Existence and multiplicity of solutions for a class of (ϕ1,ϕ2 )-Laplacian elliptic system in RN via genus theory

Computers and Mathematics with Applications 72 (2016) 110–130

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Existence and multiplicity of solutions for a class of (φ1 , φ2 )-Laplacian elliptic system in RN via genus theory Liben Wang, Xingyong Zhang ∗ , Hui Fang Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, PR China

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Article history: Received 23 January 2016 Accepted 23 April 2016 Available online 21 May 2016 Keywords: (φ1 , φ2 )-Laplacian system Genus theory Weak solution Critical point

abstract In this paper, we investigate the following nonlinear and non-homogeneous elliptic system involving (φ1 , φ2 )-Laplacian

 −div(φ1 (|∇ u|)∇ u) + V1 (x)φ1 (|u|)u = Fu (x, u, v) −div(φ2 (|∇v|)∇v) + V2 (x)φ2 (|v|)v = Fv (x, u, v)  (u, v) ∈ W 1,Φ1 (RN ) × W 1,Φ2 (RN )

in RN , in RN , with N ≥ 2,

where the functions Vi (x)(i = 1, 2) are bounded and positive in RN , the functions φi (t )t (i = 1, 2) are increasing homeomorphisms from R+ onto R+ , and the function F is of class C 1 (RN +2 , R) and has a sub-linear Orlicz–Sobolev growth. By using the least action principle, we obtain that system has at least one nontrivial solution. When F satisfies an additional symmetric condition, by using the genus theory, we obtain that system has infinitely many solutions. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction and main results In this paper, we investigate the existence and multiplicity of solutions for the following nonlinear and non-homogeneous elliptic system involving (φ1 , φ2 )-Laplacian:

 −div(φ1 (|∇ u|)∇ u) + V1 (x)φ1 (|u|)u = Fu (x, u, v) −div(φ2 (|∇v|)∇v) + V2 (x)φ2 (|v|)v = Fv (x, u, v)  (u, v) ∈ W 1,Φ1 (RN ) × W 1,Φ2 (RN )

in RN , in RN ,

(1.1)

with N ≥ 2,

where φi (i = 1, 2) : [0, ∞) → [0, ∞) are two functions which satisfy:

(φ1 ) φi ∈ C 1 [0, ∞), t φi (t ) → ∞ as t → ∞; (φ2 ) t → φi (t )t are strictly increasing; t 2 2 (φ3 ) 1 < li := inft >0 t Φφi (i (t t)) ≤ supt >0 t Φφi (i (t t)) =: mi < min{N , l∗i }, where Φi (t ) := 0 sφi (s)ds, t ∈ [0, ∞) and l∗i := Vi (i = 1, 2) : RN → R+ are continuous and (V ) there exist two constants c1 , c2 > 0 such that c1 ≤ min{V1 (x), V2 (x)} ≤ max{V1 (x), V2 (x)} ≤ c2 ,

for all x ∈ RN ,

F : RN × R × R → R is a C 1 function and F (x, 0, 0) = 0, for all x ∈ RN .

∗

Corresponding author. E-mail address: [email protected] (X. Zhang).

http://dx.doi.org/10.1016/j.camwa.2016.04.034 0898-1221/© 2016 Elsevier Ltd. All rights reserved.

li N , N −l i

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

Set φ2 = φ1 , v = u, V2 = V1 and F (x, u, v) = F (x, v, u). Let Φ1 (t ) := to the following quasilinear elliptic equation:



−div(φ1 (|∇ u|)∇ u) + V1 (x)φ1 (|u|)u = f (x, u) u ∈ W 1,Φ1 (RN )

t 0

111

sφ1 (s)ds, t ∈ [0, ∞). Then system (1.1) reduces

in RN , with N ≥ 2.

(1.2)

Under assumptions (φ1 )–(φ2 ), Eq. (1.2) may be allowed to possess more complicated nonlinear or non-homogeneous operator Φ1 , which can be used to model many phenomena (see [1,2]): (1) (2) (3) (4) (5)

p-Laplacian: Φ1 (t ) = |t |p , 1 < p < N; (p, q)-Laplacian: Φ1 (t ) = |t |p + |t |q , 1 < p < q < N; nonlinear elasticity: Φ1 (t ) = (1 + t 2 )γ − 1, γ > 12 ; plasticity: Φ1 (t ) = t α (log(1 + t ))β , α ≥ 1, β > 0; t generalized Newtonian fluids: Φ1 (t ) = 0 s1−α (sinh−1 s)β ds, 0 ≤ α ≤ 1, β > 0.

Therefore, Eq. (1.2) or equations like (1.2) have caused great interest among scholars in recent years. When operator Φ1 is not homogeneous, an Orlicz–Sobolev space setting may be applied for this type of equation (see Section 2). On a bounded domain Ω , we refer the reader to [1,3–12] and the references therein for more information. Especially, in Clément et al. [3], the authors firstly proved the existence of nontrivial solution by variational method when V1 (x) = 0 and the nonlinear term f satisfies (AR)-condition. On the whole space RN , when V1 (x) is bounded, the main difficulty for this type of equation is the lack of compactness of the Sobolev embedding. A usual way to overcome this difficulty is to reconstruct the compactness embedding theorem, which can be done by choosing the radially symmetric function space as the working space (see [2,13,14]). For the system case, to the best of our knowledge, on the bounded domain Ω , there are only two papers to consider the existence of solutions for systems like (1.1) (see [15,16]). However, on the whole space RN , there is no paper to study the existence and multiplicity of solutions for systems like (1.1). In this paper, we will study the existence and multiplicity of solutions for system (1.1) under sub-linear Orlicz–Sobolev growth. To overcome the difficulty of lacking compactness of the Sobolev embedding, we will refer to some views in [17]. In [17], Chen and Tang studied the existence and multiplicity of solutions for a class of Kirchhoff-type equation:

   − a + b RN



 |∇ u|2 dx △u + u = f (x, u)

u ∈ H 1 (RN )

in RN ,

(1.3)

with N = 2 or 3,

where constants a > 0, b > 0, and f ∈ C (RN × R, R). To be precise, they obtained the following results: Theorem A (See [17, Theorem 1.1]). Assume that f satisfies 2

(F1 )∗ f ∈ C (RN × R, R) and there exist constants 1 < γ1 < γ2 < · · · < γm < 2 and functions ai ∈ L 2−γi (RN , [0, +∞)) (i = 1, 2, . . . , m) such that m  |f (x, u)| ≤ ai (x)|u|γi −1 , for all (x, u) ∈ RN × R; i=1

(F2 )∗ there exist an open set J ⊂ RN and three constants γ0 ∈ (1, 2), δ > 0 and η > 0 such that F (x, z ) ≥ η|z |γ0 ,

for all(x, z ) ∈ J × [−δ, δ],

where F (x, z ) :=

z



f (x, s)ds,

for all (x, z ) ∈ RN × R.

0

Then Eq. (1.3) possesses at least one nontrivial weak solution. Theorem B (See [17, Theorem 1.2]). Assume that f satisfies (F1 )∗ and (F2 )∗ , then Eq. (1.3) possesses infinitely many nontrivial weak solutions provided that F satisfies

(F3 )∗ F (x, −z ) = F (x, z ),

for all(x, z ) ∈ RN × R.

To overcome the difficulty of lacking compactness, Chen and Tang assumed that the nonlinear term f satisfies a sub-linear growth condition (F1 )∗ so that they did not work in the radially symmetric function space. Motivated by such idea in [17], we assume that the nonlinear term F in system (1.1) satisfies a corresponding sub-linear growth condition in Orlicz–Sobolev space (called sub-linear Orlicz–Sobolev growth for short) so that we do not work in the radially symmetric function space,

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L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

too. By using the least action principle (see [18]), we obtain that system (1.1) has at least a nontrivial solution, and by using the genus theory (see [19]), we obtain that system (1.1) has infinitely many solutions under an additional symmetric condition. Because functions φi (i = 1, 2) in system (1.1) may be nonlinear and non-homogeneous and the system case is more complex than the scalar case, more subtle skills are needed in the process of our proofs. Next, we state our results. Throughout this paper, C is used to denote a positive constant which may be different in various places. Theorem 1.1. Assume that F satisfies the following conditions:

(F1 ) there exist constants pi ∈ [mi , l∗i ) (i = 1, 2), max{ p11 , p12 } ≤ q1 < q2 < · · · < qk < min{ pl11 , pl22 }, and functions 1

a1j , a2j , a3j , a4j ∈ L 1−qj (RN , [0, +∞)) (j = 1, 2, . . . , k) such that

|Fu (x, u, v)| ≤

k 

a1j (x)|u|p1 qj −1 +

j =1

|Fv (x, u, v)| ≤

k 

k 

a2j (x)|v|

p2 (p1 qj −1) p1

,

j=1

a3j (x)|u|

p1 (p2 qj −1) p2

j =1

+

k 

a4j (x)|v|p2 qj −1 ,

j =1

for all (x, u, v) ∈ R × R × R; (F2 ) there exist an open set Ω ⊂ RN with |Ω | > 0, and constants α0 ∈ [1, l1 ), β0 ∈ [1, l2 ), δ > 0, c > 0 and ι, κ ∈ R with ι2 + κ 2 ̸= 0 such that N

F (x, ιt , κ t ) ≥ c (|ιt |α0 + |κ t |β0 ),

for all (x, t ) ∈ Ω × [0, δ].

Then system (1.1) possesses at least one nontrivial weak solution. Theorem 1.2. Assume that F satisfies (F1 )–(F2 ) and the symmetric condition

(F3 ) F (x, −u, −v) = F (x, u, v),

for all (x, u, v) ∈ RN × R × R.

Then system (1.1) possesses a sequence of weak solutions {(un , vn )} such that I (un , vn ) :=



Φ1 (|∇ un |)dx +

RN

 + RN

 RN

Φ2 (|∇vn |)dx +

V1 (x)Φ1 (|un |)dx

 RN

V2 (x)Φ2 (|vn |)dx −

 RN

F (x, un , vn )dx < 0

and I (un , vn ) → 0 as n → ∞. Remark 1.3. Let u = ιt and v = κ t , t ∈ [0, δ]. Then assumption (F2 ) is equivalent to the following form:

(F2 )′ there exist an open set Ω ⊂ RN with |Ω | > 0, and constants α0 ∈ [1, l1 ), β0 ∈ [1, l2 ), δ > 0, c > 0 and ι, κ ∈ R with ι2 + κ 2 ̸= 0 such that one of the following cases holds: (i) F (x, u, v) ≥ c (|u|α0 + |v|β0 ) for all x ∈ Ω and v = κι u (if ι > 0) with u ∈ [0, ιδ]; (ii) (1.4) holds for all x ∈ Ω and v = κι u (if ι < 0) with u ∈ [ιδ, 0]; (iii) (1.4) holds for all x ∈ Ω and u = 0 (if ι = 0 and κ > 0) with v ∈ [0, κδ]; (iv) (1.4) holds for all x ∈ Ω and u = 0 (if ι = 0 and κ < 0) with v ∈ [κδ, 0].

(1.4)

(F2 )′ implies that (1.4) holds only for all x ∈ Ω and all points (u, v) which belong to some line segment in u–v plane, which shows that F is allowed to be sign-changing even in any ε -neighborhood U (0, ε) in u–v plane for any given x ∈ Ω (see Fig. 1 above when ι > 0 and κ > 0). Set φ2 = φ1 , v = u, V2 = V1 and F (x, u, v) = F (x, v, u). Then system (1.1) reduces to the quasilinear elliptic equation (1.2). Under the assumptions (φ1 )–(φ3 ) and (V ), the similar results for Eq. (1.2) can be established as follows, which are also new results even for the scalar case. Corollary 1.4. Assume that f satisfies the following conditions:

(f1 ) f ∈ C (RN × R, R) and there exist constants p ∈ [m1 , l∗1 ), 1 ≤ α1 < α2 < · · · < αk < l1 , and functions p

aj ∈ L p−αj (RN , [0, +∞)) (j = 1, 2, . . . , k) such that

|f (x, u)| ≤

k  j =1

aj (x)|u|αj −1 ,

for all (x, u) ∈ RN × R;

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

113

Fig. 1. When ι > 0 and κ > 0, for any given x ∈ Ω , (1.4) holds for all points in line segment S1 and F is allowed to be negative in some points of U (0, ε).

(f2 ) there exist an open set Ω ⊂ RN with |Ω | > 0, and constants α0 ∈ [1, l1 ), δ > 0 and c > 0 such that F (x, u) ≥ c |u|α0 ,

for all (x, u) ∈ Ω × [0, δ] or for all (x, u) ∈ Ω × [−δ, 0],

where F (x, u) :=

u



f (x, s)ds,

for all (x, u) ∈ RN × R.

0

Then Eq. (1.2) possesses at least one nontrivial weak solution. Corollary 1.5. Assume that f and F satisfy (f1 )–(f2 ) and the symmetric condition

(F ) F (x, −u) = F (x, u), for all (x, u) ∈ RN × R. Then Eq. (1.2) possesses a sequence of weak solutions {un } such that

¯I (un ) :=

 RN

Φ1 (|∇ un |)dx +

 RN

V1 (x)Φ1 (|un |)dx −

 RN

F (x, un )dx < 0

and ¯I (un ) → 0 as n → ∞. Remark 1.6. By using the same skills in this paper, one can relax conditions (F1 )∗ and (F2 )∗ in Theorems A and B to the following weaker conditions (F1 )′ and (F2 )′ :

(F1 )′ f ∈ C (RN × R, R) and there exist constants p ∈ [2, 2∗ ), 1 ≤ γ1 < γ2 < · · · < γm < 2, and functions ai ∈ p L p−γi (RN , [0, +∞)) (i = 1, 2, . . . , m) such that m  |f (x, u)| ≤ ai (x)|u|γi −1 , for all (x, u) ∈ RN × R; i=1

(F2 ) there exist an open set J ⊂ RN with |J | > 0, and constants γ0 ∈ [1, 2), δ > 0 and η > 0 such that ′

F (x, z ) ≥ η|z |γ0 ,

for all (x, z ) ∈ J × [0, δ] or for all (x, z ) ∈ J × [−δ, 0].

2. Preliminaries In this section, we introduce Orlicz and Orlicz–Sobolev spaces and some fundamental notions and important properties needed in our study. We refer the reader for more details to the books [20,21] and the references therein. First of all, we recall the notion of N-function. Let φ : [0, ∞) → [0, ∞) be a right continuous, monotone increasing function with (1) φ(0) = 0; (2) limt →∞ φ(t ) = ∞; (3) φ(t ) > 0 whenever t > 0.

t

Then the function defined on [0, ∞) by Φ (t ) = 0 φ(s)ds is called an N-function. An N-function Φ satisfies a ∆2 -condition globally (or near infinity) if sup t >0

Φ (2t ) <∞ Φ (t )

 or lim sup t →∞

 Φ (2t ) <∞ , Φ (t )

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L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

which implies that there exists a constant K > 0, such that Φ (2t ) ≤ K Φ (t ) for all t ≥ 0 (or t ≥ t0 > 0). We also state the equivalent form that Φ satisfies a ∆2 -condition globally (or near infinity) if and only if for any given c ≥ 1, there exists a constant Kc > 0 such that Φ (ct ) ≤ Kc Φ (t ) for all t ≥ 0 (or t ≥ t0 > 0). For N-function Φ , the complement of Φ is given by

 (t ) = max{ts − Φ (s)}, Φ

for t ≥ 0.

s≥0

  is also an N-function and Φ  = Φ . In addition, we have the Young’s inequality, that is Φ  (t ), st ≤ Φ (s) + Φ

for all s, t ≥ 0,

(2.1)

and the following inequality (see [5, Lemma A.2]), that is

 (φ(t )) ≤ Φ (2t ), Φ

for all t ≥ 0.

(2.2)

Φ

Φ

Now we recall the Orlicz space L (Ω ) associated with Φ . When Φ satisfies ∆2 -condition globally, the Orlicz space L (Ω ) is the vectorial space of the measurable functions u : Ω → R satisfying

 Ω

Φ (|u|)dx < ∞,

where Ω ⊂ RN is an open set. LΦ (Ω ) is a Banach space endowed with Luxemburg norm

     | u| ∥u∥Φ := inf λ > 0 : Φ dx < 1 , λ Ω

for u ∈ LΦ (Ω ).

Particularly, when Φ (t ) = |t |p (1 < p < +∞), the corresponding Orlicz space LΦ (Ω ) is the classical Lebesgue space Lp (Ω ) and the corresponding Luxemburg norm ∥u∥Φ is equal to the classical Lp (Ω ) norm, that is

 ∥u∥Lp (Ω ) :=

Ω

 1p

|u(x)| dx p

,

for u ∈ Lp (Ω ).

When Ω = RN , we denote ∥u∥Lp (RN ) by ∥u∥p . The fact that Φ satisfies ∆2 -condition globally implies that in LΦ (Ω ) ⇐⇒

un → u

 Ω

Φ (|un − u|)dx → 0.

(2.3)

By the above Young’s inequality (2.1), a generalized type of Hölder’s inequality (see [20,21])

     uv dx ≤ 2∥u∥Φ ∥v∥Φ ,   Ω

for all u ∈ LΦ (Ω ) and all v ∈ LΦ (Ω )

(2.4)



can be obtained. Define W 1,Φ (Ω ) =



u ∈ LΦ ( Ω ) :

∂u ∈ LΦ (Ω ), i = 1, . . . , N ∂ xi



with the norm

∥u∥1,Φ = ∥u∥Φ + ∥∇ u∥Φ . Then W 1,Φ (Ω ) is a Banach space which is called Orlicz–Sobolev space. Denote the closure of C0∞ (Ω ) in W 1,Φ (Ω ) by 1,Φ

1,Φ

(Ω ). Then by some basis properties in Orlicz–Sobolev spaces, we have W0 (RN ) = W 1,Φ (RN ) when Ω = RN . Next, we give some inequalities which will be used in our proofs. For more details, we refer the reader to the papers [22,21]. W0

Lemma 2.1 (See [22,21]). If Φ is an N-function, then the following conditions are equivalent: (1) l = inf

t >0

t φ(t )

Φ (t )

≤ sup t >0

t φ(t )

Φ (t )

= m;

(2.5)

(2) let ζ0 (t ) = min(t l , t m ), ζ1 (t ) = max(t l , t m ), for t ≥ 0. Φ satisfies

ζ0 (t )Φ (ρ) ≤ Φ (ρ t ) ≤ ζ1 (t )Φ (ρ), (3) Φ satisfies a ∆2 -condition globally.

for all ρ, t ≥ 0;

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

115

Lemma 2.2 (See [22]). Let Ω = RN . If Φ is an N-function and (2.5) holds, then Φ satisfies

ζ0 (∥u∥Φ ) ≤



Φ (|u|)dx ≤ ζ1 (∥u∥Φ ),

RN

for all u ∈ LΦ (RN ).

 be the complement of Φ and ζ2 (t ) = Lemma 2.3 (See [22]). If Φ is an N-function, l, m ∈ (1, ∞) and (2.5) holds. Let Φ      satisfies  := mm−1 . Then Φ min(t l , t m ), ζ3 (t ) = max(t l , t m ), for t ≥ 0, wherel := l−l 1 and m (1)

 = inf m

t >0

 ′ (t ) tΦ

≤ sup

 (t ) Φ

t >0

 ′ (t ) tΦ  (t ) Φ

=l;

(2)

 (ρ) ≤ Φ  (ρ t ) ≤ ζ3 (t )Φ  (ρ), ζ2 (t )Φ

for all ρ, t ≥ 0;

(3)



ζ2 (∥u∥Φ ) ≤

 (|u|)dx ≤ ζ3 (∥u∥Φ ), Φ

RN

N for all u ∈ LΦ  (R ).

∗

∗

∗

∗

Lemma 2.4 (See [22]). If Φ is an N-function, l, m ∈ (1, N ) and (2.5) holds. Let ζ4 (t ) = min(t l , t m ), ζ5 (t ) = max(t l , t m ), . Then Φ∗ satisfies for t ≥ 0, where l∗ := NlN−l , m∗ := NmN −m (1) l∗ = inf

t >0

t Φ∗′ (t )

Φ∗ (t )

≤ sup t >0

t Φ∗′ (t )

Φ∗ (t )

= m∗ ;

(2)

ζ4 (t )Φ∗ (ρ) ≤ Φ∗ (ρ t ) ≤ ζ5 (t )Φ∗ (ρ),

for all ρ, t ≥ 0;

(3)

ζ4 (∥u∥Φ∗ ) ≤

 RN

Φ∗ (|u|)dx ≤ ζ5 (∥u∥Φ∗ ),

for all u ∈ LΦ∗ (RN ),

where Φ∗ is the Sobolev conjugate function of Φ , which is defined by

Φ∗−1 (t ) =

t

 0

Φ −1 (s) s

N +1 N

ds,

for t ≥ 0.

The following important embedding proposition involving the Orlicz–Sobolev spaces will be used frequently in our study. Proposition 2.5 (See [2,21]). Under the assumptions of Lemma 2.4, the embedding W 1,Φ (RN ) ↩→ LΨ (RN ) is continuous for any N-function Ψ satisfying lim sup t →0

Ψ (t ) < ∞ and Φ (t )

lim sup t →∞

Ψ (t ) < ∞. Φ∗ (t )

Therefore, there exists a constant C such that

∥u∥Ψ ≤ C ∥u∥1,Φ ,

for all u ∈ W 1,Φ (RN ).

(2.6)

When the space R is replaced by a bounded domain D ⊂ R and Ψ increases essentially more slowly than Φ∗ near infinity, that is N

lim

t →∞

N

Ψ (ct ) =0 Φ∗ (t )

for any constant c > 0. Then the embedding W 1,Φ (D) ↩→ LΨ (D) is compact.

i (i = 1, 2) are N-functions that satisfy Remark 2.6. By Lemmas 2.1 and 2.3, (φ1 )–(φ3 ) imply that Φi (i = 1, 2) and Φ ∆2 -condition globally. Thus LΦi (RN ) (i = 1, 2) and W 1,Φi (RN ) (i = 1, 2) are separable and reflexive Banach spaces (see [20,21]).

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L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

Remark 2.7. Under assumptions (φ1 )–(φ3 ) and (F1 ), the above lemmas and Proposition 2.5 imply that the embedding W 1,Φi (RN ) ↩→ Lpi (RN )

(i = 1, 2)

(2.7)

is continuous and the embedding W 1,Φi (Br ) ↩→ Lpi (Br )

(i = 1, 2)

(2.8)

is compact, where r > 0 and Br := {x ∈ R : |x| < r }. N

1

Remark 2.8. Condition (F1 ) implies that there exist functions a5j , a6j ∈ L 1−qj (RN , [0, +∞)) (j = 1, 2, . . . , k) such that k 

|F (x, u, v)| ≤

a5j (x)|u|p1 qj +

k 

a6j (x)|v|p2 qj ,

for all (x, u, v) ∈ RN × R × R.

j =1

j=1

In fact, since F (x, u, v) =

u



Fs (x, s, v)ds + F (x, 0, v)

0 u =

Fs (x, s, v)ds +

v



0

Fs (x, 0, s)ds + F (x, 0, 0),

0

by (F1 ) and the fact that F (x, 0, 0) = 0 for all x ∈ RN , we have |u|

 |v| |Fs (x, s, v)|ds + |Ft (x, 0, t )|dt 0 0    |u|   k k p2 (p1 qj −1)  p1 qj −1 ≤ a1j (x)|s| + a2j (x)|v| p1 ds + 

|F (x, u, v)| ≤

0

j =1

k  a1j (x)

=

j =1

p1 qj

k  a1j (x)

≤

j =1

p1 qj

k  1

=

j =1

p1 qj

0

j =1

|u|p1 qj +

k 

a2j (x)|u| |v|

p2 (p1 qj −1) p1

+

k  a4j (x) j =1

j =1

|u|p1 qj +

k 

a2j (x)

j =1



k 

|v|

1 p1 q j

(a1j (x) + a2j (x))|u|p1 qj +

|u|p1 qj +

p1 qj − 1 p1 qj

k   p1 qj − 1

p1 qj

j =1

p2 qj

a4j (x)|t |p2 qj −1 dt

j =1

|v|p2 qj

|v|p2 qj

 +

k  a4j (x) j =1

a2j (x) +

1 p2 qj

p2 qj

|v|p2 qj



a4j (x) |v|p2 qj ,

for all (x, u, v) ∈ RN × R × R. Let a5j (x) =

1 p1 qj

(a1j (x) + a2j (x)) and a6j =

p1 q j − 1 p1 q j

a2j (x) +

1 p2 qj

a4j (x).

1

Then a5j , a6j ∈ L 1−qj (RN , [0, +∞)) (j = 1, 2, . . . , k) and

|F (x, u, v)| ≤

k 

a5j (x)|u|p1 qj +

j=1

k 

a6j (x)|v|p2 qj ,

for all (x, u, v) ∈ RN × R × R.

j =1

3. Proofs Define W := W 1,Φ1 (RN ) × W 1,Φ2 (RN ) with the norm

∥(u, v)∥ = ∥u∥1,Φ1 + ∥v∥1,Φ2 = ∥u∥Φ1 + ∥∇ u∥Φ1 + ∥v∥Φ2 + ∥∇v∥Φ2 . Then W is a separable and reflexive Banach space. The associated Euler–Lagrange functional I corresponding to system (1.1) is defined by I (u, v) :=

 RN

Φ1 (|∇ u|)dx +

 + RN

 RN

Φ2 (|∇v|)dx +

V1 (x)Φ1 (|u|)dx

 RN

V2 (x)Φ2 (|v|)dx −

 RN

F (x, u, v)dx,

(u, v) ∈ W .

(3.1)

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

117

Define Ii (i = 1, 2) : W → R by I1 (u, v) =

 RN

Φ1 (|∇ u|)dx +

 RN

V1 (x)Φ1 (|u|)dx +

 RN

Φ2 (|∇v|)dx +

 RN

V2 (x)Φ2 (|v|)dx

and I2 (u, v) =

 RN

F (x, u, v)dx.

Then I (u, v) = I1 (u, v) − I2 (u, v). Lemma 3.1. Suppose that (φ1 )–(φ3 ), (V ) and (F1 ) hold. Then I : W → R is well defined and of class C 1 (W , R) and

⟨I ′ (u, v), (˜u, v˜ )⟩ =



φ1 (|∇ u|)∇ u∇ u˜ dx +

RN

 + RN

 − RN

 RN



φ2 (|∇v|)∇v∇ v˜ dx +

V2 (x)φ2 (|v|)v v˜ dx

RN



Fu (x, u, v)˜udx −

V1 (x)φ1 (|u|)uu˜ dx

Fv (x, u, v)˜v dx

RN

(3.2)

for all (˜u, v˜ ) ∈ W . Proof. Under the assumptions (φ1 )–(φ3 ) and (V ), by similar arguments as [23], we can prove that I1 : W → R is well defined and of class C 1 (W , R) and

⟨I1 (u, v), (˜u, v˜ )⟩ = ′



φ1 (|∇ u|)∇ u∇ u˜ dx +

RN

 + RN

 RN

φ2 (|∇v|)∇v∇ v˜ dx +

V1 (x)φ1 (|u|)uu˜ dx

 RN

V2 (x)φ2 (|v|)v v˜ dx

for all (˜u, v˜ ) ∈ W . So, it is sufficient to prove that I2 : W → R is well defined and of class C 1 (W , R) and

⟨I2′ (u, v), (˜u, v˜ )⟩ =





Fu (x, u, v)˜udx +

RN

RN

Fv (x, u, v)˜v dx

(3.3)

for all (˜u, v˜ ) ∈ W . By Remark 2.8, (2.7), (2.6) and the Hölder’s inequality, we have

 RN

F (x, u, v)dx ≤

≤

 RN

k  

RN

j =1

≤

|F (x, u, v)|dx a5j (x)|u|p1 qj dx +

j =1

k  

+

k  

RN

j =1

=

|a5j (x)|

RN

j =1

k 

∥a5j ∥

j =1

 ≤C

1 1−qj

|a6j (x)|

j =1

∥a5j ∥

qj |u|p1 dx

dx

1−qj 

qj

|v| dx

dx k 

∥a6j ∥

j =1

∥ u∥

p2

RN

∥u∥p11 j +

1 1−qj

a6j (x)|v|p2 qj dx

1−qj 

1 1−qj

p q

1 1−qj

RN

RN

k



k  

p1 qj 1,Φ1

+

k  j =1

p qj

1 1−qj

∥v∥p22

 ∥a6j ∥

1 1−qj

∥v∥

p2 qj 1,Φ2

Then I2 is well defined in W . We now prove that (3.3) holds. For any given (u, v), (˜u, v˜ ) ∈ W , we have 1

⟨I2′ (u, v), (˜u, v˜ )⟩ = lim (I2 (u + hu˜ , v + hv˜ ) − I2 (u, v)) h→0

h

.

(3.4)

118

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

= lim

h→0

1



h

RN

= lim

RN

F (x, u, v)dx

F (x, u, v + hv˜ ) − F (x, u, v)

+ lim

h→0

h

RN

 = lim

RN

dx

h

RN



h→0





F (x, u + hu˜ , v + hv˜ ) − F (x, u, v + hv˜ )

 h→0

F (x, u + hu˜ , v + hv˜ )dx −

dx



Fu (x, u + θ1 (x)hu˜ , v + hv˜ )˜udx + lim

h→0

RN

Fv (x, u, v + θ2 (x)hv˜ )˜v dx,

(3.5)

where θ1 , θ2 : RN → (0, 1) and it is clear that Fu (x, u + θ1 (x)hu˜ , v + hv˜ )˜u → Fu (x, u, v)˜u

(3.6)

and Fv (x, u, v + θ2 (x)hv˜ )˜v → Fv (x, u, v)˜v as h → 0 for a.e. x ∈ RN . Furthermore, for all h ∈ (−1, 1), by (F1 ), we have

|Fu (x, u + θ1 (x)hu˜ , v + hv˜ )˜u| k k p2 (p1 qj −1)   |˜u| ≤ a1j (x)|u + θ1 (x)hu˜ |p1 qj −1 |˜u| + a2j (x)|v + hv˜ | p1 j =1



j=1 k



≤C

a1j (x)(|u|

p1 qj −1

p1 qj −1

+ |˜u|

)|˜u| +

k 

j =1

 ≤C

k 



a2j (x) |v|

p2 (p1 qj −1) p1

+ |˜v |

p2 (p1 qj −1) p1



 |˜u|

j =1

a1j (x)(|u|

p1 qj −1

p1 qj

|˜u| + |˜u|

)+

j =1

k 

 a2j (x)(|˜u|

p1 qj

p2 qj

+ |v|

+ |˜v |

p2 qj

)

j =1

=: g1 (x).

(3.7)

By (2.7), (2.6) and the Hölder’s inequality, we obtain

 RN

g1 (x)dx

 ≤C

k 

∥a1j ∥

j =1

 ≤C

k  j =1

∥a1j ∥

 1 1−qj

∥u∥

p1 qj −1 p1

∥˜u∥p1 + ∥˜u∥

p1 qj p1



+

k 

∥a2j ∥

j =1

 1 1−qj

∥u∥

p1 qj −1 1,Φ1

∥˜u∥1,Φ1 + ∥˜u∥

p1 qj 1,Φ1



+

k 

 1 1−qj

∥a2j ∥

j =1

∥˜u∥

1 1−qj

p1 qj p1

+ ∥v∥

p2 qj p2

+ ∥˜v ∥

p2 qj p2

 

  pq  p2 qj p2 qj 1 j ∥˜u∥1,Φ1 + ∥v∥1,Φ2 + ∥˜v ∥1,Φ2

< +∞.

(3.8)

It follows from (3.6)–(3.8) and Lebesgue’s dominated convergence theorem that

 lim

h→0

RN

Fu (x, u + θ1 (x)hu˜ , v + hv˜ )˜udx =

 RN

Fu (x, u, v)˜udx.

(3.9)

Similarly, we can obtain that

 lim

h→0

RN

Fv (x, u, v + θ2 (x)hv˜ )˜v dx =

 RN

Fv (x, u, v)˜v dx.

(3.10)

Combining (3.9)–(3.10) with (3.5), we can conclude that (3.3) holds. Next, we prove the continuity of I2′ . Let (un , vn ) → (u, v) in W . For all (˜u, v˜ ) ∈ W , by (3.3), (2.7), (2.6) and the Hölder’s inequality we have

|⟨I2′ (un , vn ) − I2′ (u, v), (˜u, v˜ )⟩|         = Fu (x, un , vn )˜udx + Fv (x, un , vn )˜v dx − Fu (x, u, v)˜udx − Fv (x, u, v)˜v dx N N N N R R R R  ≤ |Fu (x, un , vn ) − Fu (x, u, v)| |˜u|dx + |Fv (x, un , vn ) − Fv (x, u, v)| |˜v |dx RN

RN

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130



 p1 |Fu (x, un , vn ) − Fu (x, u, v)| p1 −1 dx

≤ RN



|Fu (x, un , vn ) − Fu (x, u, v)|

≤C RN

 +C 

p1 −1 p1

|Fv (x, un , vn ) − Fv (x, u, v)|

RN



≤C

p1 p1 −1

1

dx

p2 p2 −1

+ RN

|Fv (x, un , vn ) − Fv (x, u, v)|

∥˜v ∥p2

p2 p2 −1

∥˜u∥1,Φ1

 p2p−1 2

dx

p1



RN

 p2 |Fv (x, un , vn ) − Fv (x, u, v)| p2 −1 dx

 p1p−1

|Fu (x, un , vn ) − Fu (x, u, v)| p1 −1 dx

RN

 ∥˜u∥p1 +

119 p2 −1 p2

∥˜v ∥1,Φ2

 p1p−1 1



 p2p−1 2

dx

 ∥(˜u, v˜ )∥.

(3.11)

Firstly, we claim that



p1

RN

|Fu (x, un , vn ) − Fu (x, u, v)| p1 −1 dx → 0,

as n → ∞.

(3.12)

Otherwise, there exist a constant ε0 > 0 and a subsequence of {(un , vn )} denoted by {(uni , vni )} such that



p1

RN

|Fu (x, uni , vni ) − Fu (x, u, v)| p1 −1 dx ≥ ε0 ,

for all i ∈ N.

(3.13)

Since (uni , vni ) → (u, v) in W , then uni → u in W 1,Φ1 (RN ) and vni → v in W 1,Φ2 (RN ), respectively. It follows from (2.7) that uni → u in Lp1 (RN ) and vni → v in Lp2 (RN ), respectively. By [24, Theorem 4.9], there exist subsequences of {uni } and {vni }, still denoted by {uni } and {vni }, respectively, and functions h1 ∈ Lp1 (RN ) and h2 ∈ Lp2 (RN ) such that uni (x) → u(x),

vni (x) → v(x),

a.e. x ∈ RN

(3.14)

and

|uni (x)| ≤ h1 (x),

|vni (x)| ≤ h2 (x),

for all i ∈ N, a.e. x ∈ RN .

(3.15)

By (3.14) and the continuity of Fu , we have p1

|Fu (x, uni (x), vni (x)) − Fu (x, u(x), v(x))| p1 −1 → 0,

a.e. x ∈ RN .

(3.16)

By (F1 ) and (3.15), for all i ∈ N, a.e. x ∈ R , we have N

p1

|Fu (x, uni , vni ) − Fu (x, u, v)| p1 −1   p1 p1 ≤ C |Fu (x, uni , vni )| p1 −1 + |Fu (x, u, v)| p1 −1  ≤C

k 

a1j (x)|uni |p1 qj −1 +

j =1

 +

≤C

k 

k 

a2j (x)|vni |

a1j (x)|u|p1 qj −1 +

k 

a2j (x)|v|

p1

p1

|a1j (x)| p1 −1 |uni | p1 −1

(p1 qj −1)

+

≤C

j=1

1

 p p−1 1  1

 p1

p2

|a2j (x)| p1 −1 |vni | p1 −1

(p1 qj −1)

j =1

k 

k 

p2 (p1 qj −1) p1

k 

|a1j (x)|

p1

p1 −1

|u|

p1 (p q −1) p1 −1 1 j

+

k 

j =1



 p p−1 1

j=1

j=1

+

p2 (p1 qj −1) p1

j =1

j=1



k 

|a2j (x)|

p1 p1 −1

|v|

p2 (p q −1) p1 −1 1 j

j =1 p1

p1 (p q −1) p −1 1 j

|a1j (x)| p1 −1 h1 1

+

k  j =1

p1

p2 ( p q −1 ) p −1 1 j

|a2j (x)| p1 −1 h2 1



120

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

+

k 

|a1j (x)|

p1 p1 −1

| u|

p1 (p q −1) p1 −1 1 j

+

k 

j =1

|a2j (x)|

p1 p1 −1

|v|

p2 (p q −1) p1 −1 1 j



j=1

=: g2 (x).

(3.17)

By (2.7), (2.6) and the Hölder’s inequality, we obtain



 RN

g2 (x)dx ≤ C

k 

∥a1j ∥

j=1 k

+



∥a1j ∥

j =1

 ≤C

k 

∥a1j ∥

j=1

+

k 

∥h1 ∥p11

+

k 

∥a2j ∥

j =1

p1 p1 −1 1 1−qj

p1 p1 −1 1 1−qj

∥a1j ∥

j =1

p1 (p q −1) p −1 1 j

p1 p1 −1 1 1−qj

p1

∥ u∥

p1 −1 p1

(p1 qj −1)

+

k 

∥a2j ∥

j=1 p1 (p q −1) p −1 1 j

∥h1 ∥p11

+

k 

∥a2j ∥

j =1

p1 p1 −1 1 1−qj

p1 p1 −1 1 1−qj

p1

p1 −1

∥u∥1,Φ1

(p1 qj −1)

+

k 

p1 p1 −1 1 1−qj

p1 p1 −1 1 1−qj

∥a2j ∥

j=1

p2 (p q −1) p −1 1 j

∥h2 ∥p21 ∥v∥

p2 (p q −1) p1 −1 1 j p2



p2 (p q −1) p −1 1 j

∥h2 ∥p21

p1 p1 −1 1 1−qj

p2 (p q −1) p1 −1 1 j



∥v∥1,Φ2

< +∞.

(3.18)

It follows from (3.16)–(3.18) and Lebesgue’s dominated convergence theorem that



p1

RN

|Fu (x, uni , vni ) − Fu (x, u, v)| p1 −1 dx → 0,

as i → ∞,

(3.19)

which contradicts (3.13). Then (3.12) holds. Similarly, we can obtain that



p2

RN

|Fv (x, un , vn ) − Fv (x, u, v)| p2 −1 dx → 0,

as n → ∞.

(3.20)

Combining (3.12) and (3.20) with (3.11), we can conclude that I2′ is continuous.



A standard argument shows that the critical points of the functional I correspond to the weak solutions of system (1.1). Let E be a Banach space. We say that J ∈ C 1 (E , R) satisfies (PS)-condition if any (PS)-sequence {un } ⊂ E has a convergent subsequence, where (PS)-sequence {un } means that J (un ) is bounded and J ′ (un ) → 0 as n → ∞. Lemma 3.2 (See [18]). Let E be a real Banach space and J ∈ C 1 (E , R) satisfies (PS)-condition. If J is bounded from below, then c = infE J is a critical value of J. In order to prove Theorem 1.2, we will use the following genus theory (see [18,19]). Let E be a Banach space, J ∈ C 1 (E , R) and c ∈ R. Set

Σ = {A ⊂ E \ {0} : A is closed in E and symmetric with respect to 0}. Definition 3.3 (See [18]). For A ∈ Σ , define the genus of A to be n (denoted by γ (A) = n) if there is an odd map ϕ ∈ C (A, Rn \ {0}) and n is the smallest integer with this property. Lemma 3.4 (See [19]). Let J be an even C 1 functional on E which satisfies the (PS)-condition. For any n ∈ N, set

Σn = {A ∈ Σ : γ (A) ≥ n},

dn = inf sup J (u). A∈Σn u∈A

If Σn ̸= ∅ and dn ∈ R, then dn is a critical value of J. Lemma 3.5. Suppose that (φ1 )–(φ3 ), (V ) and (F1 ) hold. Then I is coercive on W , that is, I (u, v) → +∞ as ∥(u, v)∥ → +∞. Proof. By (3.1), (3.4), (V ) and Lemma 2.2, we have I (u, v) =

 RN

Φ1 (|∇ u|)dx +

 + RN



 RN

Φ2 (|∇v|)dx + l

V1 (x)Φ1 (|u|)dx



m

N

R 

V2 (x)Φ2 (|v|)dx −



l

 RN m

F (x, u, v)dx

≥ min ∥∇ u∥Φ1 1 , ∥∇ u∥Φ11 + c1 min ∥u∥Φ1 1 , ∥u∥Φ11



L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130



m

l





m

l

+ min ∥∇v∥Φ2 2 , ∥∇v∥Φ22 + c1 min ∥v∥Φ2 2 , ∥v∥Φ22   k k   p1 qj p2 qj −C ∥a5j ∥ 1 ∥u∥1,Φ1 + ∥a6j ∥ 1 ∥v∥1,Φ2 1−qj

j =1

121



1−qj

j =1

  l l l l ≥ min{1, c1 } ∥∇ u∥Φ1 1 + ∥u∥Φ1 1 + ∥∇v∥Φ2 2 + ∥v∥Φ2 2 − 4   k k   p1 qj p2 qj −C ∥a5j ∥ 1 ∥u∥1,Φ1 + ∥a6j ∥ 1 ∥v∥1,Φ2 1−qj

j =1



l

l

1−qj

j =1



≥ C ∥u∥11,Φ1 + ∥v∥12,Φ2 − 4 min{1, c1 }  k k   p q −C ∥a5j ∥ 1 ∥u∥11,Φj1 + ∥a6j ∥ 1−qj

j =1

Since max{ p1 , 1 +∞. 

1 p2

j =1

 1 1−qj

∥v∥

p2 qj 1,Φ2

.

(3.21)

} ≤ qj < min{ pl11 , pl22 }(j = 1, 2, . . . , k), (3.21) implies I (u, v) → +∞ as ∥(u, v)∥ = ∥u∥1,Φ1 + ∥v∥1,Φ2 →

Lemma 3.6. If a sequence {(un , vn )} converges to (u0 , v0 ) in W weakly, then

 RN

(Fu (x, un , vn ) − Fu (x, u0 , v0 ))(un − u0 )dx → 0

(3.22)

(Fv (x, un , vn ) − Fv (x, u0 , v0 ))(vn − v0 )dx → 0

(3.23)

and

 RN

as n → ∞. Proof. Since (un , vn ) ⇀ (u0 , v0 ) in W , it is clear that un ⇀ u0 in W 1,Φ1 (RN ) and vn ⇀ v0 in W 1,Φ2 (RN ), respectively. Thus, {∥un ∥1,Φ1 } and {∥vn ∥1,Φ2 } is bounded. By (2.7) and (2.6), there exists a constant H > 0 such that

∥un ∥1,Φ1 , ∥vn ∥1,Φ2 , ∥un ∥p1 , ∥vn ∥p2 ≤ H ,

for all n = 0, 1, 2, . . . .

(3.24)

By (F1 ), for any given number ε > 0, there exists a constant r (ε) > 0 such that

 RN \Br (ε)

|aij (x)|

1 1−qj

1−qj dx

< ε,

for all i = 1, 2 and j = 1, 2, . . . , k.

(3.25)

Then by (F1 ), (3.24)–(3.25) and the Hölder’s inequality, we have

 RN \Br (ε)

|(Fu (x, un , vn ) − Fu (x, u0 , v0 ))| |un − u0 |dx

 ≤ RN \Br (ε)

|Fu (x, un , vn )| |un |dx +

 + RN \Br (ε)

RN \Br (ε) j=1

 +

a1j (x)|un |p1 qj dx +

k 

RN \Br (ε) j=1

 +

k 

RN \Br (ε) j=1

 +

RN \Br (ε)

|Fu (x, u0 , v0 )| |un |dx +

k 

 ≤



k 

RN \Br (ε) j=1

|Fu (x, un , vn )| |u0 |dx

 RN \Br (ε)

|Fu (x, u0 , v0 )| |u0 |dx

k 



RN \Br (ε) j=1

a1j (x)|un |p1 qj −1 |u0 |dx +

a1j (x)|u0 |p1 qj −1 |un |dx +

a1j (x)|u0 |p1 qj dx +



a2j (x)|un | |vn | k 



RN \Br (ε) j=1 k 



RN \Br (ε) j=1 k 

RN \Br (ε) j=1

p2 (p1 qj −1) p1

dx

a2j (x)|u0 | |vn |

a2j (x)|un | |v0 |

a2j (x)|u0 | |v0 |

p2 (p1 qj −1) p1

dx

p2 (p1 qj −1) p1

dx

p2 (p1 qj −1) p1

dx

122

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

≤

k 

 RN \Br (ε)

j =1

 k 

+

 k 

+

|a1j (x)|

RN \Br (ε)

j =1

 k 

+

|a2j (x)|

RN \Br (ε)

j =1

 k 

+

|a1j (x)|

RN \Br (ε)

j =1

 k 

+

k 

+

|a2j (x)|

RN \Br (ε)

j =1



k 

|a2j (x)|

RN \Br (ε)

j =1

|un | dx

dx

1 1−qj

 1 −q j  

|un |p1 dx

dx

1 1−qj

 p1 pqj −1

 p1  1

RN \Br (ε)

1

|vn |p2 dx RN \Br (ε)

 1 −q j  



p1 qj −1 p1

|u0 | dx RN \Br (ε)

 p1 pqj −1

 p1  1

 1 −q j  

1

|u0 |p1 dx

dx RN \Br (ε)

1 1−qj

|vn |p2 dx RN \Br (ε)

 1 −q j  



p1 qj −1 p1

 1 −q j  

|un | dx RN \Br (ε)

|un |p1 dx

dx

 1 −q j  

1

|v0 |p2 dx RN \Br (ε)

|u0 |p1 dx RN \Br (ε)

 p1 pqj −1

 p1  1

p1

1

p2

|v0 | dx

|u0 | dx

dx

RN \Br (ε)

RN \Br (ε) p2 (p1 qj −1) p1

p q

 p1 pqj −1

 p1  1

q j

 1 −q j  



1

p1

|u0 | dx

dx

RN \Br (ε)

1 1−qj

 p1



p1

RN \Br (ε) 1 1−qj

1

p1

|un | dx RN \Br (ε)

1 1−qj

 p1



p1

dx

1

 k 

+

qj p1

|a1j (x)| 1−qj dx

RN \Br (ε)

j =1

1−qj  RN \Br (ε)

|a2j (x)|

RN \Br (ε)

j =1

≤ε

|a1j (x)|

1 1−qj

∥un ∥p11 j + ∥un ∥p1 ∥vn ∥p2

p q j −1

+ ∥un ∥p11

p2 (p1 qj −1) p1

∥u0 ∥p1 + ∥u0 ∥p1 ∥vn ∥p2

j =1 p1 qj −1 u0 p1

+∥ ∥

≤ε

k  

4H

∥un ∥p1 + ∥un ∥p1 ∥v0 ∥

p1 qj

+ 4HH

p2 (p1 qj −1) p1



p2 (p1 qj −1) p1 p2

p1 qj u0 p1

+∥ ∥

p2 (p1 qj −1) p1



+ ∥u0 ∥p1 ∥v0 ∥p2

.

(3.26)

j =1

Moreover, un ⇀ u0 in W 1,Φ1 (RN ) implies un |Br (ε) ⇀ u0 |Br (ε) in W 1,Φ1 (Br (ε) ). It follows from (2.8) that un |Br (ε) → u0 |Br (ε) in Lp1 (Br (ε) ). Then for the given ε above, we can choose a n0 ∈ N such that

 p1



1

p1

|un − u0 | dx

< ε,

for all n > n0 .

(3.27)

Br (ε)

Then by (F1 ), (3.24), (3.27) and the Hölder’s inequality, we have



|(Fu (x, un , vn ) − Fu (x, u0 , v0 ))| |un − u0 |dx Br (ε)



p1

 p1p−1  1

 p1

1

|Fu (x, un , vn ) − Fu (x, u0 , v0 )| p1 −1 dx

≤ Br (ε)

|un − u0 |p1 dx

Br (ε)

   p1p−1 1 p1 p1 ε |Fu (x, un , vn )| p1 −1 + |Fu (x, u0 , v0 )| p1 −1 dx

 ≤C Br (ε)

 ≤C

k   j =1

p1

p1

|a1j (x)| p1 −1 |un | p1 −1 Br (ε)

(p1 qj −1)

dx +

k   j =1

p1

p2

|a2j (x)| p1 −1 |vn | p1 −1 Br (ε)

(p1 qj −1)

dx

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

+

k  

≤C

k 

k 

∥a1j ∥

∥a1j ∥

j =1

 ≤C 2

dx +

k 

k   j=1

j=1

+

|u0 |

p1 (p q −1) p1 −1 1 j

Br (ε)

j =1



|a1j (x)|

p1 p1 −1

p1 p1 −1 1 1−qj

p1 p1 −1 1 1−qj

∥a1j ∥

j =1

p1 (p q −1) p −1 1 j

∥un ∥p11

∥a2j ∥

j =1

∥ u0 ∥

p1 p1 −1 1 1−qj

k 

+

p1 (p q −1) p1 −1 1 j p1

+

k 

∥a2j ∥

H

+2

p1 p1 −1 1 1−qj

k 

p1 −1 1 1−qj

∥a2j ∥

j =1

p2 (p q −1) p1 −1 1 j

|v0 |

 dx

ε

Br (ε) p2 (p q −1) p −1 1 j

∥vn ∥p21

p1

j =1

p1 (p q −1) p1 −1 1 j

|a2j (x)|

p1 p1 −1

123 p1 −1 p1

∥v0 ∥

p1 p1 −1 1 1−qj

H

p2 ( p q −1 ) p1 −1 1 j p2

p2 (p q −1) p1 −1 1 j

 p1p−1 1

ε

 p1p−1 1

ε

(3.28)

for all n > n0 . As ε is arbitrary, combining (3.26) with (3.28), we conclude that (3.22) holds. With a similar discussion as above, we can prove that (3.23) holds.  Definition 3.7 (See [25, Definition 2.97]). Let X be a real, reflexive Banach space, and denote by X ∗ its dual space. The operator A : X → X ∗ is pseudomonotone iff un ⇀ u0

lim sup⟨A(un ), un − u0 ⟩ ≤ 0

and

n→∞

implies lim⟨A(un ), un ⟩ = ⟨A(u0 ), u0 ⟩

and A(un ) ⇀ A(u0 ).

Lemma 3.8. Suppose that (φ1 )–(φ3 ) and (V ) hold. Define operators Ai (i = 1, 2) : W 1,Φ1 (RN ) → W 1,Φ1 (RN )∗ and

Bi (i = 1, 2) : W 1,Φ2 (RN ) → W 1,Φ2 (RN )∗ by

⟨A1 (u), u˜ ⟩ :=



⟨A2 (u), u˜ ⟩ :=



⟨B1 (v), v˜ ⟩ :=



⟨B2 (v), v˜ ⟩ :=



RN

RN

φ1 (|∇ u|)∇ u∇ u˜ dx,

u, u˜ ∈ W 1,Φ1 (RN ),

V1 (x)φ1 (|u|)uu˜ dx,

u, u˜ ∈ W 1,Φ1 (RN ),

and

RN

RN

φ2 (|∇v|)∇v∇ v˜ dx,

v, v˜ ∈ W 1,Φ2 (RN ),

V2 (x)φ2 (|v|)v v˜ dx,

v, v˜ ∈ W 1,Φ2 (RN ).

Then Ai (i = 1, 2) and Bi (i = 1, 2) are pseudomonotone. Proof. This lemma can be proved in a similar way to Proposition A.2 in [12]. We omit the details.



Lemma 3.9. Suppose that (φ1 )–(φ3 ), (V ) and (F1 ) hold. Then I satisfies (PS)-condition. Proof. Let {(un , vn )} be any (PS)-sequence in W for I. It follows from Lemma 3.5 that sequence {(un , vn )} is bounded in W . Therefore, going if necessary to a subsequence, we can assume that (un , vn ) ⇀ (u0 , v0 ) in W . Then un ⇀ u0 in W 1,Φ1 (RN ) and vn ⇀ v0 in W 1,Φ2 (RN ), respectively. Note that on (1) = ⟨I ′ (un , vn ) − I ′ (u0 , v0 ), (un − u0 , vn − v0 )⟩

 =

(φ1 (|∇ un |)∇ un − φ1 (|∇ u0 |)∇ u0 ) (∇ un − ∇ u0 )dx  V1 (x)(φ1 (|un |)un − φ1 (|u0 |)u0 ) (un − u0 )dx N R (φ2 (|∇vn |)∇vn − φ2 (|∇v0 |)∇v0 ) (∇vn − ∇v0 )dx RN  V2 (x)(φ2 (|vn |)vn − φ2 (|v0 |)v0 ) (vn − v0 )dx N R (Fu (x, un , vn ) − Fu (x, u0 , v0 ))(un − u0 )dx

RN

+ + + −

RN

124

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

 − RN

(Fv (x, un , vn ) − Fv (x, u0 , v0 ))(vn − v0 )dx.

(3.29)

Furthermore, (V ) and (φ2 ) imply that operators Ai , Bi (i = 1, 2) defined in Lemma 3.8 are strictly monotone. So, it follows from Lemma 3.6 that (3.29) implies that

 RN

 RN

 RN

(φ1 (|∇ un |)∇ un − φ1 (|∇ u0 |)∇ u0 ) (∇ un − ∇ u0 )dx → 0,

(3.30)

V1 (x)(φ1 (|un |)un − φ1 (|u0 |)u0 ) (un − u0 )dx → 0,

(3.31)

(φ2 (|∇vn |)∇vn − φ2 (|∇v0 |)∇v0 ) (∇vn − ∇v0 )dx → 0,

(3.32)

V2 (x)(φ2 (|vn |)vn − φ2 (|v0 |)v0 ) (vn − v0 )dx → 0,

(3.33)

and

 RN

as n → ∞. Firstly, according to (3.30), we claim that ∥∇ un − ∇ u0 ∥Φ1 → 0, as n → ∞. In fact by (2.3), it is sufficient to prove that

 RN

Φ1 (|∇ un − ∇ u0 |)dx → 0,

as n → ∞.

(3.34)

By (φ2 ) and (3.30), we have

 RN

|(φ1 (|∇ un |)∇ un − φ1 (|∇ u0 |)∇ u0 )(∇ un − ∇ u0 )|dx → 0,

which implies that

(φ1 (|∇ un (x)|)∇ un (x) − φ1 (|∇ u0 (x)|)∇ u0 (x), ∇ un (x) − ∇ u0 (x)) → 0,

a.e. x ∈ RN .

By (φ2 ), Proposition A.1 in [12] implies that

∇ un (x) → ∇ u0 (x),

a.e. x ∈ RN .

(3.35)

Thus, according to the continuity of Φ1 and the fact that Φ1 (0) = 0, we have

Φ1 (|∇ un (x) − ∇ u0 (x)|) → 0,

a.e. x ∈ RN .

(3.36)

We define fn := Φ1 (|∇ un − ∇ u0 |), gn := 2m1 −1 [φ1 (|∇ un |)|∇ un |2 + Φ1 (|∇ u0 |)], and g := 2m1 −1 [φ1 (|∇ u0 |)|∇ u0 |2 + Φ1 (|∇ u0 |)]. Then

|fn | ≤ gn ,

for all n ∈ N, (see [12, Proposition A.3]).

(3.37)

By the continuity of φ1 , (3.35) implies that gn (x) → g (x), a.e. x ∈ R and we have N

 RN

g (x)dx =

 RN

2m1 −1 [φ1 (|∇ u0 |)|∇ u0 |2 + Φ1 (|∇ u0 |)]dx

≤ 2m1 ∥φ1 (|∇ u0 |)|∇ u0 |∥Φ1 ∥∇ u0 ∥Φ1 + 2m1 −1

 RN

Φ1 (|∇ u0 |)dx

< +∞,

(3.38)

which can be obtained easily by using (2.2), (2.4) and the definition of Orlicz space. Since un ⇀ u0 in W 1,Φ1 (RN ), by (3.30), we have lim ⟨A1 (un ), un − u0 ⟩ = lim

n→∞

n→∞

 RN

φ1 (|∇ un |)∇ un (∇ un − ∇ u0 )dx

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

 = lim

n→∞

RN

125

φ1 (|∇ u0 |)∇ u0 (∇ un − ∇ u0 )dx = 0,

which together with Lemma 3.8 implies that

 lim

n→∞

RN



φ1 (|∇ un |)|∇ un |2 dx = lim⟨A1 (un ), un ⟩ = ⟨A1 (u0 ), u0 ⟩ =

RN

φ1 (|∇ u0 |)|∇ u0 |2 dx.

Then

 lim

n→∞

RN

gn (x)dx =

 RN

g (x)dx.

(3.39)

By (3.36)–(3.39), the generalized Lebesgue convergence theorem implies that (3.34) holds. Therefore, ∥∇ un − ∇ u0 ∥Φ1 → 0, as n → ∞. By similar arguments as above, combining (3.31)–(3.33) with Lemma 3.8, we can also get

∥un − u0 ∥Φ1 → 0,

∥vn − v0 ∥Φ2 → 0,

∥∇vn − ∇v0 ∥Φ2 → 0,

Then (un , vn ) → (u0 , v0 ) in W . Hence, I satisfies (PS)-condition.

as n → ∞.



Lemma 3.10. Suppose that (φ1 )–(φ3 ), (V ) and (F2 ) hold. Then there exists a point (u, v) ∈ W such that I (u, v) < 0. Proof. Without loss of generality, we can assume that ι = ̸ 0 in (F2 ). Now we choose w0 ∈ C0∞ (RN ) \ {0} with w0 (x) ≥ 0, supp(w0 ) ⊂ Ω and ∥w0 ∥∞ ≤ 1. Let u0 = ιw0 and v0 = κw0 . Then it is clear that (u0 , v0 ) ∈ W . When 0 < t < δ , by (3.1), (V ), Lemma 2.2 and (F2 ), we have I (tu0 , t v0 ) =

 RN

Φ1 (|∇ tu0 |)dx +

 + RN



V1 (x)Φ1 (|tu0 |)dx

RN

Φ2 (|∇ t v0 |)dx +

 RN

l

V2 (x)Φ2 (|t v0 |)dx − m

 RN l1

F (x, tu0 , t v0 )dx m

≤ max{t l1 ∥∇ u0 ∥Φ1 1 , t m1 ∥∇ u0 ∥Φ11 } + c2 max{t l1 ∥u0 ∥Φ1 , t m1 ∥u0 ∥Φ11 } l

m

l

m

+ max{t l2 ∥∇v0 ∥Φ2 2 , t m2 ∥∇v0 ∥Φ22 } + c2 max{t l2 ∥v0 ∥Φ2 2 , t m2 ∥v0 ∥Φ22 } − l

m

l

 Ω

F (x, t ιw0 , t κw0 )dx

m

≤ t l1 ∥∇ u0 ∥Φ1 1 + t m1 ∥∇ u0 ∥Φ11 + c2 t l1 ∥u0 ∥Φ1 1 + c2 t m1 ∥u0 ∥Φ11 l

m

l

m

+ t l2 ∥∇v0 ∥Φ2 2 + t m2 ∥∇v0 ∥Φ22 + c2 t l2 ∥v0 ∥Φ2 2 + c2 t m2 ∥v0 ∥Φ22 − l1

l1

= t ∥∇ u0 ∥Φ1 + t

m1

m1

l1

l1

∥∇ u0 ∥Φ1 + c2 t ∥u0 ∥Φ1 + c2 t

l

m

l

m1

m1

m

Ω

α0

c (|t ιw0 |α0 + |t κw0 |β0 )dx



|u0 |α0 dx  − ct β0 |v0 |β0 dx.

∥u0 ∥Φ1 − ct

+ t l2 ∥∇v0 ∥Φ2 2 + t m2 ∥∇v0 ∥Φ22 + c2 t l2 ∥v0 ∥Φ2 2 + c2 t m2 ∥v0 ∥Φ22



Ω

Ω

Since α0 ∈ [1, l1 ) and β0 ∈ [1, l2 ), we can choose t0 > 0 small enough such that I (t0 u0 , t0 v0 ) < 0.



Proof of Theorem 1.1. Let E = W and J = I. By Lemma 3.1, Lemma 3.5 and Lemma 3.9, all conditions of Lemma 3.2 hold. Then the functional I possesses a critical point (u, v) ∈ W which is a weak solution of system (1.1) satisfying I (u, v) = infW I. Lemma 3.10 implies that (u, v) ̸= 0. Thus system (1.1) possesses at least one nontrivial weak solution.  Lemma 3.11. Suppose that (φ1 )–(φ3 ), (V ) and (F2 )–(F3 ) hold. Let E = W and J = I. Then for any n ∈ N, dn defined in Lemma 3.4 exists and dn < 0. Proof. Without loss nof generality, we can assume that ι ̸= 0 in (F2 ). For any given n ∈ N, we take n disjoint nonempty open sets Ωi such that i=1 Ωi ⊂ Ω . For i = 1, 2, . . . , n, we choose wi ∈ C0∞ (RN ) \ {0} with wi (x) ≥ 0, supp(wi ) ⊂ Ωi and ∥wi ∥∞ ≤ 1. Let ui = ιwi and vi = κwi . Then ui ∈ W 1,Φ1 (RN ), vi ∈ W 1,Φ2 (RN ) and (ui , vi ) ∈ W (i = 1, 2, . . . , n). When κ ̸= 0, we define spaces W 1,Φ1 (RN )(n) := span{u1 , u2 , . . . , un }, W 1,Φ2 (RN )(n) := span{v1 , v2 , . . . , vn }, and W(n) := span{(u1 , v1 ), (u2 , v2 ), . . . , (un , vn )}.

126

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

In space W(n) , for every (u, v) ∈ W(n) , there exists a unique point (λ1 , λ2 , . . . , λn ) ∈ Rn such that

(u, v) =

n 

 λi (ui , vi ) =

i=1

n 

λi ui ,

n 

i=1

 λi vi

 =

i =1

n 

λi ιwi ,

i =1

n 

 λi κwi ,

(3.40)

i=1

which implies that u ∈ W 1,Φ1 (RN )(n) and v ∈ W 1,Φ2 (RN )(n) , respectively, and we take norm

 ∥(u, v)∥0 :=

n 

 21 λ2i

,

and let Sn := {(u, v) ∈ W(n) : ∥(u, v)∥0 = 1}.

i=1

¯ 1 , λ¯ 2 , . . . , λ¯ n ) ∈ In (n)-dimensional normed space W 1,Φ1 (RN )(n) , for every u ∈ W 1,Φ1 (RN )(n) , there exists a unique point (λ n n ¯ R such that u = i=1 λi ui , and we take norms  ∥u∥α0 =

|u| dx n 

 α0  α1  α1  n n    0 0    α 0 |λ¯ i ιwi | dx , λ¯ i ui  dx =   i =1  i=1 Ωi



0

=

RN

 ∥u∥0 :=

 α1

α0

RN

 21 λ¯ 2i

and ∥u∥1,Φ1 = ∥∇ u∥Φ1 + ∥u∥Φ1 ,

i =1

which are equivalent. In (n)-dimensional normed space W 1,Φ2 (RN )(n) , for every v ∈ W 1,Φ2 (RN )(n) , there exists a unique  ˜ 1 , λ˜ 2 , . . . , λ˜ n ) ∈ Rn such that v = ni=1 λ˜ i vi , and we take norms point (λ

∥v∥β0 =

0

|v|β0 dx RN

 ∥v∥0 :=

  =

 β1



n 

RN

β0  β10    β1 n n    0    β λ˜ i vi  dx = |λ˜ i κwi | 0 dx ,    i=1 i=1 Ωi

 12 λ˜

2 i

∥v∥1,Φ2 = ∥∇v∥Φ2 + ∥v∥Φ2 ,

and

i=1

which are equivalent. When (u, v) ∈ Sn and 0 < t < δ , using (3.1), (V ), Lemma 2.2, (F2 ), (3.40) and the properties of equivalent norms we have I (tu, t v) =

 RN

Φ1 (|∇ tu|)dx +

 + RN

 RN

Φ2 (|∇ t v|)dx + l

V1 (x)Φ1 (|tu|)dx



V2 (x)Φ2 (|t v|)dx −

RN m1

 RN

F (x, tu, t v)dx

l

m

≤ max{t l1 ∥∇ u∥Φ1 1 , t m1 ∥∇ u∥Φ1 } + c2 max{t l1 ∥u∥Φ1 1 , t m1 ∥u∥Φ11 } l

m

l

m

+ max{t l2 ∥∇v∥Φ2 2 , t m2 ∥∇v∥Φ22 } + c2 max{t l2 ∥v∥Φ2 2 , t m2 ∥v∥Φ22 }    n n   − F x, t λi ιwi , t λi κwi dx RN

l1

i=1 l1

≤ t ∥∇ u∥Φ1 + t

m1

i=1 m1

l

m

∥∇ u∥Φ1 + c2 t l1 ∥u∥Φ1 1 + c2 t m1 ∥u∥Φ11

l

l

m

m

+ t l2 ∥∇v∥Φ2 2 + t m2 ∥∇v∥Φ22 + c2 t l2 ∥v∥Φ2 2 + c2 t m2 ∥v∥Φ22 −

n  

Ωi

i=1

F (x, t λi ιwi , t λi κwi )dx

 ≤ max{1, c2 } t l1 (∥∇ u∥Φ1 + ∥u∥Φ1 )l1 + t m1 (∥∇ u∥Φ1 + ∥u∥Φ1 )m1 n    + t l2 (∥∇v∥Φ2 + ∥v∥Φ2 )l2 + t m2 (∥∇v∥Φ2 + ∥v∥Φ2 )m2 − c (|t λi ιwi |α0 + |t λi κwi |β0 )dx i =1

Ωi

   β + t ∥v∥ − c t α0 ∥u∥αα00 + t β0 ∥v∥β00 ≤ max{1, c2 } t ∥u∥ + t ∥u∥ + t ∥v∥     α β l m l m ≤ C t l1 ∥u∥01 + t m1 ∥u∥0 1 + t l2 ∥v∥02 + t m2 ∥v∥0 2 − C t α0 ∥u∥0 0 + t β0 ∥v∥0 0     = C t l1 + t m1 + t l2 + t m2 − C t α0 + t β0 . 

l1

l1 1,Φ1

m1

m1 1,Φ1

l2

l2 1,Φ2

m2

m2 1,Φ2

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

127

Since α0 ∈ [1, l1 ) and β0 ∈ [1, l2 ), we can choose t0 , ε > 0 small enough such that I (t0 u, t0 v) < −ε for all (u, v) ∈ Sn . Let Snt0

:= {t0 (u, v) : (u, v) ∈ Sn },

D

t0

:=

 

 (λ1 , λ2 , . . . , λn ) ∈ R : N



n 

 21 λ

2 i

< t0

i =1

 

,



and I −ε := {(u, v) ∈ W : I (u, v) ≤ −ε}. Then Snt0 ⊂ I −ε . It follows from (F3 ) that I is even, which, together with the fact that I ∈ C 1 (W , R) and I (0, 0) = 0, implies that I −ε ∈ Σ . t

Moreover, there exists an odd homeomorphism mapping ψ ∈ C (Sn0 , ∂ Dt0 ) which can be defined by

ψ(u, v) := ψ

n 

λi (ui , vi ) = (λ1 , λ2 , . . . , λn ).

1

By some properties of the genus (see 3° of Propositions 7.5 and 7.7 in [19]), we have

γ (I −ε ) ≥ γ (Snt0 ) = n, which, together with Lemma 3.5, implies that

−∞ < dn = inf sup I (u, v) ≤ −ε < 0. A∈Σn u∈A

When κ = 0, then vi = κwi = 0, and the space W 1,Φ2 (RN )(n) defined above is {0}, which makes the proof simpler. We omit the details.  Next, we claim that dn → 0, as n → +∞. Before stating this argument, we present a useful preliminary lemma. Since W is reflexive and separable Banach space, then there exist sequences {ei : i ∈ N} ⊂ W and {e∗i : i ∈ N} ⊂ W ∗ such that W = span{ei : i = 1, 2, . . .},

W ∗ = span{e∗i : i = 1, 2, . . .},

and e∗i (ej ) =



1 0

if i = j, if i ̸= j

(see [26, Section 17]). Define Yn := span{ei : i = 1, 2, . . . , n},

Zn := span{ei : i = n, n + 1, . . .}.

(3.41)

Lemma 3.12. Suppose that (φ1 )–(φ3 ), (V ) and (F1 ) hold. For any given constant η > 0, set

πn :=

sup

(u,v)∈Zn ,∥(u,v)∥≤η

|I2 (u, v)|,

then πn → 0 as n → ∞. Proof. Note that I2 ∈ C 1 (W , R). Then 0 < πn+1 ≤ πn < ∞ for all n ∈ N, so πn → π0 ≥ 0, as n → ∞. For every n ∈ N, there exists (un , vn ) ∈ Zn such that 1

∥(un , vn )∥ ≤ η and πn < |I2 (un , vn )| + . n

(3.42)

Then, there exist a subsequence of {(un , vn )}, still denoted by {(un , vn )}, and a point (∞u, v) ∈ W such that {(un , vn )} ⇀ (u, v) in W . Since Zn is a closed subspace in W , by Mazur’s theorem, we have (u, v) ∈ n=1 Zn = {(0, 0)}. So {(un , vn )} ⇀ (0, 0) in W , and un ⇀ 0 in W 1,Φ1 (RN ) and vn ⇀ 0 in W 1,Φ2 (RN ), respectively. We claim that |I2 (un , vn )| → 0 as n → ∞. Indeed, for any given number ε > 0, there exists a constant r (ε) > 0 such that

 RN \Br (ε)

|aij (x)|

1 1−qj

1−qj dx

< ε,

for all i = 5, 6 and j = 1, 2, . . . , k.

(3.43)

128

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

Then by Remark 2.8, the Hölder’s inequality, (2.7), (2.6) and (3.43), we have

 RN \Br (ε)





|F (x, un , vn )|dx ≤

k 

RN \Br (ε)

RN \Br (ε)

j =1

|a5j (x)|

RN \Br (ε)

j=1

 ≤ε

k 

p1 qj un p1

≤ εC ≤ εC

dx

1−qj 

q j p1

|un | dx

dx

1−qj 

1 1−qj

qj p2

|vn | dx

dx RN \Br (ε)

 p2 qj n p2

∥v ∥

j =1

k 

p1 qj un 1,Φ1

∥ ∥

+

k 

k 

 p2 qj n 1,Φ2

∥v ∥

j =1

j=1



a6j (x)|vn |

RN \Br (ε)

k 

+

∥ ∥

1 1−qj

|a6j (x)|

j =1



 p2 qj

j=1

 k 

+

+

k 

j =1

 k 

≤

a5j (x)|un |

p1 qj

η

p1 qj

k 

+

j=1

 η

.

p2 qj

(3.44)

j =1

Moreover, un ⇀ 0 in W 1,Φ1 (RN ) implies un |Br (ε) ⇀ 0 in W 1,Φ1 (Br (ε) ) and vn ⇀ 0 in W 1,Φ2 (RN ) implies vn |Br (ε) ⇀ 0 in

W 1,Φ2 (Br (ε) ). It follows from (2.8) that un |Br (ε) → 0 in Lp1 (Br (ε) ) and vn |Br (ε) → 0 in Lp2 (Br (ε) ). Then for the given ε above, we can choose a n0 ∈ N such that

q j



|un | dx

q j



< ε and

p1

< ε for all n > n0 , j = 1, 2, . . . , k.

p2

|vn | dx

Br (ε)

(3.45)

Br (ε)

Then by Remark 2.8, the Hölder’s inequality and (3.45), we have



|F (x, un , vn )|dx ≤





Br (ε)

Br (ε)

≤

k 

k 

 a6j (x)|vn |

p2 qj

|a5j (x)|

 1 −q j  

1 1−qj



k 

q j |un |p1 dx

dx Br (ε)

 k 

1 1−qj

|a6j (x)|

 1 −q j  

∥a5j ∥

j =1

q j p2

|vn | dx

dx

Br (ε)

j =1

dx

j =1

Br (ε)

j=1

≤ε

+

j =1

 k 

+

a5j (x)|un |

p1 qj

Br (ε)

1 1−qj

+

k 

 ∥a6j ∥

j =1

(3.46)

1 1−qj

for all n > n0 . Combining (3.44) with (3.46), we have

|I2 (un , vn )| ≤

 RN

|F (x, un , vn )|dx =

  ≤ε C

k  j=1

η

p1 qi

+

k  j =1

 RN \Br (ε)

|F (x, un , vn )|dx +

 η

p2 qi

+



|F (x, un , vn )|dx Br (ε)

k  j =1

∥a5j ∥

1 1−qj

+

k  j =1

 ∥a6j ∥

1 1−qj

for all n > n0 . As ε is arbitrary, then |I2 (un , vn )| → 0 as n → ∞. Thus, it follows from (3.42) that πn → 0, as n → ∞.



Proof of Theorem 1.2. Let E = W and J = I. By Lemmas 3.1, 3.9 and 3.11, all conditions of Lemma 3.4 hold. Then the functional I possesses a sequence of critical points (un , vn ) ∈ W which are weak solutions of system (1.1) satisfying I (un , vn ) = dn < 0. Now we prove that dn → 0 as n → ∞. Indeed, Lemma 3.5 implies that there exists a constant η0 > 0 such that I (u, v) > 0,

for all ∥(u, v)∥ > η0 .

(3.47)

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

For arbitrarily A ∈ Σn , by the properties of genus we know that A Set

πn :=

sup

(u,v)∈Zn ,∥(u,v)∥≤η0



129

Zn ̸= ∅, where Zn is the subspace of W defined in (3.41).

|I2 (u, v)|.

Lemma 3.12 implies that πn → 0 as n → ∞. When (u, v) ∈ Zn with ∥(u, v)∥ ≤ η0 , we have I (u, v) = I1 (u, v) − I2 (u, v) ≥ −I2 (u, v) ≥ −|I2 (u, v)| ≥ −πn , which, together with (3.47), implies that sup I (u, v) ≥ −πn

(u,v)∈A

and hence dn ≥ −πn . Thus dn → 0 as n → ∞.



4. Example Consider this elliptic system

   2 3 −div (4|∇ u|2 + 5|∇ u|3 )∇ u + (2 + sin  |x|)(4|u| + 5|u| )u    3 1 3 3   = a(x) |u|u − |u| 2 u − |u|− 2 u|v| 2    2         |∇v|3 |v|3 −div 4|∇v|2 log(1 + |∇v|) + ∇v + 2 4|v|2 log(1 + |v|) + v  1 + |∇v| 1 + |v|      3 3 3 1    = a(x) |v|v − |v| 2 v − |u| 2 |v|− 2 v   2   u(x) → 0 and v(x) → 0

in R6 ,

in R6 , as |x| → ∞,

where a(x) =

1

(1 +

x21

x22

+

+

x23

)(1 + x24 + x25 + x26 )

.

Let

φ1 (t ) := 4t 2 + 5t 3 ,

φ2 (t ) := 4t 2 log(1 + t ) +

t3 1+t

,

for t > 0.

Then

Φ1 (t ) = t 4 + |t |5 ,

Φ2 (t ) = t 4 log(1 + |t |),

for t ∈ R.

By simple computations we have that φi (i = 1, 2) satisfy (φ1 )–(φ3 ) and l1 = l2 = 4,

m1 = m2 = 5.

Let V1 (x) := 2 + sin |x|,

V2 (x) := 2,

for x ∈ RN .

Then, it is obvious that Vi (i = 1, 2) satisfy (V ). Let F (x, u, v) := a(x)



1 3

3

2

1

7 2

2

3

7 2

3 2

|u| − |u| + |v| − |v| − |u| |v| 7

3

7

3 2



,

for (x, u, v) ∈ RN × R × R.

Then F is of class C 1 and



3

Fu (x, u, v) = a(x) |u|u − |u| 2 u −



3 2

Fv (x, u, v) = a(x) |v|v − |v| v −

3 2 3 2

1

3

3 2

− 21

|u| |v|



,

for (x, u, v) ∈ RN × R × R,

v ,

for (x, u, v) ∈ RN × R × R,

|u|− 2 u|v| 2



by which, we have

  3 11 9 5 1 3 7 a(x)|u|3−1 + a(x)|u| 2 −1 + a(x)|v|3−1 , |Fu (x, u, v)| ≤ a(x) |u|2 + |u| 2 + |u| 2 |v| 2 ≤ 2

8

8

130

L. Wang et al. / Computers and Mathematics with Applications 72 (2016) 110–130

  3 9 11 5 3 1 7 |Fv (x, u, v)| ≤ a(x) |v|2 + |v| 2 + |u| 2 |v| 2 ≤ a(x)|u|3−1 + a(x)|v|3−1 + a(x)|v| 2 −1 . 2

8

8

We take p1 = p2 = 6, α1 = β1 = α2 = β2 = a11 (x) = a41 (x) = 11 a(x), a21 (x) = a31 (x) = 8 a12 = a22 = a32 = a42 = a(x). It is easily to check that (F1 ) holds. 1 , ι = 1 and κ = 0. Then We take Ω = {x ∈ RN | |x| < 1}, α0 = β0 = 3, δ = 1, c = 84 1 , 2

F (x, u, 0) = a(x)



1 3

2

7

|u|3 − |u| 2



7

≥

7 , 12

1 84

|u|3 = c |u|α0 ,

9 a 8

(x), and

for x ∈ Ω , u ∈ [0, 1] = [0, ιδ],

which implies that (F2 ) holds. However, when v = u, we have



1 4 7 F (x, u, u) = a(x) − |u|3 − |u| 2 3 7



< 0,

for x ∈ Ω , u ∈ R \ {0},

which implies that F is sign-changing in any neighborhood of (0, 0) in u–v plane for any given x ∈ Ω . Obviously, F satisfies (F3 ). Thus this system satisfies all the conditions of Theorems 1.1 and 1.2. Acknowledgments This project is supported by the National Natural Science Foundation of China (No: 11301235) and Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (No: 11226135). References [1] N. Fukagai, K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. of Math. 186 (3) (2007) 539–564. [2] C.O. Alves, G.M. Figueiredo, J.A. Santos, Strauss and Lions type results for a class of Orlicz–Sobolev spaces and applications, Topol. Methods Nonlinear Anal. 44 (2014) 435–456. [3] Ph. Clément, M. García-Huidobro, R. Manásevich, K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. 11 (2000) 33–62. [4] N. Fukagai, M. Ito, K. Narukawa, Quasilinear elliptic equations with slowly growing principal part and critical Orlicz–Sobolev nonlinear term, Proc. Roy. Soc. Edinburgh Sect. A 139 (1) (2009) 73–106. [5] V.K. Le, Some existence results and properties of solutions in quasilinear variational inequalities with general growth, Differential Equations Dynam. Systems 17 (2009) 343–364. [6] M. Mihăilescu, D. Repovš, Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz–Sobolev spaces, Appl. Math. Comput. 217 (2011) 6624–6632. [7] G. Bonanno, G. Molica Bisci, V. Rădulescu, Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces, Nonlinear Anal. 75 (12) (2012) 4441–4456. [8] F. Fang, Z. Tan, Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz–Sobolev space setting, J. Math. Anal. Appl. 389 (1) (2012) 420–428. [9] F. Cammaroto, L. Vilasi, Multiple solutions for a non-homogeneous Dirichlet problem in Orlicz–Sobolev spaces, Appl. Math. Comput. 218 (2012) 11518–11527. [10] N.T. Chung, H.Q. Toan, On a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz–Sobolev spaces, Appl. Math. Comput. 219 (2013) 7820–7829. [11] J.A. Santos, Multiplicity of solutions for a class of quasilinear equations involving critical Orlicz–Sobolev nonlinear term, Electron. J. Differential Equations 2013 (249) (2013) 1–13. [12] M.L.M. Carvalho, Jose V.A. Goncalves, E.D. da Silva, On quasilinear elliptic problems without the Ambrosetti–Rabinowitz condition, J. Math. Anal. Appl. 426 (2015) 466–483. [13] A. Azzollini, P. d’Avenia, A. Pomponio, Quasilinear elliptic equations in RN via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 (2014) 197–213. [14] Jefferson A. Santosa, Sergio H.M. Soaresb, Radial solutions of quasilinear equations in Orlicz–Sobolev type spaces, J. Math. Anal. Appl. 428 (2015) 1035–1053. [15] J. Huentutripay, R. Manásevich, Nonlinear eigenvalues for a quasilinear elliptic system in Orlicz–Sobolev spaces, J. Dynam. Differential Equations 18 (2006) 901–929. [16] F. Xia, G. Wang, Existence of solution for a class of elliptic systems, J. Hunan Agric. Univ. Nat. Sci. 33 (3) (2007) 362–366. [17] P. Chen, X.H. Tang, Existence and multiplicity results for infnitely many solutions for Kirchhoff-type problems in RN , Math. Methods Appl. Sci. 37 (2014) 1828–1837. [18] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, in: Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. [19] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS Reg. Conf. Ser. Math., vol. 65, American Mathematical Society, Providence, RI, 1986. [20] R.A. Adams, J.F. Fournier, Sobolev Spaces, Academic Press, New York, 2003. [21] M.M. Rao, Z.D. Ren, Applications of Orlicz Spaces, in: Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002. [22] N. Fukagai, M. Ito, K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on RN , Funkcial. Ekvac. 49 (2) (2006) 235–267. [23] M. García-Huidobro, V.K. Le, R. Manásevich, K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999) 207–225. [24] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, in: Universitext, Springer, New York, 2011. [25] S. Carl, V.K. Le, D. Motreanu, Nonsmooth Variational Problems and their Inequalities—Comparison Principles and Applications, Springer, New York, 2007. [26] J.F. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, 1991, (in Chinese).