following Klein-Gordon-Maxwell system:

Recently, Carrião et al. [11] studied the existence of ground state solutions of the corresponding critical problem and supposing that V is positive and periodic. After that, multiplicity results for (1.3) were obtained by He [24] and Li and Tang [28], extending the results obtained by Carrião et al. [11]. Cunha [19] also studied System (1.3), proving that the existence of nontrivial solution under potential function V satisfies the periodic condition. When V is radially symmetric, Chen and Song [13] obtained two solutions of the nonhomogeneous version of system (1.3). For other related results, we can refer [9,15,16,20,26,31] and references therein.

Unlike Klein-Gordon-Maxwell systems on R 3 , there are very few papers dealing with (1.1). In this study, as in [1,12], we are also interested in a borderline case of the Sobolev embedding theorems, commonly known as the Trudinger-Moser case. Different from the case when N ≥ 3 , the case N = 2 is very special, as clearly the Sobolev exponent 2 * becomes ∞ and the corresponding Sobolev embedding yields H 1 ( R 2 ) ⊂ L s ( R 2 ) for all s ∈ [ 1 , + ∞ ) , but H 1 ( R 2 ) ⊈ L ∞ ( R 2 ) . In this case, the Trudinger-Moser inequality in R 2 can be seen as a substitute of the Sobolev inequality, which was first established by Cao in [10], see also [4,14], and reads as follows:

Based on Lemma 1.1, we say f ( t ) has critical exponential growth at t = ± ∞ if it verifies

1. f ∈ C ( R , R ) and there exists α 0 > 0 such that

   (1.4) lim ∣ t ∣ → ∞ ∣ f ( t ) ∣ e α t 2 = 0 for all α > α 0

   and

   (1.5) lim ∣ t ∣ → ∞ ∣ f ( t ) ∣ e α t 2 = + ∞ for all α < α 0 .

It seems that the first result dealing with Klein-Gordon-Maxwell systems with critical exponential growth is due to Albuquerque and Li [1]. Replacing f ( u ) by K ( x ) f ( u ) with K ∈ C ( R 2 , R ) , by using the Mountain Pass theorem, they obtained the existence of nontrivial solutions for (1.1) when V and K satisfy

1. V ∈ C ( R 2 , R ) is positive and radial, and there exist constants a 0 , a ∞ > − 2 such that

   lim ̲ ∣ x ∣ → 0 V ( ∣ x ∣ ) ∣ x ∣ a 0 > 0 , lim ̲ ∣ x ∣ → ∞ V ( ∣ x ∣ ) ∣ x ∣ a ∞ > 0 ,

2. K ∈ C ( R 2 , R ) is positive and radial, and there exist constants b 0 > − 2 and b ∞ < a ∞ such that

   lim ¯ ∣ x ∣ → 0 K ( ∣ x ∣ ) ∣ x ∣ b 0 < ∞ , lim ¯ ∣ x ∣ → ∞ K ( ∣ x ∣ ) ∣ x ∣ b ∞ < ∞ ,

1. f ( t ) = o ( t ) as t → 0 ;

2. there exists a constant μ 0 ≥ 4 such that f ( t ) t ≥ μ 0 F ( t ) ≥ 0 for all t ∈ R ; and

3. there exist ϱ 0 > 0 large enough and κ 0 > 2 such that F ( t ) ≥ ϱ 0 ∣ t ∣ κ 0 for all t ∈ R .

Recently, Chen et al. [12] obtained the existence of nontrivial solutions for (1.1), where V satisfies the following assumptions:

1. V ∈ C ( R 2 , R ) , V ( x ) = V ( ∣ x ∣ ) for all x ∈ R 2 and inf u ∈ H 1 ( R 2 ) \ { 0 } ∫ R 2 [ ∣ ∇ u ∣ 2 + V ( x ) u 2 ] d x ∫ R 2 u 2 d x > 0 ;

2. V ∈ C 1 ( R 2 , R ) and there exists a constant β 0 ≥ 0 such that V ( x ) + 1 2 ∇ V ( x ) ⋅ x ≥ β 0 for every x ∈ R 2 ;

1. there exists a constant μ > 2 such that f ( t ) t ≥ μ F ( t ) ≥ 0 for all t ∈ R ;

2. lim ̲ ∣ t ∣ → ∞ t 2 F ( t ) e α 0 t 2 ≥ κ > 4 α 0 2 ρ 2 , where ρ > 0 such that ρ 2 [ max ∣ x ∣ ≤ ρ V ( x ) + ω 2 ] < 2 ; and

3. there exist M 0 > 0 and t 0 > 0 such that

   F ( t ) ≤ M 0 ∣ f ( t ) ∣ , ∀ ∣ t ∣ ≥ t 0 .

which is essential to prove the nontriviality of the solution of (1.1). By using the extremal function sequences related to the Moser-Trudinger inequality, they give the optimal threshold value under which the Cerami sequence weakly converges to some nontrivial solution of (1.1). Their approach is totally different from that of [1] which depends on the sufficient large parametric perturbation.

Motivated by [12,18,27,29,34], in the present study, we shall further study the existence of nontrivial solutions for (1.1) by using some new analytical approaches. In detail, we introduce a new assumption:

1. lim ̲ ∣ t ∣ → ∞ t 2 F ( t ) e α 0 t 2 ≥ κ > 2 e α 0 2 min r > 0 r − 2 e 1 2 r 2 ( V r + ω 2 ) , where V r ≔ max ∣ x ∣ ≤ r V ( x ) .

To state our results, in addition to (F1)–(F4), we continue to introduce the following assumptions:

1. V ∈ C ( R 2 , R ) , inf u ∈ H 1 ( R 2 ) \ { 0 } ∫ R 2 [ ∣ ∇ u ∣ 2 + V ( x ) u 2 ] d x ∫ R 2 u 2 d x > 0 and there exists a constant r > 0 such that

   lim ∣ y ∣ → ∞ meas { x ∈ R 2 : ∣ x − y ∣ ≤ r , V ( x ) ≤ M } = 0 , ∀ M > 0 ;

2. lim ̲ ∣ t ∣ → ∞ t 2 F ( t ) e α 0 t 2 ≥ κ > λ + ω 2 α 0 2 , where λ > 0 is a constant; and

3. F ( t ) ≥ 0 for all t ∈ R , and there exist μ > 2 and θ > 0 such that

   f ( t ) t − μ F ( t ) + θ t 2 ≥ 0 , ∀ t ∈ R .

Now, our main results can be stated as follows.

When V ( x ) = λ > 0 for all x ∈ R 2 in (1.1), it reduces to the following Klein-Gordon-Maxwell system:

The study is organized as follows. In Section 2, we give the variational setting and preliminaries. We give the proofs of Theorems 1.2–1.5 and Corollary 1.7 in Sections 3.

Throughout the study, we make use of the following notations:

• H 1 ( R 2 ) denotes the usual Sobolev space equipped with the standard norm;

• L s ( R 2 ) ( 1 ≤ s < ∞ ) denotes the Lebesgue space with the norm ‖ u ‖ s = ∫ R 2 ∣ u ∣ s d x 1 ⁄ s ;

• For any x ∈ R 2 and r > 0 , B r ( x ) ≔ { y ∈ R 2 : ∣ y − x ∣ < r } and B r = B r ( 0 ) ;

• C 1 , C 2 , ⋯ denote positive constants possibly different in different places.

2 Variational framework and preliminaries

Under (V1) or (V3), we define the Hilbert space

endowed with the norm ‖ u ‖ ≔ ( u , u ) induced by the scalar product

Then, the embedding E ↪ H 1 ( R 2 ) is continuous, i.e., there exists γ 0 > 0 such that

Under assumptions (V1) (or (V3)), (F1), and (F2), due to the variational characteristic of (1.1), its weak solutions ( u , ϕ ) ∈ E × H 1 ( R 2 ) are critical points of the functional given by

In order to avoid the difficulty originated by the strongly indefiniteness of the functional J , we apply a reduction method developed by [7], reducing the study of J to the study of a one variable functional that does not present such a strongly indefinite nature. To this end, we need the following technical results.

According to [1], for any u ∈ H 1 ( R 2 ) , there exists a unique ϕ = ϕ u ∈ H 1 ( R 2 ) which solves the equation

Moreover, the map I : u ∈ H 1 ( R 2 ) ↦ ϕ u ∈ H 1 ( R 2 ) is continuously differentiable, and

It is easy to see that

By the definition of J and using (2.5), we let

Under assumptions (F1) and (F2), fix α > α 0 , we know that for any ε > 0 and any q ≥ 0 , there exists C α , ε , q > 0 such that

and

In view of Lemma 1.1, we can demonstrate that Φ ∈ C 1 ( E , R ) , and

Following [6], ( u , ϕ u ) ∈ E × H 1 ( R 2 ) is a critical point for J if and only if u ∈ E is a critical point of Φ ; moreover, every critical point u ∈ E is a weak solution to (1.1), i.e., satisfy (1.1) with ϕ = ϕ u .

When V satisfies the radial condition (V1), to handle the lack of compactness, we restrict the working space to radial functions, in other words, we will consider the relevant problems in the following space:

which is compactly embedded in L s ( R 2 ) for s > 2 .

To prove Theorems 1.2 and 1.3, we apply Jeanjean’s monotonicity trick [25], i.e., an approximation procedure to obtain a bounded (PS)-sequence for Φ , instead of starting directly from an arbitrary (PS)-sequence. For this purpose, for every λ ∈ [ 1 ⁄ 2 , 1 ] , we introduce a family of functional on E r defined by

When V satisfies the coercive condition (V3), in view of [8, Lemma 3.1], [35], the embedding E ↪ L s ( R 2 ) is compact for any s ≥ 2 .

Inspired by [17], to overcome the difficulty arising from the critical growth of Trudinger-Moser-type, with (F4), we control the minimax level in a suitable range where we can recover the compactness.

To this end, as in [23], we define Moser-type functions θ n ( x ) supported in B ρ ( 0 ) as follows:

where ρ > 0 satisfying ρ − 2 e 1 2 ρ 2 ( V ρ + ω 2 ) = min r > 0 r − 2 e 1 2 r 2 ( V r + ω 2 ) and V r ≔ max ∣ x ∣ ≤ r V ( x ) . Through a direct computation, we have

and

where

for n ≥ 2 .