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The following fractional Klein-Gordon-Maxwell system is studied
(-Δ)^{p} stands for the fractional Laplacian, *ω* > 0 is a constant, *V* is vanishing potential and *K* is a smooth function. Under some suitable conditions on *K* and *f*, we obtain a Palais-Smale sequence by using a weaker Ambrosetti-Rabinowitz condition and prove the ground state solution for this system by employing variational methods. In particular, this kind of problem is a vast range of applications and challenges.

In this paper, the following fractional Klein-Gordon-Maxwell system is considered

{ ( − Δ ) p u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = K ( x ) f ( u ) , in ℝ 3 , ( Δ ) p ϕ = ( ω + ϕ ) u 2 , in ℝ 3 , (1.1)

where p ∈ ( 3 / 4 ,1 ) , ( − Δ ) p denotes the fractional Laplacian operator, V is zero mass potential and K is a smooth function. When ( 2 ω + ϕ ) ϕ u = 0 , system (1.1) reduces to a fractional Schrödinger equation. The fractional Schrödinger equation was first proposed by Laskin [

If p = 1 , V ( x ) = m 2 − ω 2 and K ( x ) f ( u ) = | u | q − 2 u , system (1.1) reduces to a Klein-Gordon-Maxwell equation, which was first studied by Benci and Fortunato [

When 2 < q < 4 and 0 < ω < q 2 − 1 m , D’Aprile and Mugnai [

{ − Δ u + [ m 2 − ( ω + ϕ ) 2 ] ϕ u = | u | q − 2 u , x ∈ ℝ 3 , Δ ϕ = ( ω + ϕ ) u 2 , x ∈ ℝ 3 , (1.2)

they obtained some results which complete the results obtained in [

In recent years, under various hypotheses on the potential V ( x ) and the nonlinearity f ( u ) , the existence of positive, multiple, ground state solutions for Klein-Gordon-Maxwell systems or similar systems, has been widely studied in the literature. For example, Azzollini and Pomponio [

(H4’) For all u > 0 , There exists μ > 4 such that 0 < μ F ( u ) ≤ f ( u ) u , where F ( u ) = ∫ 0 u f ( t ) d t .

Inspired mainly by the aforementioned results, we find a ground state solution for (1.1) with potential vanishing at infinity. To show our result, we make the following assumptions first:

(H1) V ∈ C ( ℝ 3 , ( 0, + ∞ ) ) , K ∈ L ∞ ( ℝ 3 ) ∩ C ( ℝ 3 , ( 0, + ∞ ) ) and

K / V ∈ L ∞ ( ℝ 3 ) , (1.3)

or for any p ∈ ( 0,1 ) , there exists s ∈ ( 2,2 p * ) , where 2 p * = 6 / ( 3 − 2 p ) , such that

lim | x | → ∞ K ( x ) V ( x ) γ = 0 , γ = 2 p s − 3 ( s − 2 ) 4 p ∈ ( 0 , 1 ) . (1.4)

(H2) f ∈ C ( ℝ , ℝ + ) and f | ℝ − = 0 . If (1.3) holds, then

lim sup t → 0 + f ( t ) t = 0.

If (1.4) holds, then

lim sup t → 0 + f ( t ) t s − 1 < + ∞ .

(H3) If (1.3) holds, then

lim sup t → + ∞ f ( t ) t 2 p * − 1 = 0.

If condition (1.4) holds, we assume that

lim sup s → + ∞ F ( t ) t p < + ∞ .

(H4) There exists μ > 2 , such that f ( u ) u ≥ μ F ( u ) > 0 for all u > 0 .

To the best of our knowledge, Ambrosetti-Rabinowitz condition (AR condition for short) plays an important role in proving the boundedness of Palais-Smale sequence (PS sequence for short). In recent years, there are many papers devoted to replacing (AR) condition with weaker condition. It is easy to see that (H4) is weaker than (H4’). In this paper, we obtain a (PS) sequence by using the weaker (AR) condition. Besides, it seems that there is only one work about the Klein-Gordon-Maxwell system involving fractional Laplacian.

Theorem 1.1. Assume that p ∈ ( 3 / 4 ,1 ) and (H1)-(H4) hold. Then problem (1.1) admits a positive solution in E, where E is defined in Section 2.

In this paper, the main difficulty is lack of compactness of Sobolev embedding in whole space because of the nonlocal term ϕ and the fractional operator. To overcome this problem, we use the reduction method introduced by Caffarelli and Silvestre [

This paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of main result.

In this section, by the local reduction derived from Caffarelli and Silvestre [

{ − div ( y 1 − 2 p ∇ w 1 ) = 0, in ℝ + 4 , w 1 = u , on ℝ 3 × { 0 } , k p y 1 − 2 p ∂ w 1 ∂ η = K ( x ) f ( u ) + ( 2 ω + ϕ ) ϕ u − V ( x ) u , on ℝ 3 × { 0 } , − div ( y 1 − 2 p ∇ w 2 ) = 0, in ℝ + 4 , w 2 = ϕ , on ℝ 3 × { 0 } , k p y 1 − 2 p ∂ w 2 ∂ η = ( ω + ϕ ) u 2 , on ℝ 3 × { 0 } , (2.1)

where div ( y 1 − 2 p ∇ w 1 ) denotes the divergence of y 1 − 2 p ∇ w 1 and k p = 2 1 − 2 p Γ ( 1 − p ) / Γ ( p ) such that

− k p l i m y → 0 + y 1 − 2 p ∂ w 1 ( x , y ) ∂ y = ( − Δ ) p u ( x ) ,

where ϕ ( x ) = w 2 ( x , 0 ) : = w ˜ 2 , u ( x ) = w 1 ( x , 0 ) : = w ˜ 1 , and

y 1 − 2 p ∂ w 1 ∂ η = − lim y → 0 + y 1 − 2 p ∂ w 1 ∂ y ,

is the outward normal derivative of w 1 . Similar definition is given for w 2 .

For p ∈ ( 3 / 4 ,1 ) and ϕ : ℝ 3 → ℝ , the fractional Laplacian ( − Δ ) p of ϕ is defined by

F ( ( − Δ ) p φ ) ( z ) = | z | 2 p F ( φ ) ( z ) , z ∈ ℝ 3 ,

where F denotes the Fourier transform, that is

F ( φ ) ( z ) = 1 ( 2 π ) 3 / 2 ∫ ℝ 3 e x p ( − 2 π j z ⋅ x ) φ ( x ) d x ,

where j denotes the imaginary unit. When φ is smooth enough, the ( − Δ ) p of φ can be obtained by the following singular integral

( − Δ ) p φ ( x ) = c α P . V . ∫ ℝ 3 φ ( x ) − φ ( y ) | x − y | 3 + 2 p d y , x ∈ ℝ 3 ,

where c α is a normalization constant and P . V . is the principle value.

For any p ∈ ( 3 / 4 ,1 ) , X 2 p ( ℝ + 4 ) and H p ( ℝ 3 ) are the completion of C 0 ∞ ( ℝ + 4 ¯ ) and C 0 ∞ ( ℝ 3 ) , and endowed with the norms

‖ u ‖ X 2 p : = ( ∫ ℝ + 4 k p y 1 − 2 p | ∇ u | 2 d x d y ) 1 / 2 ,

‖ u ‖ H p : = ( ∫ ℝ 3 | 2 π z | 2 p | F ( u ( z ) ) | 2 d z ) 1 / 2 = ( ∫ ℝ 3 | ( − Δ ) p u | 2 d x ) 1 / 2 ,

respectively. The Sobolev space D p ,2 ( ℝ + 4 ) is defined by

D p ,2 ( ℝ + 4 ) = { u ∈ L 2 p * ( ℝ + 4 ) : | z | p u ^ ∈ L 2 ( ℝ + 4 ) } ,

which is the completion of C 0 ∞ ( ℝ + 4 ) under the norm

‖ u ‖ D p ,2 ( ℝ + 4 ) 2 = ‖ ( − Δ ) p / 2 u ‖ 2 2 = ∫ ℝ + 4 y 1 − 2 p | ∇ ( u ) | 2 d x d y , u ∈ D p ,2 ( ℝ + 4 ) .

Let E be defined by

E = { u ∈ X 2 p ( ℝ + 4 ) : ∫ ℝ 3 V ( x ) u ( x ,0 ) 2 d x < ∞ } ,

which is endowed with norm

‖ u ‖ : = ( ∫ ℝ + 4 k p y 1 − 2 p | ∇ u | 2 d x d y + ∫ ℝ 3 V ( x ) u ( x ,0 ) 2 d x ) 1 / 2 , (2.2)

then E is a Hilbert space. In the following, for convenience, for any u, let u ˜ : = u ( x ,0 ) .

The functional associated to (2.1) is given by

Φ ( w 1 ) = k p 2 ∫ ℝ + 4 y 1 − 2 p | ∇ w 1 | 2 d x d y + 1 2 ∫ ℝ 3 V ( x ) w ˜ 1 2 d x − 1 2 ∫ ℝ 3 ω w ˜ 2 w ˜ 1 2 d x − ∫ ℝ 3 K ( x ) F ( w ˜ 1 ) d x , w 1 ∈ E , (2.3)

which is of C 1 by (H1)-(H3).

A vector w 1 is a weak solution of system (2.1) if 〈 Φ ′ ( w 1 ) , U 〉 = 0 for any U ∈ E , i.e.

〈 Φ ′ ( w 1 ) , U 〉 = k p ∫ ℝ + 4 y 1 − 2 p 〈 ∇ w 1 , ∇ U 〉 d x d y + ∫ ℝ 3 V ( x ) w ˜ 1 U ˜ d x − ∫ ℝ 3 [ 2 ω + w ˜ 2 ] w ˜ 2 w ˜ 1 U ˜ d x − ∫ ℝ 3 K ( x ) f ( w ˜ 1 ) U ˜ d x . (2.4)

Lemma 2.1. [

{ − div ( y 1 − 2 p ∇ w ) = 0, in ℝ + 4 , k p y 1 − 2 p ∂ w ∂ η = ( ω + ϕ ) u 2 , on ℝ 3 × { 0 } . (2.5)

Furthermore, in the set { ( x ,0 ) : u ˜ : = u ( x ,0 ) ≡ 0 } , we have − ω ≤ ϕ u ≤ 0 for ω > 0 .

Let the weighted Banach space be

L K s = { u : ℝ + 4 → ℝ is measurable and ∫ ℝ 3 K ( x ) | u ˜ | s d x < ∞ } , s ∈ ( 1, + ∞ )

under the norm

‖ u ‖ L K s = ( ∫ ℝ 3 K ( x ) | u ˜ | s d x ) 1 / s .

The following Proposition 2.2 comes from the arguments in [

Proposition 2.2. [

1)

2)

3) If u k ⇀ u in E, then up to a subsequence

l i m k → ∞ ∫ ℝ 3 K ( x ) F ( u ˜ k ) d x = ∫ ℝ 3 K ( x ) F ( u ˜ ) d x ;

4) If u k ⇀ u in E, then up to a subsequence

l i m k → ∞ ∫ ℝ 3 K ( x ) u ˜ k f ( u ˜ k ) d x = ∫ ℝ 3 K ( x ) u ˜ f ( u ˜ ) d x ;

5) If u k ⇀ u in E, then up to a subsequence, for any z ∈ E ,

l i m k → ∞ ∫ ℝ 3 K ( x ) f ( u ˜ k ) z ˜ d x = ∫ ℝ 3 K ( x ) f ( u ˜ ) z ˜ d x .

Lemma 2.3. [

Lemma 2.4. Assume that (H2) and (H3) hold. Then the functional Φ satisfies

1) There exists β , ρ > 0 such that Φ ( u ) ≥ β if ‖ u ‖ = ρ ;

2) There exists u 0 ∈ E \ { 0 } with ‖ u ‖ > ρ such that Φ ( u 0 ) ≤ 0 .

The proof of Lemma 2.4 is standard, so we omit the details here.

From Lemma 2.4, there exists a ( P S ) c sequence { u k } ⊂ E such that

Φ ( u k ) → c and ‖ Φ ′ ( u k ) ‖ ( 1 + ‖ u k ‖ ) → 0, as k → + ∞ , (2.6)

where

c = i n f γ ∈ Γ m a x t ∈ [ 0,1 ] Φ ( γ ( t ) )

with

Γ = { γ ∈ C ( [ 0,1 ] , E ) ; γ ( 0 ) = 0 and Φ ( γ ( 1 ) ) ≤ 0 } .

Lemma 3.1. Assume that (H2)-(H4) hold. Then the ( P S ) c sequence { u k } ⊂ E given in (2.6) is bounded.

Proof. Let { u k } ⊂ E be a ( P S ) c sequence of Φ . Arguing indirectly, suppose ‖ u k ‖ → ∞ such that

Φ ( u k ) → c , Φ ′ ( u k ) → 0, as k → ∞ , (3.1)

after passing to a subsequence. Denote v k : = u k / ‖ u k ‖ . Then ‖ v k ‖ = 1 , v k ⇀ v 0 in E and v k ( x ) → v 0 ( x ) for a.e. x ∈ ℝ 3 . If v 0 = 0 , by the fact v k → 0 in L 2 ( ℝ 3 ) , (2.2), (2.3), (2.4), (3.1) and Lemma 2.1, there are two cases to consider.

Case (1): μ ∈ [ 4, ∞ ) . From (2.2), (2.3), (2.4) and (3.1), we derive

o ( 1 ) = μ Φ ( u k ) − 〈 Φ ′ ( u k ) , u k 〉 ‖ u k ‖ 2 = ( θ 2 − 1 ) + ( 2 − μ 2 ) ∫ ℝ 3 ω ϕ u k ( x , 0 ) u ˜ k 2 ‖ u k ‖ 2 d x + ∫ ℝ 3 ϕ u k 2 ( x , 0 ) u ˜ k 2 ‖ u k ‖ 2 d x + ∫ ℝ 3 K ( x ) [ f ( u ˜ k ) u ˜ k − μ F ( u ˜ k ) ] ‖ u k ‖ 2 d x ≥ μ 2 − 1 + o ( 1 ) ,

then 0 ≥ μ 2 − 1 , which contradict μ ≥ 4 .

Case (2): μ ∈ ( 2,4 ) . In this case, by (2.2), (2.3), (2.4), (3.1) and Lemma 2.1, one gets

o ( 1 ) = μ Φ ( u k ) − 〈 Φ ′ ( u k ) , u k 〉 ‖ u k ‖ 2 ≥ ( μ 2 − 1 ) + ( 2 − μ 2 ) ∫ ℝ 3 ω ϕ u k ( x , 0 ) u ˜ k 2 ‖ u k ‖ 2 d x ≥ ( μ 2 − 1 ) − ( 2 − μ 2 ) ω 2 | v k | 2 2 = θ 2 − 1 + o ( 1 ) ,

then 0 ≥ μ 2 − 1 , which contradict μ > 2 .

If v 0 ≠ 0 , then meas { Ω 1 } > 0 , where Ω 1 : = { x ∈ ℝ 3 : v 0 ( x ,0 ) ≠ 0 } . For x ∈ Ω 1 , we have | u ˜ k | → ∞ as k → ∞ , and then, from (H4), we get

F ( u ˜ k ) u ˜ k 2 v ˜ k 2 → + ∞ , as k → ∞ . (3.2)

From (3.2) and Fatou’s Lemma, we obtain

∫ Ω 1 F ( u ˜ k ) u ˜ k 2 v ˜ k 2 d x → + ∞ , as k → ∞ . (3.3)

From (2.2), (2.3), (3.1), (3.3) and Lemma 2.1, we have

0 = lim k → ∞ Φ ( u k ) ‖ u k ‖ 2 = lim k → ∞ [ 1 2 − 1 2 ∫ ℝ 3 ω ϕ u k ( x , 0 ) u ˜ k 2 ‖ u k ‖ 2 d x − ∫ ℝ 3 K ( x ) F ( u ˜ k ) ‖ u k ‖ 2 d x ] = 1 2 + o ( 1 ) − lim k → ∞ ∫ ℝ 3 K ( x ) F ( u ˜ k ) ‖ u k ‖ 2 d x ≤ 1 2 + o ( 1 ) − lim k → ∞ ∫ Ω 1 K ( x ) F ( u ˜ k ) u ˜ k 2 v ˜ k 2 d x = − ∞ ,

a contradiction. Hence, the boundedness of { u n } in E is obtained.

Proof of Theorem 1.1. Let { u k } be a ( P S ) c sequence given in (2.6). It follows from Lemma 3.1 that { u k } is bounded, passing to a subsequence, one can assume that there is u ∈ E such that

u k ⇀ u , weakly in E , as k → ∞ .

It suffices to show that u k → u , as k → ∞ . By Proposition 2.2, one has

l i m k → ∞ ∫ ℝ 3 K ( x ) f ( u ˜ k ) u ˜ k d x = ∫ ℝ 3 K ( x ) f ( u ˜ ) u ˜ d x .

From (2.4), we have

〈 Φ ′ ( u k ) , U 〉 = k s ∫ ℝ + 4 y 1 − 2 s 〈 ∇ u k , ∇ U 〉 d x d y + ∫ ℝ 3 V ( x ) u ˜ k U ˜ d x − ∫ ℝ 3 [ 2 ω + w ˜ 2 ] w ˜ 2 u ˜ k U ˜ d x − ∫ ℝ 3 K ( x ) f ( u ˜ k ) U ˜ d x .

By 〈 Φ ′ ( u k ) , u k 〉 = o n ( 1 ) , one gets

l i m k → ∞ ‖ u k ‖ 2 = l i m k → ∞ [ ∫ ℝ 3 ( 2 ω + w ˜ 2 ) w ˜ 2 u ˜ k 2 d x − ∫ ℝ 3 K ( x ) f ( u ˜ k ) u ˜ k d x ] . (3.4)

By Proposition 2.2, one obtains

l i m k → ∞ ∫ ℝ 3 K ( x ) f ( u ˜ k ) u ˜ d x = ∫ ℝ 3 K ( x ) f ( u ˜ ) u ˜ d x .

From the proof of Lemma 2.3 in [

ϕ u k ⇀ z in L r ( ℝ 3 × { 0 } ) as k → ∞ , (3.5)

ϕ u k ⇀ z in L l o c r ( ℝ 3 × { 0 } ) as k → ∞ , r ∈ [ 2, 6 / ( 3 − 2 p ) ] , (3.6)

and ϕ u = z . Hence, from Lemma 2.3, we obtain that

l i m n → ∞ ∫ ℝ 3 ( 2 ω + w ˜ 2 ) w ˜ 2 u ˜ k 2 d x = ∫ ℝ 3 ( 2 ω + z ˜ ) z ˜ u ˜ 2 d x .

Then

l i m n → ∞ ‖ u k ‖ 2 = ∫ ℝ 3 ( 2 ω + z ˜ ) z ˜ u ˜ 2 d x + ∫ ℝ 3 K ( x ) f ( u ˜ ) u ˜ d x . (3.7)

Otherwise, since 〈 Φ ′ ( u ) , u 〉 = o ( 1 ) , one has

‖ u ‖ 2 = ∫ ℝ 3 ( 2 ω + z ˜ ) z ˜ u ˜ 2 d x + ∫ ℝ 3 K ( x ) f ( u ˜ ) u ˜ d x . (3.8)

Hence, from (3.7) and (3.8), we have

l i m n → ∞ ‖ u k ‖ 2 = ‖ u ‖ 2 ,

which shows that

u k → u in E , as k → ∞ .

Hence, we conclude that

Φ ( u ) = c and Φ ′ ( u ) = 0.

Thus, u is a ground state solution for Φ . It follows from u k ≥ 0 that u ≥ 0 . Since there is a ( P S ) c sequence { u k } , we can obtain that u is positive from Lemma 2.1 by contradiction.

In this paper, we first reformulated the system (1.1) into a local system by using the local reduction. Then, we take advantage of the interaction of the behaviour of the potential and nonlinearity to recover the compactness. Meanwhile, we obtained a Palais-Smale sequence by using a weaker Ambrosetti-Rabinowitz condition. Finally, the existence of positive solution is proved by the mountain pass theorem. Obviously, the weaker Ambrosetti-Rabinowitz condition has been successfully applied to find the solution of the fractional Klein-Gordon-Maxwell system with potential vanishing at infinity. We hope that this result can be widely used in other systems.

The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.

No potential conflict of interest was reported by the authors.

This work is supported by the National Natural Science Foundation of China (No. 11961014, No. 61563013) and Guangxi Natural Science Foundation (2016 GXNSFAA380082, 2018GXNSFAA281021).

Gan, C.L., Xiao, T. and Zhang, Q.F. (2020) Existence Result for Fractional Klein-Gordon-Maxwell System with Quasicritical Potential Vanishing at Infinity. Journal of Applied Mathematics and Physics, 8, 1318-1327. https://doi.org/10.4236/jamp.2020.87101