Our main purpose in this paper is to study the existence of nontrivial solution for the following system:

( − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = u ln | u | + | u | p − 2 u , in   Ω , − Δ ϕ = − ( ω + ϕ ) u 2 , on   ∂ Ω , (1.1)

where Ω ⊂ ℝ 3 is a smooth bounded domain, ω > 0 is parameters. We assume:

(V) V ∈ L 3 2 ( Ω ) and V 0 = inf x ∈ Ω V ( x ) > − ∞ .

In recent years, the following Klein-Gordon-Maxwell system:

( − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) , in   ℝ 3 , − Δ ϕ = − ( ω + ϕ ) u 2 , in   ℝ 3 , (1.2)

has been object of interest for many researchers, ω > 0 is a parameter, ϕ , u : ℝ 3 → ℝ , V : ℝ 3 → ℝ and f : ℝ 3 × ℝ → ℝ . Such a system was first introduced by Benci and Fortunato [1] as a model describing solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field. The unknowns of the system are the field u associated to the particle and the electric potential ϕ . The presence of the nonlinear term f simulates the interaction between many particles or external nonlinear perturbations. By applying a well known equivariant version of mountain pass theorem, Benci and Fortunato [1] [2] first studied the following special Klein-Gordon-Maxwell system with constant potential m 0 2 − ω 2 ,

( − Δ u + [ m 0 2 − ( ω + ϕ ) 2 ] u = | u | p − 2 u , x ∈ ℝ 3 , Δ ϕ = ( ω + ϕ ) u 2 , x ∈ ℝ 3 . (1.3)

They considered | ω | < | m 0 | and f ( u ) = | u | p − 2 u , 4 < p < 2 ∗ = 6 , and proved that system (1.3) has infinitely many radially symmetric solutions. In [3], D’Aprile and Mugnai extended the interval of definition of the power in the nonlinearity for the case 2 < p ≤ 4 . A nonexistence result has been established by the same authors in [4]. In [5] [6] the existence of ground state solutions of (1.3) was established.

Furthermore, if system (1.3) is added by a lower order perturbation, in the year 2004, Cassani [7] studied this kind of Klein-Gordon-Maxwell system:

( − Δ u + [ m 0 2 − ( ω + ϕ ) 2 ] u = μ | u | p − 2 u + | u | 4 u , x ∈ ℝ 3 , Δ ϕ = ( ω + ϕ ) u 2 , x ∈ ℝ 3 , (1.4)

he obtained the existence of a radially symmetric solution. Later, Wang [8] improved the result of [7]. Without need of symmetry, other related results about the autonomous Klein-Gordon-Maxwell system with a more general function f ( x , u ) can be found in [9] [10] [11] and references therein. Very recently, the existence result for the (1.2) with a nonconstant potential V ( x ) can be found in [12] [13] [14] [15] [16] [17] and so on. Furthermore, on bounded domains about Klein-Gordon-Maxwell system we refer to research in [18] [19].

Moreover, logarithmic nonlinearity is widely used in partial differential equations which describe the mathematical and physical phenomena. For elliptic equations with logarithmic nonlinearity, we can refer to [20] [21] [22] [23] [24] and the references therein. Compared with polynomial nonlinearity, logarithmic nonlinearity has both advantages and disadvantages. However, because logarithmic nonlinearity didn’t satisfy the monotonicity condition and Ambroseti-Rabinowitz condition which is quite different from these in the polynomial case.

Remark 1.1. In this paper, we have:

lim t → 0 t ln | t | + | t | p − 2 t t = − ∞ , (1.5)

it is obviously different from the usual conditions,

lim t → 0 f ( x , t ) t = 0. (1.6)

In other words, t ln | t | + | t | p − 2 t cannot be a special case of f ( x , t ) in general.

Now we state our main results as following.

Theorem 1.1. Assume (V) hold. Then when 4 < p < 6 problem (1.1) has a nontrivial solution ( u , ϕ ) ∈ E × D 1,2 ( Ω ) .

The plan of the paper is as follows. In Section 2, we give the variational framework for problem (1.1) and some preliminary results. In Section 3, we prove the some basic lemmas. In Section 4, we complete the proof of Theorem 1.1.

Throughout the paper, we give the following natations:

・ C and C k ( k = 1 , 2 , ⋯ ) for psositive constants;

・ ‖ u ‖ = ‖ u ‖ D 1,2 ( Ω ) = ( ∫ Ω | ∇ u | 2 d x ) 1 2 denote the norm of D 1,2 ( Ω ) ;

・ ‖ u ‖ H 1 = ‖ u ‖ H 1 ( Ω ) = ( ∫ Ω ( | ∇ u | 2 + u 2 ) d x ) 1 2 denote the norm of H 1 ( Ω ) ;

・ | u | s = ( ∫ Ω     u s d x ) 1 s denotes the norm of L s ( Ω ) ,   ( 1 ≤ s ≤ ∞ ) ;

・ S denotes the Sobolev constant.

Let

E : = { u ∈ H 1 ( Ω ) : ∫ Ω ( | ∇ u | 2 + V ( x ) u 2 ) d x < ∞ } ,

E is a Hilbert space with the inner product,

( u , v ) E = ∫ Ω ( ∇ u ⋅ ∇ v + V ( x ) u v ) d x

and the norm,

‖ u ‖ E = ( ∫ Ω | ∇ u | 2 + V ( x ) u 2 d x ) 1 2 = ( ‖ u ‖ 2 + ∫ Ω     V ( x ) u 2 d x ) 1 2 ,

which is equivalent to the standard norm in H 1 ( Ω ) . Obviously, the embedding E ↪ L s ( Ω ) is continuous, for any s ∈ [ 2,2 ∗ ] , where 2 ∗ = 6 . Consequently, for each s ∈ [ 2,6 ] , there exists a constant S s > 0 such that,

| u | s = S s ‖ u ‖ E ,   ∀   u ∈ E . (2.1)

The solutions ( u , ϕ ) ∈ E × D 1,2 ( Ω ) of the (1.1) system are critical points of the functional J : E × D 1,2 ( Ω ) → ℝ defined as:

J ( u , ϕ ) = 1 2 ∫ Ω ( | ∇ u | 2 + V ( x ) u 2 − | ∇ ϕ | 2 − ( 2 ω + ϕ ) ϕ u 2 ) d x     − 1 2 ∫ Ω     u 2 ln | u | d x + 1 4 ∫ Ω     u 2 d x − 1 p | u | p d x . (2.2)

By standard argument we can see that J ∈ C 1 ( E × D 1,2 ( Ω ) , ℝ ) and that weak solutions of (1.1) turn out to be critical points for the energy functional J.

To obtain our main result, we have to overcome several difficulties. The first difficulty is that J is strongly indefinite, that is, is unbounded both from below and from above on infinite-dimensional subspaces. In order to avoid this indefiniteness, which rules out many of the usual tools of critical point theory, a reduction method is performed in [2] which we now recall.

Similar to [ [4], Lemma 2.1], which deal with in the case of entire domain ℝ 3 , for u and ϕ defined above, we have the following lemmas.

Lemma 2.1. For every u ∈ H 1 ( Ω ) , there exists a unique ϕ = ϕ u ∈ D 1 , 2 ( Ω ) which solves the equation:

− Δ ϕ + u 2 ϕ = − ω u 2 . (2.3)

Moreover, the map Φ : u ∈ H 1 ( Ω ) ↦ ϕ u ∈ D 1,2 ( Ω ) is continuously differentiable, and

− ω ≤ ϕ u ≤ 0   a .e .   in   Ω . (2.4)

Proof. Its proof is similar to [19] [25]. □

Multiplying (2.3) by ϕ u and integrating by parts, one has

∫ Ω | ∇ ϕ u | 2 d x = − ∫ Ω     ω ϕ u u 2 d x − ∫ Ω     ϕ u 2 u 2 d x . (2.5)

Using (2.2) and (2.5), the functional I ( u ) : = J ( u , ϕ ) reduces to the following form

I ( u ) = 1 2 ∫ Ω | ∇ u | 2 d x + 1 2 ∫ Ω     V ( x ) u 2 d x − 1 2 ∫ Ω     ω ϕ u u 2 d x     − 1 2 ∫ Ω     u 2 ln | u | d x + 1 4 ∫ Ω     u 2 d x − 1 p ∫ Ω | u | p d x , (2.6)

and we have for any u , v ∈ E

〈 I ′ ( u ) , v 〉 = ∫ Ω     ∇ u ∇ v d x + ∫ Ω     V ( x ) u v d x − ∫ Ω ( 2 ω + ϕ u ) ϕ u u v d x     − ∫ Ω     u v ln | u | d x − ∫ Ω | u | p − 2 u v d x . (2.7)

Then, ( u , ϕ ) ∈ E × D 1,2 ( Ω ) is a weak solution of (1.1) if, and only if, ϕ = ϕ u and u ∈ E is a critical point of I, that is, a weak solution of

− Δ u + V ( x ) u 2 − ( 2 ω + ϕ u ) ϕ u u = u ln | u | + | u | p − 2 u , (2.8)

for any x ∈ Ω .

The second, to deal with logarithmic nonlinearity u ln | u | . We shall also need a logarithmic Sobolev inequality [26] which holds for all u ∈ H 1 ( ℝ N ) \ { 0 } and b > 0 , we have

2 ∫ ℝ N     u 2 ln | u | | u | 2 d x + N ( 1 + ln b ) | u | 2 2 ≤ b 2 π ∫ ℝ N | ∇ u | 2 d x . (2.9)

For u ∈ H 0 1 ( Ω ) , we can define u ( x ) = 0 for x ∈ ℝ N \ Ω . Then it holds, for any positive number b,

2 ∫ Ω     u 2 ln | u | | u | 2 d x + N ( 1 + ln b ) | u | 2 2 ≤ b 2 π ∫ Ω | ∇ u | 2 d x . (2.10)

Lemma 2.2. (Sobolev imbedding theorem) [ [27], Theorem 1.8] The following imbeddings are continuous:

H 1 ( ℝ N ) ⊂ L p ( ℝ N ) , 2 ≤ p < ∞ , N = 1 , 2 H 1 ( ℝ N ) ⊂ L p ( ℝ N ) , 2 ≤ p ≤ 2 * , N ≥ 3 D 1 , 2 ( ℝ N ) ⊂ L 2 * ( ℝ N ) , N ≥ 3

In particular, the Sobolev inequality holds:

S : = inf u ∈ D 1,2 ( ℝ N ) | u | 2 * = 1 | ∇ u | 2 2 > 0.

Lemma 2.3. [ [27], Lemma 2.13] Set N ≥ 3 , a ∈ L N 2 ( Ω ) , then functional

χ : H 0 1 ( Ω ) → ℝ ,

χ ( u ) = ∫ Ω     a u 2 d x ,   u ∈ H 0 1 ( Ω ) , (2.11)

is weakly continuous.

Lemma 2.4 (Hölder inequality) [28] Assume 1 ≤ p , q ≤ ∞ , 1 p + 1 q = 1 . Then

if u ∈ L p ( Ω ) , v ∈ L q ( Ω ) , we have

∫ Ω | u v | d x ≤ | u | p | v | q . (2.12)

In this section, we prove that the functional I satisfy the Palais-Smale condition in the cases 4 < p < 6 and b ∈ ( 0, 2 π ) . First, we recall that a C 1 functional I satisfies the Palais-Smale condition at level c ((PS)_c in short) if every sequence { u n } n ⊂ H 1 ( Ω ) satisfying I ( u n ) → c and I ( u n ) → 0   ( n → ∞ ) has a convergent subsequence.

We first begin giving the following general mountain pass theorem (see [29]).

Theorem 3.1. Let X is a real Banach space and I ∈ C 1 ( X , ℝ ) , with I ( 0 ) = 0 . Assume that

1) there exist r , α > 0 such that I ( u ) ≥ α for all u ∈ X , with ‖ u ‖ = r ;

2) there exist ‖ e ‖ > r satisfying ‖ u ‖ X > r such that I ( e ) < 0 .

Define Γ : = { γ ∈ C ( [ 0,1 ] , X ) : γ ( 0 ) = 0   and   I ( γ ( 1 ) ) < 0 } ,

c = inf γ ∈ Γ max t ∈ [ 0 , 1 ] I ( γ ( t ) ) ≥ α (3.1)

and there exists a (PS)_c sequence { u n } ∈ X .

Next, we begin proving that I satisfies the assumptions of the mountain pass theorem.

Lemma 3.1. Suppose that 4 < p < 6 and b ∈ ( 0, 2 π ) are satisfied. Then the functional I satisfies the mountain pass geometry, that is,

1) there exist r , α > 0 such that I ( u ) ≥ α for any u ∈ E such that ‖ u ‖ = r ;

2) there exists e ∈ E with ‖ u ‖ > r such that I ( e ) < 0 .

Proof. By (V), (2.4) and (2.10), one has

I ( u ) = 1 2 ∫ Ω | ∇ u | 2 d x + 1 2 ∫ Ω     V ( x ) u 2 d x − 1 2 ∫ Ω     ω ϕ u u 2 d x     − 1 2 ∫ Ω     u 2 ln | u | d x + 1 4 ∫ Ω     u 2 d x − 1 p ∫ Ω | u | p d x = 1 2 ∫ Ω | ∇ u | 2 d x + 1 4 ∫ Ω ( 2 V ( x ) + 1 ) u 2 d x − 1 2 ∫ Ω     ω ϕ u u 2 d x     − 1 2 ∫ Ω     u 2 ln | u | d x − 1 p ∫ Ω | u | p d x

= 1 2 ∫ Ω | ∇ u | 2 d x + 1 4 ∫ Ω ( 2 V ( x ) + 1 ) u 2 d x − 1 2 ∫ Ω     ω ϕ u u 2 d x     − 1 2 ∫ Ω     u 2 ln | u | | u | 2 d x − 1 2 ∫ Ω     u 2 ln | u | 2 d x − 1 p ∫ Ω | u | p d x ≥ 1 2 ∫ Ω | ∇ u | 2 d x + 1 4 ∫ Ω ( 2 V 0 + 1 ) u 2 d x + 3 4 ( 1 + ln b ) | u | 2 2     − b 2 4 π ∫ Ω | ∇ u | 2 d x − 1 2 ∫ Ω     u 2 ln | u | 2 d x − 1 p ∫ Ω | u | p d x = ( 1 2 − b 2 4 π ) ‖ u ‖ 2 + 1 4 ( 2 V 0 + 4 + 3 ln b − 2 ln | u | 2 ) | u | 2 2 − 1 p | u | p p .

(3.2)

When u ∈ E \ { 0 } and | u | 2 ≤ b 3 2 e V 0 + 2 , we get 2 V 0 + 4 + 3 ln b − 2 ln | u | 2 ≥ 0 .

Then, by Sobolev imbedding theorem, one has

I ( u ) ≥ ( 1 2 − b 2 4 π ) ‖ u ‖ 2 − 1 p | u | p p ≥ ( 1 2 − b 2 4 π ) ‖ u ‖ 2 − S p − p 2 p ‖ u ‖ p .

When b ∈ ( 0, 2 π ) , we can choose r , α > 0 such that I ( u ) ≥ α for ‖ u ‖ = r .

On the other hand, let u ∈ E \ { 0 } and 4 < p < 6 , using (2.4), we get

I ( t u ) = t 2 2 ‖ u ‖ 2 − 1 2 ∫ Ω     ω ϕ t u ( t u ) 2 d x + t 2 2 ∫ Ω     V ( x ) u 2 d x     − t 2 2 ∫ Ω     u 2 ln | t u | d x + t 2 4 ∫ Ω     u 2 d x − t p p ∫ Ω | u | p d x ≤ t 2 2 ‖ u ‖ 2 + t 2 ω 2 2 | u | 2 + t 2 2 ∫ Ω     V ( x ) u 2 d x     − t 2 ln t 2 | u | 2 2 − t 2 2 ∫ Ω     u 2 ln | u | d x + t 2 4 | u | 2 2 − t p p | u | p p → − ∞ ,   as   t → + ∞ . (3.3)

Thus, there exists e ∈ E \ { 0 } such that I ( e ) < 0 . This completes the proof of Lemma 3.1.

Lemma 3.2. Assume (V) and 4 < p < 6 hold. Then I satisfies the (PS)_c condition, that is, for any (PS)_c sequence { u n } ⊂ E , there admits a subsequence strongly convergent in E.

Proof. First we show that { u n } is bounded. We assume that { u n } ⊂ E satisifies

c + o n ( 1 ) = I ( u n ) = 1 2 ∫ Ω | ∇ u n | 2 d x + ∫ Ω     V ( x ) u n 2 d x − 1 2 ∫ Ω     ω ϕ u n u n 2 d x     − 1 2 ∫ Ω     u n 2 ln | u n | d x + 1 4 ∫ Ω     u n 2 d x − 1 p ∫ Ω | u n | p d x (3.4)

and

o n ( 1 ) ‖ u n ‖ = 〈 I ′ ( u n ) , u n 〉 = ∫ Ω     ∇ u n 2 d x + ∫ Ω     V ( x ) u n 2 d x − ∫ Ω ( 2 ω + ϕ u n ) ϕ u n u n 2 d x     − ∫ Ω     u n 2 ln | u n | d x − ∫ Ω | u n | p d x . (3.5)

One has

c + o n ( 1 ) ‖ u n ‖ = I ( u n ) − 1 4 〈 I ′ ( u n ) , u n 〉 = 1 4 ∫ Ω | ∇ u n | 2 d x + 1 4 ∫ Ω     V ( x ) u n 2 d x + 1 4 ∫ Ω     ϕ u n 2 u n 2 d x     − 1 4 ∫ Ω     u n 2 ln | u n | d x + 1 4 ∫ Ω     u n 2 d x + ( 1 4 − 1 p ) ∫ Ω | u n | p d x

≥ 1 4 ∫ Ω | ∇ u n | 2 d x + 1 4 ∫ Ω     V ( x ) u n 2 d x − 1 4 ∫ Ω     u n 2 ln | u n | d x     + 1 4 ∫ Ω     u n 2 d x + ( 1 4 − 1 p ) ∫ Ω | u n | p d x ≥ 1 4 ‖ u n ‖ 2 + 1 4 ∫ Ω     V 0 u n 2 d x − 1 4 ∫ Ω | u n | 3 d x + ( 1 4 − 1 p ) ∫ Ω | u n | p d x ≥ C 1 ‖ u n ‖ 2 − C 2 ‖ u n ‖ 3 + C 3 ‖ u n ‖ p , (3.6)

where C 1 , C 2 , C 3 are positive constants. Since p ∈ ( 4,6 ) , then { u n } is bounded in E. Going if necessary to a subsequence (still denoted by { u n } ), we can assume that

u n ⇀ u   in   E , u n → u   in   L s ( Ω ) ,   s ∈ [ 2,6 ) , u n ( x ) → u ( x )   a .e .   in   Ω .

According to (2.7), one obtains

〈 I ′ ( u n ) , u n − u 〉 = ∫ Ω     ∇ u n ∇ ( u n − u ) + ∫ Ω     V ( x ) u n ( u n − u ) d x     − ∫ Ω ( 2 ω + ϕ u n ) ϕ u n u n ( u n − u ) d x     − ∫ Ω ln | u n | u n ( u n − u ) d x − ∫ Ω | u n | p − 2 u n ( u n − u ) d x . (3.7)

Similarly, one gets

〈 I ′ ( u ) , u n − u 〉 = ∫ Ω     ∇ u ∇ ( u n − u ) + ∫ Ω     V ( x ) u ( u n − u ) d x     − ∫ Ω ( 2 ω + ϕ u ) ϕ u u ( u n − u ) d x     − ∫ Ω ln | u | u ( u n − u ) d x − ∫ Ω | u | p − 2 u ( u n − u ) d x . (3.8)

We easily get that

〈 I ′ ( u n ) − I ′ ( u ) , u n − u 〉 = ∫ Ω | ∇ ( u n − u ) | 2 + ∫ Ω     V ( x ) ( u n − u ) 2 d x     − 2 ω ∫ Ω ( ϕ u n u n − ϕ u u ) ( u n − u ) d x     − ∫ Ω ( ϕ u n 2 u n − ϕ u 2 u ) ( u n − u ) d x     − ∫ Ω ( u n ln | u n | − u ln | u | ) ( u n − u ) d x     − ∫ Ω ( | u n | p − 2 u n − | u | p − 2 u ) ( u n − u ) d x . (3.9)

It is clear that

〈 I ′ ( u n ) − I ′ ( u ) , u n − u 〉 → 0,   as   n → ∞ (3.10)

In fact, by (2.5), we get

‖ ϕ u n ‖ 2 = − ∫ Ω     ω ϕ u n u n 2 d x − ∫ Ω     ϕ u n 2 u n 2 d x ≤ − ∫ Ω     ω ϕ u u 2 d x ≤ C ‖ ϕ u n ‖ | u n | 12 5 2 ≤ C ‖ ϕ u n ‖ ‖ u n ‖ H 1 2 , (3.11)

So { ϕ u n } is bounded in D 1,2 ( Ω ) . By the Hölder inequality and the Sobolev inequality, one has

| ∫ Ω ( ϕ u n − ϕ u ) u n ( u n − u ) d x | ≤ | ϕ u n − ϕ u | 6 | u n − u | 3 | u n | 2 ≤ C ‖ ϕ u n − ϕ u ‖ 6 | u n − u | 3 | u n | 2 . (3.12)

Because u n → u in L s ( Ω ) for any s ∈ [ 2,6 ) , we have

∫ Ω ( ϕ u n − ϕ u ) u n ( u n − u ) d x → 0,   as   n → ∞ , (3.13)

and

| ∫ Ω     ϕ u ( u n − u ) 2 d x | ≤ | ϕ u | 6 | u n − u | 3 | u n − u | 2 → 0   as   n → ∞ . (3.14)

Thus, we get

∫ Ω ( ϕ u n u n − ϕ u u ) ( u n − u ) d x = ∫ Ω ( ϕ u n − ϕ u ) u n ( u n − u ) d x + ∫ Ω     ϕ u ( u n − u ) 2 d x → 0,   as   n → ∞ . (3.15)

Observe that the sequence { ϕ u n 2 u n } is bounded in L 3 2 ( Ω ) , since

| ϕ u n 2 u n | 3 2 ≤ | ϕ u n | 6 2 | u n | 3 , (3.16)

so

| ∫ Ω ( ϕ u n 2 u n − ϕ u 2 u ) ( u n − u ) d x | ≤ | ϕ u n 2 u n − ϕ u 2 u | 3 2 | u n − u | 3 ≤ ( | ϕ u n 2 u n | 3 2 + | ϕ u 2 u | 3 2 ) | u n − u | 3 → 0   as   n → ∞ . (3.17)

Because V ( x ) ∈ L 3 2 ( Ω ) , by Lemma 2.3, we get

∫ Ω     V ( x ) ( u n − u ) 2 d x → 0   as   n → ∞ . (3.18)

By | t ln | t | | ≤ C | 1 + t 2 | , t ∈ ℝ , and Hölder inequality, one has

| ∫ Ω ( u n − u ) ( u n ln | u n | − u ln | u | ) d x | ≤ ∫ Ω | u n − u | | u n ln | u n | | d x + ∫ Ω | u n − u | | u ln | u | | d x ≤ C ∫ Ω | u n − u | | 1 + u n 2 | d x + C ∫ Ω | u n − u | | 1 + u 2 | d x → 0   as   n → ∞ . (3.19)

Moreover, by the Hölder inequality, we have that

| ∫ Ω ( | u n | p − 2 u n − | u | p − 2 u ) ( u n − u ) d x | ≤ | | u n | p − 2 u n − | u | p − 2 u | p p − 1 | u n − u | p → 0   as   n → ∞ . (3.20)

Therefore, according to (3.10)-(3.20), we obtain that

‖ u n − u ‖ 2 → 0   as   n → ∞ . (3.21)

Thus { u n } has a strongly convergent subsequence in E.

Next, we only need to prove that u ≠ 0 . Suppose by contradiction that u = 0 , and hence ϕ u = 0 . Since as n → ∞ , 〈 I ′ ( u n ) , u n 〉 → 0 , u n → 0 in L p ( Ω ) ( p ∈ ( 2,2 * ) ) . Thus we get

∫ Ω     V ( x ) u n 2 d x → 0,

∫ Ω     ϕ u n 2 u n 2 d x + 2 ω ∫ Ω     ϕ u n u n 2 d x → 0,

∫ Ω     u n 2 ln | u n | d x → 0,

∫ Ω | u n | p d x → 0.

We may assume

∫ Ω | ∇ u n | 2 d x = ‖ u n ‖ 2 → l ,   l ≥ 0.

Obviously, l = 0 ⇔ u n → 0 in E. As a consequence we obtain that

I ( u n ) → l 2 ,   as   n → ∞ .

According to I ( u n ) → c > 0 , we get

c = l 2 > 0 ,

which implies that l = 0 is impossible, this is, which contradicts with u = 0 . Therefore, u is a nontrivial solution of system (1.1). We have completed the proof of Theorem 1.1.

The author declares no conflicts of interest.

Shi, Q.Y. (2022) Existence of Nontrivial Solution for Klein-Gordon-Maxwell System with Logarithmic Nonlinearity. Open Access Library Journal, 9: e9120. https://doi.org/10.4236/oalib.1109120