Startseite Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
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Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

  • Cuiling Liu und Xingyong Zhang EMAIL logo
Veröffentlicht/Copyright: 2. März 2023
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Advances in Nonlinear Analysis
Aus der Zeitschrift Advances in Nonlinear Analysis Band 12 Heft 1

Abstract

We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ for a.e. x ∈ G , where G is a domain in R N , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ for a.e. x ∈ R N . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

Keywords: Orlicz-Sobolev spaces; mountain pass theorem; symmetric mountain pass theorem; locally super-(m 1, m 2) condition
MSC 2010: 35J50; 35J62; 35J92

1 Introduction

In this article, we are dedicated to studying the existence and multiplicity of weak solutions for the following generalized nonlinear and nonhomogeneous Kirchhoff type elliptic system in Orlicz-Sobolev spaces:

(1.1) − M 1 ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x Δ Φ 1 u + V 1 ( x ) ϕ 1 ( ∣ u ∣ ) u = F u ( x , u , v ) , x ∈ R N , − M 2 ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x Δ Φ 2 v + V 2 ( x ) ϕ 2 ( ∣ v ∣ ) v = F v ( x , u , v ) , x ∈ R N , u ∈ W 1 , Φ 1 ( R N ) , v ∈ W 1 , Φ 2 ( R N ) ,

where Δ Φ i ( u ) = div ( ϕ i ( ∣ ∇ u ∣ ) ∇ u ) , ( i = 1 , 2 ) , ϕ i : ( 0 , + ∞ ) → ( 0 , + ∞ ) are two functions which satisfy the following conditions:

  1. ϕ i ∈ C 1 [ ( 0 , + ∞ ) , R + ] , t ϕ i ( t ) → 0 as t → 0 , t ϕ i ( t ) → + ∞ as t → + ∞ ;

  2. t → t ϕ i ( t ) are strictly increasing;

  3. 1 < l i ≔ inf t > 0 t 2 ϕ i ( t ) Φ i ( t ) ≤ sup t > 0 t 2 ϕ i ( t ) Φ i ( t ) ≕ m i < min { N , l i ∗ } , where Φ i ( t ) ≔ ∫ 0 ∣ t ∣ s ϕ i ( s ) d s , t ∈ R and l i ∗ = l i N N − l i ;

  4. there exist positive constants c i , 1 and c i , 2 , i = 1 , 2 , such that

    c i , 1 ∣ t ∣ l i ≤ Φ i ( t ) ≤ c i , 2 ∣ t ∣ l i , ∀ ∣ t ∣ < 1 ;

Moreover, we introduce the following conditions on F , V i , and M i :

  1. F : R N × R × R → R is a C 1 function such that F ( x , 0 , 0 ) = 0 for all x ∈ R N and F ( x , u , v ) ≥ 0 for all ( x , u , v ) ∈ R N × R × R ;

  2. V i ∈ C ( R N , R ) and inf R N V i ( x ) > 1 , i = 1 , 2 ;

  3. there exist constants C i , 1 > 0 such that

    lim ∣ z ∣ → ∞ meas { x ∈ R N : ∣ x − z ∣ ≤ C i , 1 , V i ( x ) ≤ C i , 2 } = 0 for every C i , 2 > 0 , i = 1 , 2 ,

    where meas( ⋅ ) denotes the Lebesgue measure in R N ;

  1. M i ∈ C ( R + , R + ) and C i , 3 ≤ M i ( t ) ≤ C i , 4 , ∀ t ≥ 0 for some C i , 3 , C i , 4 > 0 , i = 1 , 2 ;

  2. M i ^ ( t ) ≔ ∫ 0 t M i ( s ) d s ≥ M i ( t ) t , i = 1 , 2 .

Let ϕ 1 = ϕ 2 ≕ ϕ , v = u , M 1 = M 2 ≕ M , V 1 = V 2 ≕ V , and F ( x , u , v ) = F ( x , v , u ) . Then the system (1.1) reduces to the following nonhomogeneous and nonlocal quasilinear elliptic equation:

(1.2) − M ∫ Ω Φ ( ∣ ∇ u ∣ ) d x Δ Φ u + V ( x ) ϕ ( ∣ u ∣ ) u = f ( x , u ) , x ∈ Ω , u ∈ W 1 , Φ ( Ω ) ,

where Ω is a domain in R N and f ( x , u ) = F u ( x , u , u ) .

In recent years, many authors are concerned with nonlocal problems like (1.2), which can been seen as a generalization of the second-order semilinear elliptic equations, p -Laplacian equations, and ( p , q ) -Laplacian equations (see [2–6,9,12,14,15,18–20,22,23,26–28,30,36,37] and the references therein).

Next, we emphasize the results in [29], which mainly inspires our works in this article. In [29], Tang et al. investigated the existence of nontrivial solutions for the following semilinear Schrödinger equation:

(1.3) − Δ u + V ( x ) u = f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) ,

where the potential V ∈ C ( R N , R ) is a sign-changing function, which satisfies the periodic or coercive conditions. If f satisfies a subcritical condition and the following locally super-quadratic condition:

( f 1 ) There exists a domain G ⊂ R N such that

lim t → ∞ ∫ 0 t f ( x , s ) d s t 2 = + ∞ , a.e. x ∈ G ,

which is weaker than the following naturally super-quadratic condition:

( f 1 ′ ) lim t → ∞ F ( x , t ) ∣ t ∣ 2 = + ∞ uniformly for x ∈ R N ,

by using the linking geometry theorem, they obtained that the problem (1.3) has a nontrivial weak solution.

There have been some contributions devoted to the study of system (1.1) with M i = 1 involving the existence and multiplicity of weak solutions. In [32], Wang et al. considered the following quasilinear elliptic system in Orlicz-Sobolev spaces:

(1.4) − div ( ϕ 1 ( ∣ ∇ u ∣ ) ∇ u ) = F u ( x , u , v ) , in Ω , − div ( ϕ 2 ( ∣ ∇ v ∣ ) ∇ v ) = F v ( x , u , v ) , in Ω , u = v = 0 , on ∂ Ω ,

where Ω is a bounded domain in R N ( N ≥ 2 ) with smooth boundary ∂ Ω . When F satisfies some appropriate conditions including ( ϕ 1 , ϕ 2 ) -superlinear and subcritical growth conditions at infinity as well as symmetric condition, by using the mountain pass theorem and the symmetric mountain pass theorem, they obtained that system (1.4) has a nontrivial weak solution and infinitely many weak solutions, respectively. Subsequently, more works were obtained for systems like (1.1) (see [31,33,34,39,40]). For example, in [31], Wang et al. considered the quasilinear elliptic system like (1.1) with M i ( t ) = 1 in R N . When the potential functions are bounded, F satisfies sub-linear growth condition, by using the least action principle, they obtained that system has at least one nontrivial weak solution. If F also satisfies a symmetric condition, by using the genus theory, they obtained that system has infinitely many weak solutions. In [39,40], we developed the Moser iteration technique, and then by using the mountain pass theorem and cut-off technique, we obtain that system like (1.1) with M i ( t ) = 1 and a parameter λ has a nontrivial weak solution ( u λ , v λ ) with ‖ ( u λ , v λ ) ‖ ∞ ≤ 2 for every λ large enough if the nonlinear term F satisfies some growth conditions only in a circle with center 0 and radius 4.

Inspired by [5,9,29,32], in this article, we shall investigate the existence and multiplicity of weak solutions for system (1.1) with locally super- ( m 1 , m 2 ) growth in R N . We assume that F satisfies the following local condition:

lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ for a.e. x ∈ G ,

where G is a domain in R N , by using the mountain pass theorem, we obtain that system (1.1) has a nontrivial weak solution. Besides, if F also satisfies a symmetric condition and the following naturally global restriction,

lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ for a.e. x ∈ R N ,

by using the symmetric mountain pass theorem, we obtain that system (1.1) has infinitely many weak solutions of high energy. We develop some results in some known references in the following sense:

  1. Our local condition extends the locally super-quadratic condition of [29] (see ( f 1 ) mentioned earlier).

  2. Different from those in [31–34,39,40], we consider the nonlocal Kirchhoff-type problems.

  3. Our conditions are weaker than the Ambrosetti-Rabinowitz ((A-R) for short) condition in [5,23].

  4. Different from that in [32], we work in the whole-space R N rather than a bounded domain Ω ⊂ R N . Especially, we introduce a new ( ϕ 1 , ϕ 2 ) -superlinear condition (see ( F 4 ) and Remark 3.4 for details), which is different from the following condition in [32] even if we restrict ( F 4 ) to the bounded domain Ω : ( f 2 ), there exists a continuous function γ : [ 0 , ∞ ) → R and it satisfies that Γ ( t ) ≔ ∫ 0 ∣ t ∣ γ ( s ) d s , t ∈ R is an N -function with

    1 < l Γ ≔ inf t > 0 t γ ( t ) Γ ( t ) ≤ sup t > 0 t γ ( t ) Γ ( t ) ≕ m Γ < + ∞ ,

    such that

    Γ F ( x , u , v ) ∣ u ∣ l 1 + ∣ v ∣ l 2 ≤ d 1 F ¯ ( x , u , v ) , x ∈ Ω , ∣ ( u , v ) ∣ ≥ r 1 ,

    where constants d 1 , r 1 > 0 and

    F ¯ ( x , u , v ) ≔ 1 m 1 F u ( x , u , v ) u + 1 m 2 F v ( x , u , v ) v − F ( x , u , v ) , ∀ ( x , u , v ) ∈ Ω × R × R ,

    and the following ( ϕ 1 , ϕ 2 ) -superlinear growth conditions hold:

    lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) Φ 1 ( u ) + Φ 2 ( v ) = + ∞ uniformly for all x ∈ Ω ,

    where Ω is a bounded domain in R N ;

  5. Similar to (IV), for the scalar equation, we also obtain a new ϕ -superlinear condition, which is different from those in [9,32] even if we restrict it to the bounded domain Ω . One can see the Remark 4.3 for details.

  6. Because of the coupling relationship of u and v and the inhomogeneous properties of Φ i , i = 1 , 2 , our proofs become more difficult and complex than those in [29]. Especially, such difficulty and complexity can be embodied in the proofs of the compactness of Cerami sequence. Moreover, because the new condition ( F 4 ) is different from ( f 2 ) and we consider the problem (1.1) on the whole space R N rather than a bounded domain Ω , our proofs on the compactness of Cerami sequence are different from those in [32].

The remainder of this article focuses on some preliminaries, the main results of this article and their proofs, and an example that illustrates our results. Finally, a remark on semi-trivial solutions of (1.1) is given.

2 Preliminaries

In this section, to deal with such problem for system (1.1), we need to briefly list some fundamental definitions and essential properties of Orlicz and Orlicz-Sobolev spaces and introduce some classical results from variational methods. For a deeper understanding of these concepts, we refer readers for more details to the books in [1,21,24,25].

Definition 2.1

[1] Let b : [ 0 , + ∞ ) → [ 0 , + ∞ ) be a right continuous, monotone increasing function with

  1. b ( 0 ) = 0 ;

  2. lim t → + ∞ b ( t ) = + ∞ ;

  3. b ( t ) > 0 whenever t > 0 .

Then the function defined on R by B ( t ) = ∫ 0 ∣ t ∣ b ( s ) d s is called as an N -function.

By the definition of N -function B , it is obvious that B ( 0 ) = 0 and B is strictly convex. We call that an N -function B satisfies a Δ 2 -condition globally (or near infinity) if

sup t > 0 B ( 2 t ) B ( t ) < + ∞ or limsup t → ∞ B ( 2 t ) B ( t ) < + ∞ ,

which implies that there exists a constant K > 0 such that B ( 2 t ) ≤ K B ( t ) for all t ≥ 0 (or t ≥ t 0 > 0 ). We also state the equivalent form that B satisfies a Δ 2 -condition globally (or near infinity) if and only if for any c ≥ 1 , there exists a constant K c > 0 such that B ( c t ) ≤ K c B ( t ) for all t ≥ 0 (or t ≥ t 0 > 0 ).

Definition 2.2

[1] For an N -function B , we define

B ˜ ( t ) = ∫ 0 ∣ t ∣ b − 1 ( s ) d s , t ∈ R ,

where b − 1 is the right inverse of the right derivative b of B . Then B ˜ is an N -function called as the complement of B .

It holds that Young’s inequality (see [1,25])

(2.1) s t ≤ B ( s ) + B ˜ ( t ) , s , t ≥ 0

and the inequality (see [16, Lemma A.2])

(2.2) B ˜ ( b ( t ) ) ≤ B ( 2 t ) , t ≥ 0 .

Now, we recall the Orlicz space L B ( Ω ) associated with B . The Orlicz space L B ( Ω ) is the vectorial space of the measurable functions u : Ω → R satisfying

∫ Ω B ( ∣ u ∣ ) d x < + ∞ ,

where Ω ⊂ R N is an open set. L B ( Ω ) is a Banach space endowed with Luxemburg norm:

‖ u ‖ B ≔ inf λ > 0 : ∫ Ω B u λ d x ≤ 1 .

The fact that B satisfies Δ 2 -condition globally implies that

(2.3) u n → u in L B ( Ω ) ⇔ ∫ Ω B ( u n − u ) d x → 0 .

Moreover, a generalized type of Hölder’s inequality (see [1,25])

∫ Ω u v d x ≤ 2 ‖ u ‖ Φ ‖ v ‖ Φ ˜ , for all u ∈ L Φ ( Ω ) and v ∈ L Φ ˜ ( Ω )

can be gained by applying Young’s inequality (2.1).

The corresponding Orlicz-Sobolev space (see [1,25]) is defined by

W 1 , B ( Ω ) ≔ u ∈ L B ( Ω ) : ∂ u ∂ x i ∈ L B ( Ω ) , i = 1 , … , N

with the norm

‖ u ‖ 1 , B ≔ ‖ u ‖ B + ‖ ∇ u ‖ B .

Consider the subspace X of W 1 , B ( R N ) ,

(2.4) X = u ∈ W 1 , B ( R N ) ∫ R N V ( x ) B ( ∣ u ∣ ) d x < ∞

with the norm

(2.5) ‖ u ‖ X = ‖ ∇ u ‖ B + ‖ u ‖ B , V ,

where

‖ u ‖ B , V = inf α > 0 ∫ R N V ( x ) B ∣ u ∣ α d x ≤ 1

and inf x ∈ R N V ( x ) > 0 . Then ( X , ‖ ⋅ ‖ ) is a separable and reflexive Banach space (see [23]).

Lemma 2.3

[1,16] If B is an N-function, then the following conditions are equivalent:

  1. (2.6) 1 ≤ l = inf t > 0 t b ( t ) B ( t ) ≤ sup t > 0 t b ( t ) B ( t ) = m < + ∞ .

  2. Let ζ 0 ( t ) = min { t l , t m } and ζ 1 ( t ) = max { t l , t m } , t ≥ 0 . B satisfies

    ζ 0 ( t ) B ( ρ ) ≤ B ( ρ t ) ≤ ζ 1 ( t ) B ( ρ ) , ∀ ρ , t ≥ 0 .

  3. B satisfies a Δ 2 -condition globally.

Lemma 2.4

[16] If B is an N-function and (2.6) holds, then B satisfies

ζ 0 ( ‖ u ‖ B ) ≤ ∫ Ω B ( u ) d x ≤ ζ 1 ( ‖ u ‖ B ) , ∀ u ∈ L B ( Ω ) .

Lemma 2.5

[16] If B is an N-function and (2.6) holds with l > 1 . Let B ˜ be the complement of B and ζ 2 ( t ) = min { t l ˜ , t m ˜ } , ζ 3 ( t ) = max { t l ˜ , t m ˜ } for t ≥ 0 , where l ˜ ≔ l l − 1 and m ˜ ≔ m m − 1 . Then B ˜ satisfies

  1. m ˜ = inf t > 0 t B ˜ ′ ( t ) B ˜ ( t ) ≤ sup t > 0 t B ˜ ′ ( t ) B ˜ ( t ) = l ˜ ;

  2. ζ 2 ( t ) B ˜ ( ρ ) ≤ B ˜ ( ρ t ) ≤ ζ 3 ( t ) B ˜ ( ρ ) , ∀ ρ , t ≥ 0 ;

  3. ζ 2 ( ‖ u ‖ B ˜ ) ≤ ∫ Ω B ˜ ( u ) d x ≤ ζ 3 ( ‖ u ‖ B ˜ ) , ∀ u ∈ L B ˜ ( Ω ) .

Lemma 2.6

[16] If B is an N-function and (2.6) holds with l , m ∈ ( 1 , N ) . Let ζ 4 ( t ) = min { t l ∗ , t m ∗ } , ζ 5 ( t ) = max { t l ∗ , t m ∗ } for t ≥ 0 , where l ∗ ≔ l N N − l and m ∗ ≔ m N N − m . Then B ∗ satisfies

  1. l ∗ = inf t > 0 t B ∗ ′ ( t ) B ∗ ( t ) ≤ sup t > 0 t B ∗ ′ ( t ) B ∗ ( t ) = m ∗ ;

  2. ζ 4 ( t ) B ∗ ( ρ ) ≤ B ∗ ( ρ t ) ≤ ζ 5 ( t ) B ∗ ( ρ ) , ∀ ρ , t ≥ 0 ;

  3. ζ 4 ( ‖ u ‖ B ∗ ) ≤ ∫ Ω B ∗ ( u ) d x ≤ ζ 5 ( ‖ u ‖ B ∗ ) , ∀ u ∈ L B ∗ ( Ω ) ,

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

    B ∗ − 1 ( t ) = ∫ 0 t B − 1 ( s ) s N + 1 N d s f o r t ≥ 0 a n d B ∗ ( t ) = B ∗ ( − t ) for t ≤ 0 .

Next, we recall some embeddings. Let Ψ be an N -function verifying Δ 2 -condition. If

(2.7) lim t → 0 ¯ Ψ ( t ) B ( t ) < + ∞ and lim ∣ t ∣ → + ∞ ¯ Ψ ( t ) B ∗ ( t ) < + ∞ ,

then we have a continuous embedding W 1 , B ( R N ) ↪ L Ψ ( R N ) . Moreover, if

(2.8) lim ∣ t ∣ → 0 Ψ ( t ) B ( t ) < + ∞ and lim ∣ t ∣ → + ∞ Ψ ( t ) B ∗ ( t ) = 0 ,

then the embedding W 1 , B ( R N ) ↪ L loc Ψ ( R N ) is compact, and we call that such Ψ satisfies the subcritical condition.

Lemma 2.7

[23] Assume that b : [ 0 , ∞ ) → [ 0 , ∞ ) ∈ C 1 and V satisfies the following conditions:

  1. the function t → b ( t ) t is increasing in ( 0 , ∞ ) ;

  2. there exist l , m ∈ ( 1 , N ) such that

    l ≤ b ( ∣ t ∣ ) t 2 B ( t ) ≤ m for a l l t ≠ 0 ,

    where l ≤ m < l ∗ and B ( t ) = ∫ 0 ∣ t ∣ b ( s ) s d s ;

  3. V ∈ C ( R N , R ) and inf x ∈ R N V ( x ) = V 0 > 0 ;

  4. for all C 0 > 0 , μ ( V − 1 ( − ∞ , C 0 ] ) < ∞ , where μ is the Lebesgue measure in R N .

Then for any N-function Ψ satisfying Δ 2 -condition and (2.8), the embedding from X into L Ψ ( R N ) is compact. Specifically, X into L B ( R N ) is compact, where X is defined by (2.4).

Remark 2.8

By Lemmas 2.3 and 2.5, assumptions ( ϕ 1 )–( ϕ 3 ) show that Φ i ( i = 1 , 2 ) and Φ ˜ i ( i = 1 , 2 ) are N -functions satisfying Δ 2 -condition globally. Thus, L Φ i ( R N ) ( i = 1 , 2 ) and W 1 , Φ i ( R N ) ( i = 1 , 2 ) are separable and reflexive Banach spaces (see [1, 25]).

By the end of this section, we recall the mountain pass theorem (see [24, Theorem 2.2]) and the symmetric mountain pass theorem (see [24, Theorem 9.12]), which will be used to prove Theorems 3.1 and 3.9 in Section 3, respectively.

We first recall that I ∈ C 1 ( E , R ) satisfies the Palais-Smale condition ((PS)-condition for short) if any (PS)-sequence { u n } ⊂ E has a convergent subsequence, where (PS)-sequence { u n } means that

I ( u n ) is bounded , ‖ I ′ ( u n ) ‖ → 0 , as n → ∞ ,

and we call that I ∈ C 1 ( E , R ) satisfies the Cerami-condition ((C)-condition for short) if any (C)-sequence { u n } ⊂ E has a convergent subsequence, where (C)-sequence { u n } means that

(2.9) I ( u n ) is bounded and ( 1 + ‖ u n ‖ ) ‖ I ′ ( u n ) ‖ → 0 , as n → ∞ .

By the discussion in [7], the (PS)-condition can be substituted with (C)-condition in the following Lemmas 2.9 and 2.10.

Lemma 2.9

[24, Theorem 2.2] (Mountain pass theorem). Let E be a real Banach space and I ∈ C 1 ( E , R ) satisfying (PS)-condition. Suppose I ( 0 ) = 0 and

  1. there are constants ρ , α > 0 such that I ∣ ∂ B ρ ≥ α , and

  2. there is an e ∈ E \ B ρ such that I ( e ) ≤ 0 .

Then, I possesses a critical value c ≥ α .

Lemma 2.10

[24, Theorem 9.12] (Symmetric mountain pass theorem). Let E be an infinite-dimensional Banach space and let I ∈ C 1 ( E , R ) be even, satisfy (PS)-condition, and I ( 0 ) = 0 . If E = V ⊕ X , where V is finite dimensional, and I satisfies

  1. there are constants ρ , α > 0 such that I ∣ ∂ B ρ ∩ X ≥ α , and

  2. for each finite dimensional subspace E ˜ ⊂ E , there is an R = R ( E ˜ ) such that I ≤ 0 on E ˜ \ B R ( E ˜ ) , where B R = { u ∈ E : ‖ u ‖ < R } .

Then I possesses an unbounded sequence of critical values.

3 Main results and proofs

Define

(3.1) W i = u ∈ W 1 , Φ i ( R N ) ∫ R N V ( x ) Φ i ( ∣ u ∣ ) d x < ∞

with the norm

‖ u ‖ i = ‖ ∇ u ‖ Φ i + ‖ u ‖ Φ i , V i , i = 1 , 2 .

Throughout this article, we work in the subspace W ≕ W 1 × W 2 of W 1 , Φ 1 ( R N ) × W 1 , Φ 2 ( R N ) with the norm

‖ ( u , v ) ‖ = ‖ u ‖ 1 + ‖ v ‖ 2 = ‖ ∇ u ‖ Φ 1 + ‖ u ‖ Φ 1 , V 1 + ‖ ∇ v ‖ Φ 2 + ‖ v ‖ Φ 2 , V 2 .

Then ( W , ‖ ⋅ ‖ ) is a separable and reflexive Banach space.

The energy functional I on W corresponding to system (1.1) is defined as follows:

(3.2) I ( u , v ) ≔ M 1 ^ ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x + M 2 ^ ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u ∣ ) d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v ∣ ) d x − ∫ R N F ( x , u , v ) d x , ( u , v ) ∈ W .

Under the assumptions ( ϕ 1 ) – ( ϕ 3 ) , ( F 1 ), ( V 0 ) , ( V 1 ) , ( M 0 ) , and ( M 1 ) , by using the standard arguments as in [17,32], we can prove that I is well defined and of class C 1 ( W , R ) with

(3.3) ⟨ I ′ ( u , v ) , ( u ˜ , v ˜ ) ⟩ = M 1 ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x ∫ R N ϕ 1 ( ∣ ∇ u ∣ ) ∇ u ∇ u ˜ d x + M 2 ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x ∫ R N ϕ 2 ( ∣ ∇ v ∣ ) ∇ v ∇ v ˜ d x + ∫ R N V 1 ( x ) ϕ 1 ( ∣ u ∣ ) u u ˜ d x + ∫ R N V 2 ( x ) ϕ 2 ( ∣ v ∣ ) v v ˜ d x − ∫ R N F u ( x , u , v ) u ˜ d x − ∫ R N F v ( x , u , v ) v ˜ d x

for all ( u ˜ , v ˜ ) ∈ W . Then the critical points of I on W are the weak solutions of system (1.1).

3.1 Existence

In this subsection, we present the following existence result and prove it by using the mountain pass theorem.

Theorem 3.1

Assume that ( ϕ 1 )–( ϕ 4 ), ( F 0 ) , ( V 0 ) , ( V 1 ) , ( M 0 ) , ( M 1 ) , and the following conditions hold:

  1. There exist two continuous functions ψ i ( i = 1 , 2 ) : [ 0 , + ∞ ) → R and a constant C 2 > 0 such that

    (3.4) ∣ F u ( x , u , v ) ∣ ≤ C 2 ( ∣ u ∣ l 1 − 1 + ψ 1 ( ∣ u ∣ ) + Ψ ˜ 1 − 1 ( Ψ 2 ( ∣ v ∣ ) ) ) , ∣ F v ( x , u , v ) ∣ ≤ C 2 ( ∣ v ∣ l 2 − 1 + Ψ ˜ 2 − 1 ( Ψ 1 ( ∣ u ∣ ) ) + ψ 2 ( ∣ v ∣ ) )

    for all ( x , u , v ) ∈ R N × R × R , where Ψ i ( t ) ≔ ∫ 0 ∣ t ∣ ψ i ( s ) d s , t ∈ R ( i = 1 , 2 ) are two N-functions satisfying

    (3.5) m i < l Ψ i ≔ inf t > 0 t ψ i ( t ) Ψ i ( t ) ≤ sup t > 0 t ψ i ( t ) Ψ i ( t ) ≕ m Ψ i < l i ∗ ,

    Ψ ˜ i denotes the complements of Ψ i ( i = 1 , 2 ) , respectively.

  2. There exists a constant C 3 ∈ [ 0 , 1 ) such that

    limsup ∣ ( u , v ) ∣ → 0 F ( x , u , v ) Φ 1 ( ∣ u ∣ ) + Φ 2 ( ∣ v ∣ ) = C 3 uniformly i n x ∈ R N .

  3. There exists a domain G ⊂ R N such that

    lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ , for a.e. x ∈ G .

  4. There exist a continuous function γ ¯ : [ 0 , ∞ ) → R + and positive constants σ i ∈ l i ( m Γ ¯ − 1 ) m Γ ¯ , min l i , l i ∗ ( l Γ ¯ − 1 ) l Γ ¯ , i = 1 , 2 , C 4 , r > 0 such that

    (3.6) Γ ¯ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≤ C 4 F ¯ ( x , u , v ) , for a l l x ∈ R N and ( u , v ) ∈ R 2 with ∣ ( u , v ) ∣ ≥ r ,

    where Γ ¯ ( t ) ≔ ∫ 0 ∣ t ∣ γ ¯ ( s ) d s , t ∈ R , is an N-function with

    (3.7) 1 < l Γ ¯ ≔ inf t > 0 t γ ¯ ( t ) Γ ¯ ( t ) ≤ sup t > 0 t γ ¯ ( t ) Γ ¯ ( t ) ≕ m Γ ¯ < + ∞

    and

    F ¯ ( x , u , v ) ≔ 1 m 1 F u ( x , u , v ) u + 1 m 2 F v ( x , u , v ) v − F ( x , u , v ) ≥ 0 , ∀ ( x , u , v ) ∈ R N × R × R .

Then system (1.1) possesses a nontrivial weak solution.

Remark 3.2

By Lemmas 2.7 and 3.1 in [8], the assumptions ( ϕ 1 )–( ϕ 4 ), ( V 0 ) , ( V 1 ) , and (3.5) imply that the following embeddings are compact:

W i ↪ L Ψ i ( R N ) , W i ↪ L Φ i ( R N ) , and W i ↪ L p i ( R N ) , i = 1 , 2 ,

where p i ∈ [ l i , l i ∗ ) . As a result, there exist some positive constants C i , 5 , C i , 6 , i = 1 , 2 , such that

(3.8) ‖ u ‖ L p i ≤ C i , 5 ‖ u ‖ i , ‖ u ‖ L Ψ i ≤ C i , 6 ‖ u ‖ i ,

where p i ∈ [ l i , l i ∗ ) . In particular, we have σ i l ˜ Γ ¯ , σ i m ˜ Γ ¯ ∈ [ l i , l i ∗ ) , where σ i , i = 1 , 2 , l ˜ Γ ¯ = l Γ ¯ l Γ ¯ − 1 and m ˜ Γ ¯ = m Γ ¯ m Γ ¯ − 1 . Hence, the following embeddings are compact:

W i ↪ L σ i l ˜ Γ ¯ ( R N ) and W i ↪ L σ i m ˜ Γ ¯ ( R N ) , i = 1 , 2 .

Remark 3.3

By Young’s inequality (2.1), (3.4), and F ( x , 0 , 0 ) = 0 , the fact

F ( x , u , v ) = ∫ 0 u F s ( x , s , v ) d s + ∫ 0 v F t ( x , 0 , t ) d t + F ( x , 0 , 0 ) , ∀ ( x , u , v ) ∈ R N × R × R

shows that there exists a constant C 5 > 0 such that

(3.9) ∣ F ( x , u , v ) ∣ ≤ C 5 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 + Ψ 1 ( ∣ u ∣ ) + Ψ 2 ( ∣ v ∣ ) ) , ∀ ( x , u , v ) ∈ R N × R × R .

Remark 3.4

If we consider the system (1.1) on a bounded domain Ω with Dirichlet boundary condition, then it is natural that we restrict those assumptions of Theorem 3.1 on the bounded domain Ω . Thus, we can claim that ( F 4 ) and ( f 2 ) are complementary. First, we claim that if l Γ ≥ m Γ ¯ , ( F 4 ), and ( F 1 ) imply that the condition ( f 2 ) holds. To be specific, choosing r 1 ≥ 2 r with r > 1 and taking the suitable values of l Γ and m Γ such that 1 < l Γ ≤ m Γ < + ∞ , max 1 , ( m Ψ 1 − l 1 ) m Γ m Ψ 1 , ( m Ψ 2 − l 2 ) m Γ m Ψ 2 < l Γ ¯ ≤ m Γ ¯ < l Γ ≤ m Γ and

m Γ < min l Γ ¯ m Γ ¯ ( m Ψ 1 − l 1 ) + l Γ ¯ l 1 m Γ ¯ ( m Ψ 1 − l 1 ) , l 1 l Γ ¯ m Γ ¯ ( m Ψ 2 − l 2 ) + l Γ ¯ l 1 l 2 l 1 m Γ ¯ ( m Ψ 2 − l 2 ) , l Γ ¯ m Γ ¯ ( m Ψ 2 − l 2 ) + l Γ ¯ l 2 m Γ ¯ ( m Ψ 2 − l 2 ) , l 2 l Γ ¯ m Γ ¯ ( m Ψ 1 − l 1 ) + l Γ ¯ l 1 l 2 l 2 m Γ ¯ ( m Ψ 1 − l 1 ) ,

we can see that ( F 4 ) also holds for ∣ ( u , v ) ∣ ≥ r 1 . Obviously, ( F 1 ) implies that

(3.10) ∣ F ( x , u , v ) ∣ ≤ d 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 + ∣ u ∣ m Ψ 1 + ∣ v ∣ m Ψ 2 ) for ∣ ( u , v ) ∣ ≥ r ,

where d 2 > 0 . Moreover, by (3.10) and Young’s inequality, we have the following inequality:

(3.11) ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ [ d 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 + ∣ u ∣ m Ψ 1 + ∣ v ∣ m Ψ 2 ) ] m Γ − l Γ ¯ ≤ C m Γ − l Γ ¯ d 2 m Γ − l Γ ¯ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ − l Γ ¯ + C l Γ ¯ C m Γ − l Γ ¯ 2 d 2 m Γ − l Γ ¯ ( ∣ u ∣ σ 1 l Γ ¯ + m Ψ 1 ( m Γ − l Γ ¯ ) + ∣ v ∣ σ 2 l Γ ¯ + m Ψ 2 ( m Γ − l Γ ¯ ) ) + C l Γ ¯ C m Γ − l Γ ¯ 2 d 2 m Γ − l Γ ¯ 1 ξ 1 ∣ u ∣ σ 1 l Γ ¯ ξ 1 + ξ 1 − 1 ξ 1 ∣ v ∣ ξ 1 m Ψ 2 ( m Γ − l Γ ¯ ) ξ 1 − 1 + ξ 2 − 1 ξ 2 ∣ u ∣ ξ 2 m Ψ 1 ( m Γ − l Γ ¯ ) ξ 2 − 1 + 1 ξ 2 ∣ v ∣ σ 2 l Γ ¯ ξ 2 ≤ d 3 ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ − l Γ ¯ + ∣ u ∣ σ 1 l Γ ¯ + m Ψ 1 ( m Γ − l Γ ¯ ) + ∣ v ∣ σ 2 l Γ ¯ + m Ψ 2 ( m Γ − l Γ ¯ ) + ∣ u ∣ ξ 2 m Ψ 1 ( m Γ − l Γ ¯ ) ξ 2 − 1 + ∣ v ∣ ξ 1 m Ψ 2 ( m Γ − l Γ ¯ ) ξ 1 − 1 + ∣ u ∣ σ 1 l Γ ¯ ξ 1 + ∣ v ∣ σ 2 l Γ ¯ ξ 2 ,

for some ξ 1 > l 2 m Γ l 2 m Γ − m Ψ 2 ( m Γ − l Γ ¯ ) , ξ 2 > l 1 m Γ l 1 m Γ − m Ψ 1 ( m Γ − l Γ ¯ ) , where C m Γ − l Γ ¯ = 2 m Γ − l Γ ¯ − 1 , if m Γ − l Γ ¯ > 1 , 1 , if m Γ − l Γ ¯ ≤ 1 ,   C l Γ ¯ = 2 l Γ ¯ − 1 , d 3 > 0 ,

σ 1 ∈ l 1 ( m Γ ¯ − 1 ) m Γ ¯ , min l 1 , l 1 m Γ − m Ψ 1 ( m Γ − l Γ ¯ ) l Γ ¯ , l 1 l 2 m Γ − l 1 m Ψ 2 ( m Γ − l Γ ¯ ) l Γ ¯ l 2 , l 1 ∗ ( l Γ ¯ − 1 ) l Γ ¯

and

σ 2 ∈ l 2 ( m Γ ¯ − 1 ) m Γ ¯ , min l 2 , l 2 m Γ − m Ψ 2 ( m Γ − l Γ ¯ ) l Γ ¯ , l 2 l 1 m Γ − l 2 m Ψ 1 ( m Γ − l Γ ¯ ) l Γ ¯ l 1 , l 2 ∗ ( l Γ ¯ − 1 ) l Γ ¯ ,

which is reasonable by our example in the last section with m Γ = 2 . Hence, in virtue of ( F 4 ), (3.10), (3.11), Young’s inequality, and Lemma 2.3, we have

(3.12) C 4 F ¯ ( x , u , v ) ≥ Γ ¯ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ Γ ¯ ( 1 ) min ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 l Γ ¯ , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 m Γ ¯ max ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 l Γ , ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 m Γ max ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 l Γ , ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 m Γ ≥ Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ ¯ ∣ F ( x , u , v ) ∣ m Γ − m Γ ¯ , if ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ≥ 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 < 1 , Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ ∣ F ( x , u , v ) ∣ m Γ − l Γ ¯ , if ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ≥ 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1 , Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ ¯ ∣ F ( x , u , v ) ∣ l Γ − m Γ ¯ , if ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 < 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 < 1 , Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ ∣ F ( x , u , v ) ∣ l Γ − l Γ ¯ , if ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 < 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1

≥ Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ ¯ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ − m Γ ¯ , if ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ≥ 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 < 1 , Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ [ d 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 + ∣ u ∣ m Ψ 1 + ∣ v ∣ m Ψ 2 ) ] m Γ − l Γ ¯ , if ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ≥ 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1 , Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ ¯ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ − m Γ ¯ , if ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 < 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 < 1 , Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ − l Γ ¯ , if ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 < 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1 ≥ d 4 Γ ¯ ( 1 ) Γ ( 1 ) Γ ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2

for ∣ ( u , v ) ∣ ≥ r 1 , where d 4 > 0 . To be specific, for ∣ u ∣ , ∣ v ∣ ≥ r , we have

(3.13) ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ ¯ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ − m Γ ¯ ≥ min { r m Γ ( l 1 − σ 1 ) , r m Γ ( l 2 − σ 2 ) } ,

(3.14) ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ ¯ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ − m Γ ¯ ≥ min { r m Γ ¯ ( l 1 − σ 1 ) , r m Γ ¯ ( l 2 − σ 2 ) } ,

(3.15) ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) l Γ − l Γ ¯ ≥ min { r l Γ ¯ ( l 1 − σ 1 ) , r l Γ ¯ ( l 2 − σ 2 ) } ,

(3.16) ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) l Γ ¯ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ − l Γ ¯ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ≤ 1 max { r l Γ ¯ ( l 1 − σ 1 ) , r l Γ ¯ ( l 2 − σ 2 ) } ,

(3.17) ∣ u ∣ σ 1 l Γ ¯ + m Ψ 1 ( m Γ − l Γ ¯ ) + ∣ v ∣ σ 2 l Γ ¯ + m Ψ 2 ( m Γ − l Γ ¯ ) ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ≤ 1 max { r l 1 m Γ − σ 1 l Γ ¯ − m Ψ 1 ( m Γ − l Γ ¯ ) , r l 2 m Γ − σ 2 l Γ ¯ − m Ψ 2 ( m Γ − l Γ ¯ ) } ,

(3.18) ∣ u ∣ ξ 2 m Ψ 1 ( m Γ − l Γ ¯ ) ξ 2 − 1 + ∣ v ∣ ξ 1 m Ψ 2 ( m Γ − l Γ ¯ ) ξ 1 − 1 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ≤ 1 max r l 1 m Γ − ξ 2 m Ψ 1 ( m Γ − l Γ ¯ ) ξ 2 − 1 , r l 2 m Γ − ξ 1 m Ψ 2 ( m Γ − l Γ ¯ ) ξ 1 − 1 ,

and

(3.19) ∣ u ∣ σ 1 l Γ ¯ ξ 1 + ∣ v ∣ σ 2 l Γ ¯ ξ 2 ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ≤ 1 max { r l 1 m Γ − σ 1 l Γ ¯ ξ 1 , r l 2 m Γ − σ 2 l Γ ¯ ξ 2 } .

For ∣ u ∣ ≥ r , 0 ≤ ∣ v ∣ < r , and ∣ v ∣ ≥ r , 0 ≤ ∣ u ∣ < r , the inequalities (3.13)–(3.19) also hold with different values on the right-hand side of the inequalities. Hence, (3.12) holds for ∣ ( u , v ) ∣ ≥ r 1 , and then it is easy to see that ( f 2 ) holds.

Next, we claim that ( f 2 ) and the following assumption (A) imply that condition ( F 4 ) holds if we take m Γ ≥ l Γ ≥ m i m Γ ¯ m i − l i m Γ ¯ .

(A) F ¯ ( x , u , v ) ≥ C ( ∣ u ∣ ϱ 1 + ∣ v ∣ ϱ 2 ) for all ∣ ( u , v ) ∣ ≥ r 1 , where ϱ i ∈ ( 0 , m i ] with m i > l i m Γ ¯ .

In fact, choosing r = 2 r 1 with r 1 > 1 , we can see that ( f 2 ) also holds for ( u , v ) ∈ R 2 with ∣ ( u , v ) ∣ ≥ r and ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1 . Next, there are four possible cases, that is, ①   ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 < 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1 , ϱ i ≥ l i m Γ m Γ ¯ m Γ − m Γ ¯ ; ②   ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 < 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 < 1 , ϱ i ≥ l i m Γ l Γ ¯ m Γ − l Γ ¯ ; ③   ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ≥ 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1 , ϱ i ≥ l i l Γ m Γ ¯ l Γ − m Γ ¯ and ④   ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ≥ 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 < 1 , ϱ i ≥ l i l Γ l Γ ¯ l Γ − l Γ ¯ . Without loss of generality, we only focus on the proof of the first case.

① Since ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 < 1 , ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1 and ϱ i ≥ l i m Γ m Γ ¯ m Γ − m Γ ¯ , by ( f 2 ) , (A), and Lemma 2.3, we have

d 1 F ¯ ( x , u , v ) = d 1 ( F ¯ ( x , u , v ) ) m Γ ¯ m Γ ( F ¯ ( x , u , v ) ) m Γ − m Γ ¯ m Γ ≥ d 1 m Γ − m Γ ¯ m Γ Γ F ( x , u , v ) ∣ u ∣ l 1 + ∣ v ∣ l 2 m Γ ¯ m Γ ( F ¯ ( x , u , v ) ) m Γ − m Γ ¯ m Γ ≥ d 1 m Γ − m Γ ¯ m Γ Γ ( 1 ) ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 m Γ ¯ ( ∣ u ∣ ϱ 1 + ∣ v ∣ ϱ 2 ) m Γ − m Γ ¯ m Γ = d 1 m Γ − m Γ ¯ m Γ Γ ( 1 ) ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 m Γ ¯ ( ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ) m Γ ¯ ( ∣ u ∣ ϱ 1 + ∣ v ∣ ϱ 2 ) m Γ − m Γ ¯ m Γ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ¯ ≥ d 1 m Γ − m Γ ¯ m Γ Γ ( 1 ) Γ ¯ ( 1 ) Γ ¯ ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ( ∣ u ∣ ϱ 1 + ∣ v ∣ ϱ 2 ) m Γ − m Γ ¯ m Γ ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) m Γ ¯ ≥ Γ ¯ ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 d 1 m Γ − m Γ ¯ m Γ Γ ( 1 ) C m Γ ¯ Γ ¯ ( 1 ) ∣ u ∣ ϱ 1 ( m Γ − m Γ ¯ ) m Γ + ∣ v ∣ ϱ 2 ( m Γ − m Γ ¯ ) m Γ ∣ u ∣ l 1 m Γ ¯ + ∣ v ∣ l 2 m Γ ¯ ≥ d 5 Γ ¯ ∣ F ( x , u , v ) ∣ ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ,

where d 5 > 0 and C m Γ ¯ = 2 m Γ ¯ − 1 .

Hence, ( F 4 ) and ( f 2 ) are complementary. We will give an example, which satisfies ( F 4 ) but not satisfies ( f 2 ) in Section 5.

Lemma 3.5

Suppose that ( ϕ 1 ) – ( ϕ 3 ) , ( M 0 ) , ( V 0 ) , and ( F 1 ) hold. Then there are constants ρ , α > 0 such that I ∣ ∂ B ρ ≥ α .

Proof

By ( ϕ 1 ) – ( ϕ 3 ) and ( V 0 ) , we have

(3.20) min { ‖ u ‖ Φ 1 , V 1 l 1 , ‖ u ‖ Φ 1 , V 1 m 1 } ≤ ∫ R N V 1 ( x ) Φ 1 ( ∣ u ∣ ) d x ≤ max { ‖ u ‖ Φ 1 , V 1 l 1 , ‖ u ‖ Φ 1 , V 1 m 1 }

and

(3.21) min { ‖ v ‖ Φ 2 , V 2 l 2 , ‖ v ‖ Φ 2 , V 2 m 2 } ≤ ∫ R N V 2 ( x ) Φ 2 ( ∣ v ∣ ) d x ≤ max { ‖ v ‖ Φ 2 , V 2 l 2 , ‖ v ‖ Φ 2 , V 2 m 2 }

for all u ∈ W 1 and v ∈ W 2 (the details is in Lemma 2.1 of [23]). Moreover, by (3.9) and ( F 2 ) , there exist constants C 6 ∈ ( 0 , 1 ) and C 7 > 0 such that

(3.22) ∣ F ( x , u , v ) ∣ ≤ ( 1 − C 6 ) ( Φ 1 ( ∣ u ∣ ) + Φ 2 ( ∣ v ∣ ) ) + C 7 ( Ψ 1 ( ∣ u ∣ ) + Ψ 2 ( ∣ v ∣ ) ) , ∀ ( x , u , v ) ∈ R N × R × R .

Choosing ρ > 0 such that ρ = ‖ ( u , v ) ‖ = ‖ u ‖ 1 + ‖ v ‖ 2 < min 1 max { C 1 , 6 , C 2 , 6 } , 1 . Then ‖ u ‖ Ψ 1 ≤ C 1 , 6 ‖ u ‖ 1 < 1 and ‖ v ‖ Ψ 2 ≤ C 2 , 6 ‖ v ‖ 2 < 1 . By (3.20), (3.21), (3.22), ( M 0 ) , Lemma 2.4, and Remark 3.2, we obtain

I ( u , v ) ≥ C 1 , 3 ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u ∣ ) d x − ( 1 − C 6 ) ∫ R N V 1 ( x ) Φ 1 ( u ) d x − C 7 ∫ R N Ψ 1 ( u ) d x + C 2 , 3 ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v ∣ ) d x − ( 1 − C 6 ) ∫ R N V 2 ( x ) Φ 2 ( v ) d x − C 7 ∫ R N Ψ 2 ( v ) d x ≥ C 1 , 3 min { ‖ ∇ u ‖ Φ 1 l 1 , ‖ ∇ u ‖ Φ 1 m 1 } + C 6 min { ‖ u ‖ Φ 1 , V 1 l 1 , ‖ u ‖ Φ 1 , V 1 m 1 } − C 7 max { ‖ u ‖ Ψ 1 l Ψ 1 , ‖ u ‖ Ψ 1 m Ψ 1 } + C 2 , 3 min { ‖ ∇ v ‖ Φ 2 l 2 , ‖ ∇ v ‖ Φ 2 m 2 } + C 6 min { ‖ v ‖ Φ 2 , V 2 l 2 , ‖ v ‖ Φ 2 , V 2 m 2 } − C 7 max { ‖ v ‖ Ψ 2 l Ψ 2 , ‖ v ‖ Ψ 2 m Ψ 2 } ≥ C 1 , 3 ‖ ∇ u ‖ Φ 1 m 1 + C 6 ‖ u ‖ Φ 1 , V 1 m 1 − C 7 C 1 , 6 ‖ u ‖ 1 l Ψ 1 + C 2 , 3 ‖ ∇ v ‖ Φ 2 m 2 + C 6 ‖ v ‖ Φ 2 , V 2 m 2 − C 7 C 2 , 6 ‖ v ‖ 2 l Ψ 2 ≥ ‖ u ‖ 1 m 1 min { C 1 , 3 , C 6 } 2 m 1 − 1 − C 7 C 1 , 6 ‖ u ‖ 1 l Ψ 1 − m 1 + ‖ v ‖ 2 m 2 min { C 2 , 3 , C 6 } 2 m 2 − 1 − C 7 C 2 , 6 ‖ v ‖ 2 l Ψ 2 − m 2 .

Since 1 < m i < l Ψ i , we can choose positive constants ρ and α small enough such that I ( u , v ) ≥ α for all ( u , v ) ∈ W with ‖ ( u , v ) ‖ = ρ .□

Lemma 3.6

Suppose that ( ϕ 1 ) – ( ϕ 3 ) , ( M 0 ) , ( V 0 ) , and ( F 3 ) hold. Then there is a point ( u , v ) ∈ W \ B ρ such that I ( u , v ) ≤ 0 .

Proof

By ( F 3 ) and the continuity of F , there exist two constants C 8 > 0 and C 9 > 0 such that

(3.23) F ( x , u , v ) ≥ C 8 ( ∣ u ∣ m 1 + ∣ v ∣ m 2 ) − C 9 , ∀ ( u , v ) ∈ R × R and a.e. x ∈ G .

Choose u 0 ∈ C 0 ∞ ( R N ) ⧹ { 0 } with 0 < u 0 ( x ) ≤ 1 and supp ( u 0 ) ⊂ ⊂ G . Obviously, ( t u 0 , 0 ) ∈ W for all t ∈ R . By ( M 0 ) , (2) in Lemma 2.3, (3.20), (3.21), and (3.23), when t > 1 , we have

I ( t u 0 , 0 ) = M 1 ^ ∫ R N Φ 1 ( t ∣ ∇ u 0 ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( t ∣ u 0 ∣ ) d x − ∫ G F ( x , t u 0 , 0 ) d x ≤ C 1 , 4 t m 1 ∫ R N Φ 1 ( ∣ ∇ u 0 ∣ ) d x + t m 1 ∫ R N V 1 ( x ) Φ 1 ( ∣ u 0 ∣ ) d x − C 8 ∫ G t m 1 ∣ u 0 ∣ m 1 d x + C 9 ∣ supp u 0 ∣ ≤ t m 1 ( C 1 , 4 ‖ ∇ u 0 ‖ Φ 1 l 1 + C 1 , 4 ‖ ∇ u 0 ‖ Φ 1 m 1 + ‖ u 0 ‖ Φ 1 , V 1 l 1 + ‖ u 0 ‖ Φ 1 , V 1 m 1 − C 8 ‖ u 0 ‖ L m 1 ( G ) m 1 ) + C 9 ∣ supp u 0 ∣ .

If we choose

C 8 > C 1 , 4 ‖ ∇ u 0 ‖ Φ 1 l 1 + C 1 , 4 ‖ ∇ u 0 ‖ Φ 1 m 1 + ‖ u 0 ‖ Φ 1 , V 1 l 1 + ‖ u 0 ‖ Φ 1 , V 1 m 1 ‖ u 0 ‖ L m 1 ( G ) m 1 ,

then there exists t large enough such that I ( t u 0 , 0 ) ≤ 0 and ‖ ( t u 0 , 0 ) ‖ > ρ .□

Lemma 3.7

Suppose that ( ϕ 1 )–( ϕ 4 ), ( V 0 ) , ( V 1 ) , ( M 1 ) , ( F 1 ), ( F 3 ), and ( F 4 ) hold. Then (C)-sequence in W is bounded.

Proof

Let { ( u n , v n ) } be a ( C ) -sequence of I in W . Then, for n large enough, by ( ϕ 3 ) , ( M 1 ) , and (2.9), there exists a c > 0 such that

(3.24) c + 1 ≥ I ( u n , v n ) − I ′ ( u n , v n ) , 1 m 1 u n , 1 m 2 v n = M 1 ^ ∫ R N Φ 1 ( ∣ ∇ u n ∣ ) d x − 1 m 1 M 1 ∫ R N Φ 1 ( ∣ ∇ u n ∣ ) d x ∫ R N ϕ 1 ( ∣ ∇ u n ∣ ) ∣ ∇ u n ∣ 2 d x + M 2 ^ ∫ R N Φ 2 ( ∣ ∇ v n ∣ ) d x − 1 m 2 M 2 ∫ R N Φ 2 ( ∣ ∇ v n ∣ ) d x ∫ R N ϕ 2 ( ∣ ∇ v n ∣ ) ∣ ∇ v n ∣ 2 d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u n ∣ ) − 1 m 1 V 1 ( x ) ϕ 1 ( ∣ u n ∣ ) ∣ u n ∣ 2 d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v n ∣ ) − 1 m 2 V 2 ( x ) ϕ 2 ( ∣ v n ∣ ) ∣ v n ∣ 2 d x + ∫ R N 1 m 1 F u ( x , u n , v n ) u n + 1 m 2 F v ( x , u n , v n ) v n − F ( x , u n , v n ) d x ≥ ∫ R N F ¯ ( x , u n , v n ) d x .

Arguing by contradiction, we assume that there exists a subsequence of { ( u n , v n ) } , still denoted by { ( u n , v n ) } , such that ‖ ( u n , v n ) ‖ = ‖ u n ‖ 1 + ‖ v n ‖ 2 → + ∞ . Then we discuss this problem in two situations.

Case 1. Suppose that ‖ u n ‖ 1 → + ∞ and ‖ v n ‖ 2 → + ∞ . Let u ¯ n = u n ‖ u n ‖ 1 and v ¯ n = v n ‖ v n ‖ 2 . Then { ( u ¯ n , v ¯ n ) } is bounded in W . Passing to a subsequence { ( u ¯ n , v ¯ n ) } , by Remark 3.2, there exists a point ( u ¯ , v ¯ ) ∈ W such that

  1. u ¯ n ⇀ u ¯ in W 1 , u ¯ n → u ¯ in L l 1 ( R N ) , in L σ 1 l Γ ¯ ˜ ( R N ) and in L σ 1 m Γ ¯ ˜ ( R N ) , u ¯ n ( x ) → u ¯ ( x ) a.e. in R N ;

  2. v ¯ n ⇀ v ¯ in W 2 ,   v ¯ n → v ¯ in L l 2 ( R N ) , in L σ 2 l Γ ¯ ˜ ( R N ) and in L σ 2 m Γ ¯ ˜ ( R N ) , v ¯ n ( x ) → v ¯ ( x ) a.e. in R N .

To obtain the contradiction, we will first assume that both [ u ¯ ≠ 0 ] ≔ { x ∈ R N : u ( x ) ¯ ≠ 0 } and [ v ¯ ≠ 0 ] ≔ { x ∈ R N : v ( x ) ¯ ≠ 0 } have zero Lebesgue measure, that is, u ¯ = 0 a.e. in R N and v ¯ = 0 a.e. in R N . By Lemma 2.4 and the inequality (66) in [38], we have

(3.25) min { C 1 , 3 , 1 } 2 m 1 − 1 min { ‖ u n ‖ 1 l 1 , ‖ u n ‖ 1 m 1 } + min { C 2 , 3 , 1 } 2 m 2 − 1 min { ‖ v n ‖ 2 l 2 , ‖ v n ‖ 2 m 2 } − min { C 1 , 3 , 1 } − min { C 2 , 3 , 1 } ≤ C 1 , 3 min { ‖ ∇ u n ‖ Φ 1 l 1 , ‖ ∇ u n ‖ Φ 1 m 1 } + C 2 , 3 min { ‖ ∇ v n ‖ Φ 2 l 2 , ‖ ∇ v n ‖ Φ 2 m 2 } + min { ‖ u n ‖ Φ 1 , V 1 l 1 , ‖ u n ‖ Φ 1 , V 1 m 1 } + min { ‖ v n ‖ Φ 2 , V 2 l 2 , ‖ v n ‖ Φ 2 , V 2 m 2 } ≤ C 1 , 3 ∫ R N Φ 1 ( ∣ ∇ u n ∣ ) d x + C 2 , 3 ∫ R N Φ 2 ( ∣ ∇ v n ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u n ∣ ) d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v n ∣ ) d x ≤ M ^ 1 ∫ R N Φ 1 ( ∣ ∇ u n ∣ ) d x + M ^ 2 ∫ R N Φ 2 ( ∣ ∇ v n ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u n ∣ ) d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v n ∣ ) d x = I ( u n , v n ) + ∫ R N F ( x , u n , v n ) d x .

When n is large enough, we have

(3.26) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 ≤ D 1 I ( u n , v n ) + D 1 ∫ R N F ( x , u n , v n ) d x + D 2 ,

where D 1 = 1 min min { C 1 , 3 , 1 } 2 m 1 − 1 , min { C 2 , 3 , 1 } 2 m 2 − 1 , and D 2 = min { C 1 , 3 , 1 } + min { C 2 , 3 , 1 } min min { C 1 , 3 , 1 } 2 m 1 − 1 , min { C 2 , 3 , 1 } 2 m 2 − 1 . Then

(3.27) 1 ≤ D 1 I ( u n , v n ) + D 2 ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 + ∫ ∣ ( u n , v n ) ∣ ≤ R + ∫ ∣ ( u n , v n ) ∣ > R D 1 F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x = o n ( 1 ) + ∫ ∣ ( u n , v n ) ∣ ≤ R D 1 F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x + ∫ ∣ ( u n , v n ) ∣ > R D 1 F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x ,

where R is a positive constant with R > r . By ( F 2 ), there exists a constant 0 < δ < 1 such that

(3.28) ∣ F ( x , u , v ) ∣ Φ 1 ( ∣ u ∣ ) + Φ 2 ( ∣ v ∣ ) < C 3 + 1 , ∀ x ∈ R N , 0 < ∣ ( u , v ) ∣ ≤ δ .

By ( ϕ 4 ) and (3.28), we have

(3.29) ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 = ∣ F ( x , u , v ) ∣ Φ 1 ( ∣ u ∣ ) + Φ 2 ( ∣ v ∣ ) ⋅ Φ 1 ( ∣ u ∣ ) + Φ 2 ( ∣ v ∣ ) ∣ u ∣ l 1 + ∣ v ∣ l 2 < ( C 3 + 1 ) max { c 12 , c 22 } , ∀ x ∈ R N , 0 < ∣ ( u , v ) ∣ ≤ δ .

By the fact that F and Φ i are continuous, there exist two constants C ¯ 10 > 0 and R > 0 such that

(3.30) ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ≤ max δ ≤ ∣ ( u , v ) ∣ ≤ R ∣ F ( x , u , v ) ∣ min δ ≤ ∣ ( u , v ) ∣ ≤ R ∣ u ∣ l 1 + ∣ v ∣ l 2 ≕ C ¯ 10 , ∀ x ∈ R N , δ ≤ ∣ ( u , v ) ∣ ≤ R ,

combining with (3.29), which implies that there exists a positive constant C 10 such that

(3.31) ∣ F ( x , u , v ) ∣ ∣ u ∣ l 1 + ∣ v ∣ l 2 ≤ C 10 , ∀ x ∈ R N , 0 < ∣ ( u , v ) ∣ ≤ R .

Set

B ¯ n , R = { x ∈ R N ∣ ∣ ( u n ( x ) , v n ( x ) ) ∣ ≤ R } , Ω 1 n = { x ∈ B ¯ n , R ∣ u n ( x ) = 0 and v n ( x ) = 0 } , Ω 2 n = { x ∈ B ¯ n , R ∣ u n ( x ) ≠ 0 and v n ( x ) = 0 } , Ω 3 n = { x ∈ B ¯ n , R ∣ u n ( x ) = 0 and v n ( x ) ≠ 0 } , Ω 4 n = { x ∈ B ¯ n , R ∣ u n ( x ) ≠ 0 and v n ( x ) ≠ 0 } .

Then by ( F 0 ) ,

(3.32) ∫ Ω 1 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x = 0 .

Note that v n ( x ) = 0 on Ω 2 n . We have

(3.33) ∫ Ω 2 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x ≤ ∫ Ω 2 n F ( x , u n , 0 ) ‖ u n ‖ 1 l 1 d x = ∫ Ω 2 n F ( x , u n , 0 ) ∣ u n ∣ l 1 ⋅ ∣ u ¯ n ∣ l 1 d x = ∫ Ω 2 n F ( x , u n , v n ( x ) ) ∣ u n ∣ l 1 + ∣ v n ∣ l 1 ⋅ ∣ u ¯ n ∣ l 1 d x ≤ C 10 ∫ R N ∣ u ¯ n ∣ l 1 d x → 0 , as n → ∞ .

Similarly, we also have

(3.34) ∫ Ω 3 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x → 0 , as n → ∞ .

Moreover,

(3.35) ∫ Ω 4 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x = ∫ Ω 4 n F ( x , u n , v n ) ∣ u n ∣ l 1 ∣ u ¯ n ∣ l 1 + ∣ v n ∣ l 2 ∣ v ¯ n ∣ l 2 d x ≤ ∫ Ω 4 n F ( x , u n , v n ) ( ∣ u n ∣ l 1 + ∣ v n ∣ l 2 ) min 1 ∣ u ¯ n ∣ l 1 , 1 ∣ v ¯ n ∣ l 2 d x ≤ ∫ Ω 4 n F ( x , u n , v n ) max { ∣ u ¯ n ∣ l 1 , ∣ v ¯ n ∣ l 2 } ∣ u n ∣ l 1 + ∣ v n ∣ l 2 d x ≤ ∫ Ω 4 n F ( x , u n , v n ) ∣ u n ∣ l 1 + ∣ v n ∣ l 2 ( ∣ u ¯ n ∣ l 1 + ∣ v ¯ n ∣ l 2 ) d x ≤ C 10 ∫ R N ( ∣ u ¯ n ∣ l 1 + ∣ v ¯ n ∣ l 2 ) d x → 0 , as n → ∞ .

Then by (3.32)–(3.35), we have

(3.36) ∫ B ¯ n , R F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x = ∑ i = 1 4 ∫ Ω i n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x = o ( 1 ) .

Next, we set

Ω 2 n ′ = { x ∈ R N / B ¯ n , R ∣ u n ( x ) ≠ 0 and v n ( x ) = 0 } , Ω 3 n ′ = { x ∈ R N / B ¯ n , R ∣ u n ( x ) = 0 and v n ( x ) ≠ 0 } , Ω 4 n ′ = { x ∈ R N / B ¯ n , R ∣ u n ( x ) ≠ 0 and v n ( x ) ≠ 0 } .

Then for large n , we have

(3.37) ∫ Ω 4 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x ≤ ∫ Ω 4 n ′ ∣ F ( x , u n , v n ) ∣ ‖ u n ‖ 1 σ 1 + ‖ v n ‖ 2 σ 2 d x ≤ ∫ Ω 4 n ′ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ( ∣ u ¯ n ∣ σ 1 + ∣ v ¯ n ∣ σ 2 ) d x .

If ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ≥ 1 , then by Hölder’s inequality, (3.24) and Remark 3.2, we have

(3.38) ∫ Ω 4 n ′ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ( ∣ u ¯ n ∣ σ 1 + ∣ v ¯ n ∣ σ 2 ) d x ≤ ∫ Ω 4 n ′ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 l Γ ¯ d x 1 l Γ ¯ ∫ Ω 4 n ′ ( ∣ u ¯ n ∣ σ 1 + ∣ v ¯ n ∣ σ 2 ) l ˜ Γ ¯ d x 1 l ˜ Γ ¯ ≤ 2 l ˜ Γ ¯ − 1 l ˜ Γ ¯ 1 Γ ¯ ( 1 ) ∫ Ω 4 n ′ Γ ¯ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 d x 1 l Γ ¯ ( ‖ u ¯ n ‖ L σ 1 l ˜ Γ ¯ σ 1 + ‖ v ¯ n ‖ L σ 2 l ˜ Γ ¯ σ 2 ) ≤ 2 l ˜ Γ ¯ − 1 l ˜ Γ ¯ C 4 Γ ¯ ( 1 ) ∫ R N F ¯ ( x , u n , v n ) d x 1 l Γ ¯ ( ‖ u ¯ n ‖ L σ 1 l ˜ Γ ¯ σ 1 + ‖ v ¯ n ‖ L σ 2 l ˜ Γ ¯ σ 2 ) ≤ 2 l ˜ Γ ¯ − 1 l ˜ Γ ¯ C 4 ( c + 1 ) Γ ¯ ( 1 ) 1 l Γ ¯ ( ‖ u ¯ n ‖ L σ 1 l ˜ Γ ¯ σ 1 + ‖ v ¯ n ‖ L σ 2 l ˜ Γ ¯ σ 2 ) → 0 , as n → ∞ .

Similarly, if ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 < 1 , then, by Hölder’s inequality, (3.24), and Remark 3.2, we obtain that

(3.39) ∫ Ω 4 n ′ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ( ∣ u ¯ n ∣ σ 1 + ∣ v ¯ n ∣ σ 2 ) d x ≤ 2 m ˜ Γ ¯ − 1 m ˜ Γ ¯ C 4 Γ ¯ ( 1 ) ∫ R N F ¯ ( x , u n , v n ) d x 1 m Γ ¯ ( ‖ u ¯ n ‖ L σ 1 m ˜ Γ ¯ σ 1 + ‖ v ¯ n ‖ L σ 2 m ˜ Γ ¯ σ 2 ) ≤ 2 m ˜ Γ ¯ − 1 m ˜ Γ ¯ C 4 ( c + 1 ) Γ ¯ ( 1 ) 1 m Γ ¯ ( ‖ u ¯ n ‖ L σ 1 m ˜ Γ ¯ σ 1 + ‖ v ¯ n ‖ L σ 2 m ˜ Γ ¯ σ 2 ) → 0 , as n → ∞ .

Hence, (3.37)–(3.39) imply that

(3.40) ∫ Ω 4 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x → 0 , as n → ∞ .

Note that v n ( x ) = 0 on Ω 2 n ′ . Then we have

(3.41) ∫ Ω 2 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x ≤ ∫ Ω 2 n ′ ∣ F ( x , u n , 0 ) ∣ ‖ u n ‖ 1 σ 1 + ‖ v n ‖ 2 σ 2 d x ≤ ∫ Ω 2 n ′ ∣ F ( x , u n , 0 ) ∣ ‖ u n ‖ 1 σ 1 d x = ∫ Ω 2 n ′ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ⋅ ∣ u ¯ n ∣ σ 1 d x .

Similar to the arguments of (3.38) and (3.39), we can also obtain

∫ Ω 2 n ′ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ⋅ ∣ u ¯ n ∣ σ 1 d x → 0 , as n → ∞ ,

which shows that

(3.42) ∫ Ω 2 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x → 0 , as n → ∞ .

Similarly, we also obtain

(3.43) ∫ Ω 3 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x → 0 , as n → ∞ .

Hence, (3.40), (3.42), and (3.43) imply that

(3.44) ∫ R N / B ¯ n , R F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + ‖ v n ‖ 2 l 2 d x = o ( 1 ) .

By combining (3.36) and (3.44) with (3.27), we obtain a contradiction.

Next, we assume that [ u ¯ ≠ 0 ] or [ v ¯ ≠ 0 ] has nonzero Lebesgue measure. It is clear that

∣ u n ∣ = ∣ u ¯ n ∣ ‖ u n ‖ 1 → + ∞ for all [ u ¯ ≠ 0 ]

or

∣ v n ∣ = ∣ v ¯ n ∣ ‖ v n ‖ 2 → + ∞ for all [ v ¯ ≠ 0 ] .

Moreover, by (2) in Lemma 2.3, F ( x , u , v ) ≥ 0 and the assumption ( F 3 ) show that

(3.45) lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 = lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 , if ∣ u ∣ ≥ 1 , ∣ v ∣ ≥ 1 , ∣ ( u , v ) ∣ → + ∞ , lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) 1 + ∣ v ∣ σ 2 , if 0 ≤ ∣ u ∣ < 1 , ∣ v ∣ → + ∞ , lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) 1 + ∣ u ∣ σ 1 , if 0 ≤ ∣ v ∣ < 1 , ∣ u ∣ → + ∞

≥ lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 , if ∣ u ∣ ≥ 1 , ∣ v ∣ ≥ 1 , lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ v ∣ m 2 + ∣ v ∣ m 2 ≥ 1 2 lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 , if 0 ≤ ∣ u ∣ < 1 , ∣ v ∣ > 1 , lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ u ∣ m 1 ≥ 1 2 lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 , if 0 ≤ ∣ v ∣ < 1 , ∣ u ∣ > 1 = + ∞ for a.e. x ∈ G .

Hence, by combining with ( F 4 ), it is easy to see that

(3.46) lim ∣ ( u , v ) ∣ → + ∞ F ¯ ( x , u , v ) ≥ 1 C 4 lim ∣ ( u , v ) ∣ → + ∞ Γ ¯ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 ≥ 1 C 4 lim ∣ ( u , v ) ∣ → + ∞ Γ ¯ ( 1 ) min F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 l Γ ¯ , F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 m Γ ¯ ≥ 1 C 4 Γ ¯ ( 1 ) min lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 l Γ ¯ , lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ σ 1 + ∣ v ∣ σ 2 m Γ ¯ = + ∞ for a.e. x ∈ G .

Then, by (3.24), Fatou lemma and the aforementioned formula, we have

c + 1 ≥ lim n → + ∞ ∫ R N F ¯ ( x , u n , v n ) d x ≥ lim n → + ∞ ∫ G F ¯ ( x , u n , v n ) d x ≥ ∫ G lim n → + ∞ F ¯ ( x , u n , v n ) d x = + ∞ ,

which is a contradiction. So, both ‖ u n ‖ 1 → + ∞ and ‖ v n ‖ 2 → + ∞ do not hold.

Case 2. Suppose that ‖ u n ‖ 1 ≤ D 3 or ‖ v n ‖ 2 ≤ D 3 for some D 3 > 0 and all n ∈ N . Without loss of generality, we assume that ‖ u n ‖ 1 → + ∞ and ‖ v n ‖ 2 ≤ D 3 for some D 3 > 0 and all n ∈ N . Let u ¯ n = u n ‖ u n ‖ 1 and v ¯ n = v n ‖ u n ‖ 1 . Then ‖ u ¯ n ‖ 1 = 1 and ‖ v ¯ n ‖ 2 → 0 . Passing to a subsequences { ( u ¯ n , v ¯ n ) } , by Remark 3.2, there exist u ¯ ∈ W 1 and v ∈ W 2 such that

  1. u ¯ n ⇀ u ¯ in W 1 , u ¯ n → u ¯ in L l 1 ( R N ) , in L σ 1 l Γ ¯ ˜ ( R N ) and in L σ 1 m Γ ¯ ˜ ( R N ) , u ¯ n ( x ) → u ¯ ( x ) a.e. in R N ;

  2. v ¯ n → 0 in W 2 , v ¯ n → 0 in L l 2 ( R N ) , in L σ 2 l Γ ¯ ˜ ( R N ) and in L σ 2 m Γ ¯ ˜ ( R N ) , v ¯ n ( x ) → 0 a.e. in R N ;

  3. v n ⇀ v in W 2 , v n → v in L l 2 ( R N ) , in L σ 2 l Γ ¯ ˜ ( R N ) and in L σ 2 m Γ ¯ ˜ ( R N ) , v n ( x ) → v ( x ) a.e. in R N .

First, we assume that [ u ¯ ≠ 0 ] has nonzero Lebesgue measure. We can see that

∣ u n ∣ = ∣ u ¯ n ∣ ‖ u n ‖ 1 → + ∞ for all [ u ¯ ≠ 0 ] .

Then, being analogue to Case 1, we obtain a contradiction by

c + 1 ≥ ∫ R N F ¯ ( x , u n , v n ) d x → + ∞ .

Next, we suppose that [ u ¯ ≠ 0 ] has zero Lebesgue measure, that is, u ¯ = 0 a.e. in R N . By (3.24), we can see that

(3.47) ∫ R N F ¯ ( x , u n , v n ) d x ≤ c + 1 .

Then when n large enough, we can choose two positive constants D 4 and D 5 such that (3.25) is changed into

(3.48) ‖ u n ‖ 1 l 1 ≤ D 4 I ( u n , v n ) + D 4 ∫ R N F ( x , u n , v n ) d x + D 5 .

Similar to the arguments of (3.32)–(3.34), for any given

D 6 > max 5 C 10 C 2 , 5 l 2 D 3 l 2 , 5 C 4 ( c + 1 ) Γ ¯ ( 1 ) 1 l Γ ¯ C 2 , 5 σ 2 D 3 σ 2 , 5 C 4 ( c + 1 ) Γ ¯ ( 1 ) 1 m Γ ¯ C 2 , 5 σ 2 D 3 σ 2 ,

we obtain that

(3.49) ∫ Ω 1 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ ∫ Ω 1 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 d x = 0 ,

(3.50) ∫ Ω 2 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ ∫ Ω 2 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 d x → 0 , as n → ∞

and

(3.51) ∫ Ω 3 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ ∫ Ω 3 n F ( x , u n , v n ) ∣ v n ∣ l 2 ⋅ ∣ v n ∣ l 2 D 6 d x = ∫ Ω 3 n F ( x , u n , v n ) ∣ u n ∣ l 1 + ∣ v n ∣ l 2 ⋅ ∣ v n ∣ l 2 D 6 d x ≤ C 10 D 6 ∫ R N ∣ v n ∣ l 2 d x ≤ C 10 C 2 , 5 l 2 D 3 l 2 D 6 < 1 5 , as n → ∞ .

Moreover, we have

(3.52) ∫ Ω 4 n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ ∫ Ω 4 n F ( x , u n , v n ) ∣ u n ∣ l 1 ∣ u ¯ n ∣ l 1 + D 6 ∣ v n ∣ l 2 ∣ v n ∣ l 2 d x ≤ ∫ Ω 4 n F ( x , u n , v n ) ( ∣ u n ∣ l 1 + ∣ v n ∣ l 2 ) min 1 ∣ u ¯ n ∣ l 1 , D 6 ∣ v n ∣ l 2 d x ≤ ∫ Ω 4 n F ( x , u n , v n ) max ∣ u ¯ n ∣ l 1 , ∣ v n ∣ l 2 D 6 ∣ u n ∣ l 1 + ∣ v n ∣ l 2 d x ≤ ∫ Ω 4 n F ( x , u n , v n ) ∣ u n ∣ l 1 + ∣ v n ∣ l 2 ∣ u ¯ n ∣ l 1 + ∣ v n ∣ l 2 D 6 d x ≤ C 10 ∫ R N ∣ u ¯ n ∣ l 1 d x + C 10 D 6 ∫ R N ∣ v n ∣ l 2 d x ≤ C 10 ∫ R N ∣ u ¯ n ∣ l 1 d x + C 10 C 2 , 5 l 2 D 3 l 2 D 6 ≤ o ( 1 ) + 1 5 .

Hence, (3.49)–(3.52) imply that

(3.53) ∫ B ¯ n , R F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x = ∑ i = 1 4 ∫ Ω i n F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x = o ( 1 ) + 2 5 .

Moreover,

(3.54) ∫ Ω 4 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ ∫ Ω 4 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 σ 1 + D 6 d x = ∫ Ω 4 n ′ F ( x , u n , v n ) ∣ u n ∣ σ 1 ∣ u ¯ n ∣ σ 1 + D 6 ∣ v n ∣ σ 2 ∣ v n ∣ σ 2 d x ≤ ∫ Ω 4 n ′ F ( x , u n , v n ) ( ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ) min 1 ∣ u ¯ n ∣ σ 1 , D 6 ∣ v n ∣ σ 2 d x = ∫ Ω 4 n ′ F ( x , u n , v n ) ( ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ) max ∣ u ¯ n ∣ σ 1 , ∣ v n ∣ σ 2 D 6 d x ≤ ∫ Ω 4 n ′ F ( x , u n , v n ) ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ∣ u ¯ n ∣ σ 1 + ∣ v n ∣ σ 2 D 6 d x = ∫ Ω 4 n ′ F ( x , u n , v n ) ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ∣ u ¯ n ∣ σ 1 d x + 1 D 6 ∫ Ω 4 n ′ F ( x , u n , v n ) ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ∣ v n ∣ σ 2 d x ≤ max C 4 ( c + 1 ) Γ ¯ ( 1 ) 1 l Γ ¯ ‖ u ¯ n ‖ L σ 1 l ˜ Γ ¯ σ 1 , C 4 ( c + 1 ) Γ ¯ ( 1 ) 1 m Γ ¯ ‖ u ¯ n ‖ L σ 1 m ˜ Γ ¯ σ 1 + 1 D 6 max C 4 ( c + 1 ) Γ ¯ ( 1 ) 1 l Γ ¯ ‖ v n ‖ L σ 2 l ˜ Γ ¯ σ 2 , C 4 ( c + 1 ) Γ ¯ ( 1 ) 1 m Γ ¯ ‖ v n ‖ L σ 2 m ˜ Γ ¯ σ 2 ≤ o ( 1 ) + 1 5 .

Note that v n ( x ) = 0 on Ω 2 n ′ . Similar to the arguments of (3.38) and (3.39), we can also obtain

(3.55) ∫ Ω 2 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ ∫ Ω 2 n ′ ∣ F ( x , u n , 0 ) ∣ ‖ u n ‖ 1 σ 1 d x ≤ ∫ Ω 2 n ′ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ⋅ ∣ u ¯ n ∣ σ 1 d x → 0 , as n → ∞ .

Note that v ¯ n ( x ) = v n ( x ) ‖ u n ‖ 1 and u n ( x ) = 0 on Ω 3 n ′ . Similar to the argument of (3.54), we have

(3.56) ∫ Ω 3 n ′ F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ ∫ Ω 3 n ′ ∣ F ( x , 0 , v n ) ∣ D 6 ∣ v n ( x ) ∣ σ 2 ∣ v n ( x ) ∣ σ 2 d x = 1 D 6 ∫ Ω 3 n ′ ∣ F ( x , u n , v n ) ∣ ∣ u n ∣ σ 1 + ∣ v n ∣ σ 2 ⋅ ∣ v n ∣ σ 2 d x ≤ 1 5 .

Hence, (3.54), (3.55), and (3.56) imply that

(3.57) ∫ R N / B ¯ n , R F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ o ( 1 ) + 2 5 .

Then, by (3.48), (3.53), and (3.57), we have

1 ≤ D 4 I ( u n , v n ) + D 5 + D 6 ‖ u n ‖ 1 l 1 + D 6 + D 4 ∫ R N F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x = o ( 1 ) + ∫ B ¯ n , R F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x + ∫ R N / B ¯ n , R F ( x , u n , v n ) ‖ u n ‖ 1 l 1 + D 6 d x ≤ o ( 1 ) + 4 5 ,

which is a contradiction. On the basis of these advantages, we could obtain the conclusion of boundedness for sequence { ( u n , v n ) } .□

Lemma 3.8

Suppose that ( ϕ 1 )–( ϕ 4 ), ( V 0 ) , ( V 1 ) , ( M 1 ) , ( F 1 ), ( F 3 ), and ( F 4 ) hold. Then I satisfies the (C)-condition.

Proof

Let { ( u n , v n ) } be any (C)-sequence of I in W . Lemma 3.7 shows that { ( u n , v n ) } is bounded. Passing to a subsequence { ( u n , v n ) } , by Remark 3.2, there exists a point ( u , v ) ∈ W such that

  1. u n ⇀ u in W 1 , u n → u in L Ψ 1 ( R N ) , in L Φ 1 ( R N ) , in L m 1 ( R N ) , u n ( x ) → u ( x ) a.e. in R N ;

  2. v n ⇀ v in W 2 , v n → v in L Ψ 2 ( R N ) , in L Φ 2 ( R N ) , in L m 2 ( R N ) , v n ( x ) → v ( x ) a.e. in R N .

Now, we define the operators ℱ : W 1 → ( W 1 ) ∗ by

⟨ ℱ ( u ) , u ˜ ⟩ ≔ M 1 ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x ∫ R N ϕ 1 ( ∣ ∇ u ∣ ) ∇ u ∇ u ˜ d x + ∫ R N V 1 ( x ) ϕ 1 ( ∣ u ∣ ) u u ˜ d x , u , u ˜ ∈ W 1

and G : W 2 → ( W 2 ) ∗ by

⟨ G ( v ) , v ˜ ⟩ ≔ M 2 ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x ∫ R N ϕ 2 ( ∣ ∇ v ∣ ) ∇ v ∇ v ˜ d x + ∫ R N V 2 ( x ) ϕ 2 ( ∣ v ∣ ) v v ˜ d x , v , v ˜ ∈ W 2 .

Then, we have

(3.58) ⟨ ℱ ( u n ) , u n − u ⟩ = M 1 ∫ R N Φ 1 ( ∣ ∇ u n ∣ ) d x ∫ R N ϕ 1 ( ∣ ∇ u n ∣ ) ∇ u n ∇ ( u n − u ) d x + ∫ R N V 1 ( x ) ϕ 1 ( ∣ u n ∣ ) u n ( u n − u ) d x = ⟨ I ′ ( u n , v n ) , ( u n − u , 0 ) ⟩ + ∫ R N F u ( x , u n , v n ) ( u n − u ) d x .

(2.9) and the boundedness of { u n } show that

(3.59) ∣ ⟨ I ′ ( u n , v n ) , ( u n − u , 0 ) ⟩ ∣ ≤ ‖ I ′ ( u n , v n ) ‖ W ∗ ‖ u n − u ‖ 1 → 0 .

By ( F 1 ) and Hölder’s inequality, we obtain

(3.60) ∫ R N F u ( x , u n , v n ) ( u n − u ) d x ≤ C 2 ∫ R N ( ∣ u n ∣ l 1 − 1 + ψ 1 ( ∣ u n ∣ ) + Ψ ˜ 1 − 1 ( Ψ 2 ( ∣ v n ∣ ) ) ) ∣ u n − u ∣ d x = C 2 ∫ R N ∣ u n ∣ l 1 − 1 ∣ u n − u ∣ d x + C 2 ∫ R N ( ψ 1 ( ∣ u n ∣ ) + Ψ ˜ 1 − 1 ( Ψ 2 ( ∣ v n ∣ ) ) ) ∣ u n − u ∣ d x ≤ 2 C 2 ‖ ∣ u n ∣ l 1 − 1 ‖ Φ ˜ 1 ‖ u n − u ‖ Φ 1 + 2 C 2 ‖ ψ 1 ( ∣ u n ∣ ) + Ψ ˜ 1 − 1 ( Ψ 2 ( ∣ v n ∣ ) ) ‖ Ψ ˜ 1 ‖ u n − u ‖ Ψ 1 .

Condition ( F 1 ) shows that functions Ψ 1 and Ψ ˜ 1 are N -functions satisfying Δ 2 -condition globally, which together with the convexity of N -function, ( ϕ 4 ) , Lemma 2.4, Remark 2.8, Remark 3.2, inequality (A.9) in [16], inequality (2.2), and the boundedness of { ( u n , v n ) } , imply that the boundedness of the following integrals:

(3.61) ∫ R N Φ ˜ 1 ( ∣ u n ∣ l 1 − 1 ) d x = ∫ ∣ u n ∣ = 0 Φ ˜ 1 ( ∣ u n ∣ l 1 − 1 ) d x + ∫ 0 < ∣ u n ∣ < 1 Φ ˜ 1 ( ∣ u n ∣ l 1 − 1 ) d x + ∫ ∣ u n ∣ ≥ 1 Φ ˜ 1 ( ∣ u n ∣ l 1 − 1 ) d x ≤ ∫ R N Φ ˜ 1 Φ 1 ( ∣ u n ∣ ) c 1 , 1 ∣ u n ∣ d x + Φ ˜ 1 ( 1 ) ∫ R N ∣ u n ∣ l 1 d x ≤ ∫ R N 1 c 1 , 1 + K 1 c 1 , 1 Φ 1 ( ∣ u n ∣ ) d x + Φ ˜ 1 ( 1 ) ∫ R N ∣ u n ∣ m 1 d x ≤ 1 c 1 , 1 + K 1 c 1 , 1 ( ‖ u n ‖ Φ 1 l 1 + ‖ u n ‖ Φ 1 m 1 ) + Φ ˜ 1 ( 1 ) ‖ u n ‖ L m 1 m 1

and

(3.62) ∫ R N Ψ ˜ 1 ( ψ 1 ( ∣ u n ∣ ) + Ψ ˜ 1 − 1 ( Ψ 2 ( ∣ v n ∣ ) ) ) d x ≤ K ˜ 2 ∫ R N Ψ ˜ 1 ψ 1 ( ∣ u n ∣ ) 2 + Ψ ˜ 1 − 1 ( Ψ 2 ( ∣ v n ∣ ) ) 2 d x ≤ K ˜ 2 2 ∫ R N ( Ψ ˜ 1 ( ψ 1 ( ∣ u n ∣ ) ) + Ψ ˜ 1 ( Ψ ˜ 1 − 1 ( Ψ 2 ( ∣ v n ∣ ) ) ) ) d x ≤ K ˜ 2 2 ∫ R N ( Ψ 1 ( 2 ∣ u n ∣ ) + Ψ 2 ( ∣ v n ∣ ) ) d x ≤ K ˜ 2 2 ∫ R N ( K 2 Ψ 1 ( ∣ u n ∣ ) + Ψ 2 ( ∣ v n ∣ ) ) d x ≤ D 7 ∫ R N ( Ψ 1 ( ∣ u n ∣ ) + Ψ 2 ( ∣ v n ∣ ) ) d x ≤ D 7 ( ‖ u n ‖ Ψ 1 l Ψ 1 + ‖ u n ‖ Ψ 1 m Ψ 1 + ‖ v n ‖ Ψ 2 l Ψ 2 + ‖ v n ‖ Ψ 2 m Ψ 2 ) ,

where D 7 = K ˜ 2 2 max { K 2 , 1 } , K ˜ 2 > 0 and K 2 > 0 , which shows that

(3.63) ‖ ∣ u n ∣ l 1 − 1 ‖ Φ ˜ 1 ≤ D 8

and

(3.64) ‖ ψ 1 ( ∣ u n ∣ ) + Ψ ˜ 1 − 1 ( Ψ 2 ( ∣ v n ∣ ) ) ‖ Ψ ˜ 1 ≤ D 9

for some D 8 , D 9 > 0 . Moreover, ⋆ shows that

(3.65) ‖ u n − u ‖ Φ 1 → 0 and ‖ u n − u ‖ Ψ 1 → 0 .

Then, by combining (3.59), (3.60), and (3.63)–(3.65) with (3.58), we obtain

⟨ ℱ ( u n ) , u n − u ⟩ → 0 as n → ∞ .

By [9, Proposition A.3], ℱ is of the class ( S + ) , that is, if a sequence { u n } ⊂ W 1 satisfying

u n ⇀ u and limsup n → ∞ ⟨ ℱ ( u n ) , u n − u ⟩ ≤ 0 ,

then u n → u in W 1 . Similarly, we can also obtain that v n → v in W 2 . Therefore, { ( u n , v n ) } → ( u , v ) in W .□

Proof of Theorem 3.1

It is obvious that I ( 0 ) = 0 . By Lemmas 3.5, 3.6, and 3.8, all conditions of Lemma 2.9 hold. Then system (1.1) possesses a nontrivial weak solution, which is a critical point of I .□

3.2 Multiplicity

In this section, by using the symmetric mountain pass theorem, we can obtain the following multiplicity result.

Theorem 3.9

Assume that ( ϕ 1 )–( ϕ 4 ), ( V 0 ) , ( V 1 ) , ( M 0 ) , ( M 1 ) , ( F 0 ) , ( F 1 ), ( F 4 ), and the following conditions hold:

( F 3 ′ )

lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ uniformly i n x ∈ R N ;

( F 6 ) F ( x , − u , − v ) = F ( x , u , v ) for all ( x , u , v ) ∈ R N × R × R .

Then system (1.1) possesses infinitely many weak solutions { ( u k , v k ) } such that

I ( u k , v k ) ≔ M 1 ^ ∫ R N Φ 1 ( ∣ ∇ u k ∣ ) d x + M 2 ^ ∫ R N Φ 2 ( ∣ ∇ v k ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u k ∣ ) d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v k ∣ ) d x − ∫ R N F ( x , u k , v k ) d x → + ∞ as k → ∞ .

To apply the symmetric mountain pass theorem (i.e., Lemma 2.10), we need the following knowledge. One can see the details in [13,32,41]. Since W is a reflexive and separable Banach spaces, there exist two sequences { e i j : j ∈ N } ⊂ W i ( i = 1 , 2 ) and { e i j ∗ : j ∈ N } ⊂ W i ∗ ( i = 1 , 2 ) such that

W i = span { e i j : j = 1 , 2 , … } ¯ , W i ∗ = span { e i j ∗ : j = 1 , 2 , … } ¯ , i = 1 , 2 ,

and

e i n ∗ ( e i m ) = 1 if n = m , 0 if n ≠ m , i = 1 , 2 .

Let Y i ( k ) and Z i ( k ) be the subsets of W i defined by

Y i ( k ) ≔ span { e i j : j = 1 , … , k } , Z i ( k ) ≔ span { e i j : j = k + 1 , … } ¯ , i = 1 , 2 .

Then

W i = Y i ( k ) ⊕ Z i ( k ) , i = 1 , 2 , k ∈ N .

Moreover, since the embeddings W i ↪ L Ψ i ( R N ) ( i = 1 , 2 ) and W i ↪ L l i ( R N ) ( i = 1 , 2 ) are compact, with a similar discussion as [13,32], we can obtain

(3.66) α i ( k ) ≔ sup { ‖ z ‖ Ψ i : ‖ z ‖ i = 1 , z ∈ Z i ( k ) } → 0

and

(3.67) β i ( k ) ≔ sup { ‖ z ‖ L l i : ‖ z ‖ i = 1 , z ∈ Z i ( k ) } → 0 , i = 1 , 2 , as k → ∞ .

In addition, for Banach space W = W 1 × W 2 , there exists a sequence { η ( j ) } ⊂ W defined by

η ( j ) = ( e 1 n , 0 ) if j = 2 n − 1 , ( 0 , e 2 n ) if j = 2 n , for n ∈ N ,

such that

  1. W = span { η ( j ) : j = 1 , 2 , … } ¯ ,

  2. W = Y k ⊕ Z k ,

    where

    Y k ≔ span { η ( j ) : j = 1 , … , k } and Z k ≔ span { η ( j ) : j = k + 1 , … } ¯ .

Lemma 3.10

Suppose that ( ϕ 1 ) – ( ϕ 3 ) , ( M 0 ) , ( V 0 ) , and ( F 1 ) hold. Then there are two constants ρ , α > 0 , and k ∈ N such that I ∣ ∂ B ρ ∩ Z 2 k ≥ α .

Proof

For ( u , v ) ∈ Z 2 k , in view of the inequality (3.9), (3.20), (3.21), (3.66), (3.70), ( M 0 ) , Young’s inequality, Lemma 2.4, Remark 3.2, and the inequality (66) in [38], we have

I ( u , v ) = M ^ 1 ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x + M ^ 2 ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x + ∫ R N ( V 1 ( x ) Φ 1 ( ∣ u ∣ ) + V 2 ( x ) Φ 2 ( ∣ v ∣ ) − F ( x , u , v ) ) d x ≥ C 1 , 3 ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x + C 2 , 3 ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u ∣ ) d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v ∣ ) d x − C 5 ∫ R N ∣ u ∣ l 1 d x − C 5 ∫ R N ∣ v ∣ l 2 d x − C 5 ∫ R N Ψ 1 ( u ) d x − C 5 ∫ R N Ψ 2 ( v ) d x ≥ C 1 , 3 min { ‖ ∇ u ‖ Φ 1 l 1 , ‖ ∇ u ‖ Φ 1 m 1 } + min { ‖ u ‖ Φ 1 , V 1 l 1 , ‖ u ‖ Φ 1 , V 1 m 1 } − C 5 ‖ u ‖ L l 1 l 1 − C 5 max { ‖ u ‖ Ψ 1 l Ψ 1 , ‖ u ‖ Ψ 1 m Ψ 1 } + C 2 , 3 min { ‖ ∇ v ‖ Φ 2 l 2 , ‖ ∇ v ‖ Φ 2 m 2 } + min { ‖ v ‖ Φ 2 , V 2 l 2 , ‖ v ‖ Φ 2 , V 2 m 2 } − C 5 ‖ v ‖ L l 2 l 2 − C 5 max { ‖ v ‖ Ψ 2 l Ψ 2 , ‖ v ‖ Ψ 2 m Ψ 2 } ≥ min { C 1 , 3 , 1 } 2 m 1 − 1 ‖ u ‖ 1 l 1 + min { C 2 , 3 , 1 } 2 m 2 − 1 ‖ v ‖ 2 l 2 − min { C 1 , 3 , 1 } − min { C 2 , 3 , 1 } − C 5 β 1 ( k ) l 1 ‖ u ‖ 1 l 1 − C 5 β 2 ( k ) l 2 ‖ v ‖ 2 l 2 − C 5 max { α 1 ( k ) l Ψ 1 ‖ u ‖ 1 l Ψ 1 , α 1 ( k ) m Ψ 1 ‖ u ‖ 1 m Ψ 1 } − C 5 max { α 2 ( k ) l Ψ 2 ‖ v ‖ 2 l Ψ 2 , α 2 ( k ) m Ψ 2 ‖ v ‖ 2 m Ψ 2 } .

Since α i ( k ) → 0 , β i ( k ) → 0 , i = 1 , 2 , as k → ∞ , then aforementioned inequality implies that for a large constant ρ > 0 , there exists a large k ∈ N such that I ∣ ∂ B ρ ∩ Z 2 k ≥ α for some α > 0 .□

Lemma 3.11

Suppose that ( ϕ 1 ) – ( ϕ 3 ) , ( M 1 ) , ( V 0 ) , ( F 1 ), and ( F 3 ′ ) hold. Then for each finite dimensional subspace W ˜ ⊂ W , there exists a positive constant R = R ( W ˜ ) such that I ≤ 0 on W ˜ \ B R ( W ˜ ) .

Proof

For each finite dimensional subspace W ˜ ⊂ W , one has W ˜ ⊆ W 1 , 1 × W 2 , 1 , where W 1 , 1 and W 2 , 1 are finite dimensional subspaces of W 1 and W 2 , respectively. Since any two norms in finite dimensional space are equivalent, there exist positive constants h 1 , h 2 , h 3 , h 4 , h 5 , h 6 , h 7 , and h 8 such that

(3.68) h 1 ‖ u ‖ 1 ≤ ‖ u ‖ L m 1 ≤ h 2 ‖ u ‖ 1 , ∀ u ∈ W 1 , 1 ,

(3.69) h 3 ‖ v ‖ 2 ≤ ‖ v ‖ L m 2 ≤ h 4 ‖ v ‖ 2 , ∀ v ∈ W 2 , 1 ,

(3.70) h 5 ‖ u ‖ 1 ≤ ‖ u ‖ L l 1 ≤ h 6 ‖ u ‖ 1 , ∀ u ∈ W 1 , 1 ,

(3.71) h 7 ‖ v ‖ 2 ≤ ‖ v ‖ L l 2 ≤ h 8 ‖ v ‖ 2 , ∀ v ∈ W 2 , 1 ,

where h 1 , h 2 , h 5 , h 6 , and h 3 , h 4 , h 7 , h 8 depend on the spatial dimension of W 1 , 1 and W 2 , 1 , respectively. Moreover, ( F 1 ), ( F 3 ′ ) , and the continuity of F imply that for any given constant

L > max 4 max { C 1 , 4 , 1 } + C ( L ) h 6 l 1 h 1 m 1 , 4 max { C 2 , 4 , 1 } + C ( L ) h 8 l 2 h 3 m 2 ,

there exists a constant C ( L ) > 0 such that

(3.72) F ( x , u , v ) ≥ L ( ∣ u ∣ m 1 + ∣ v ∣ m 2 ) − C ( L ) ( ∣ u ∣ l 1 + ∣ v ∣ l 2 ) , ∀ ( x , u , v ) ∈ R N × R × R .

Then, by ( M 0 ) , (3.20), (3.21), (3.68)–(3.72), Lemma 2.4, and inequality (67) in [38], we have

I ( u , v ) = M ^ 1 ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x + M ^ 2 ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u ∣ ) d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v ∣ ) d x − ∫ R N F ( x , u , v ) d x ≤ C 1 , 4 ∫ R N Φ 1 ( ∣ ∇ u ∣ ) d x + ∫ R N V 1 ( x ) Φ 1 ( ∣ u ∣ ) d x − L ∫ R N ∣ u ∣ m 1 d x + C ( L ) ∫ R N ∣ u ∣ l 1 d x + C 2 , 4 ∫ R N Φ 2 ( ∣ ∇ v ∣ ) d x + ∫ R N V 2 ( x ) Φ 2 ( ∣ v ∣ ) d x − L ∫ R N ∣ v ∣ m 2 d x + C ( L ) ∫ R N ∣ v ∣ l 2 d x ≤ C 1 , 4 ‖ ∇ u ‖ Φ 1 l 1 + C 1 , 4 ‖ ∇ u ‖ Φ 1 m 1 + ‖ u ‖ Φ 1 , V 1 l 1 + ‖ u ‖ Φ 1 , V 1 m 1 − L ‖ u ‖ L m 1 m 1 + C ( L ) ‖ u ‖ L l 1 l 1 + C 2 , 4 ‖ ∇ v ‖ Φ 2 l 2 + C 2 , 4 ‖ ∇ v ‖ Φ 2 m 2 + ‖ v ‖ Φ 2 , V 2 l 2 + ‖ v ‖ Φ 2 , V 2 m 2 − L ‖ v ‖ L m 2 m 2 + C ( L ) ‖ v ‖ L l 2 l 2 ≤ 2 max { C 1 , 4 , 1 } ‖ u ‖ 1 l 1 + 2 max { C 1 , 4 , 1 } ‖ u ‖ 1 m 1 + 4 max { C 1 , 4 , 1 } − L h 1 m 1 ‖ u ‖ 1 m 1 + C ( L ) h 6 l 1 ‖ u ‖ 1 l 1 + 2 max { C 2 , 4 , 1 } ‖ v ‖ 2 l 2 + 2 max { C 2 , 4 , 1 } ‖ v ‖ 2 m 2 + 4 max { C 2 , 4 , 1 } − L h 3 m 2 ‖ v ‖ 2 m 2 + C ( L ) h 8 l 2 ‖ v ‖ 2 l 2 = ( 2 max { C 1 , 4 , 1 } + C ( L ) h 6 l 1 ) ‖ u ‖ 1 l 1 − ( L h 1 m 1 − 2 max { C 1 , 4 , 1 } ) ‖ u ‖ 1 m 1 + 4 max { C 1 , 4 , 1 } + ( 2 max { C 2 , 4 , 1 } + C ( L ) h 8 l 2 ) ‖ v ‖ 2 l 2 − ( L h 3 m 2 − 2 max { C 2 , 4 , 1 } ) ‖ v ‖ 2 m 2 + 4 max { C 2 , 4 , 1 } .

Note that l i ≤ m i ( i = 1 , 2 ) . Then the aforementioned inequality implies that

lim r → ∞ sup ( u , v ) ∈ ∂ B r ∩ W ˜ I ( u , v ) = − ∞ .

Thus, there exists an R = R ( W ˜ ) such that I ≤ 0 on W ˜ \ B R ( W ˜ ) .□

Proof of Theorem 3.9

By ( F 6 ) , it is obvious that I is even in W . By Lemmas 3.10, 3.11, and 3.8, all conditions of Lemma 2.10 hold. Then system (1.1) possesses infinitely many weak solutions { ( u k , v k ) } and I ( u k , v k ) → + ∞ as k → + ∞ .□

4 Results for the scalar equation

In this section, we study the existence and multiplicity of solutions for the following generalized Kirchhoff elliptic equation in Orlicz-Sobolev spaces:

(4.1) − M ∫ R N Φ ( ∣ ∇ u ∣ ) d x Δ Φ u + V ( x ) ϕ ( ∣ u ∣ ) u = f ( x , u ) , x ∈ R N , u ∈ W 1 , Φ ( R N ) ,

where ϕ : ( 0 , + ∞ ) → ( 0 , + ∞ ) is a function which satisfies:

  1. ϕ ∈ C 1 ( 0 , + ∞ ) , t ϕ ( t ) → 0 as t → 0 , t ϕ ( t ) → + ∞ as t → + ∞ ;

  2. t → t ϕ ( t ) are strictly increasing;

  3. 1 < l ≔ inf t > 0 t 2 ϕ ( t ) Φ ( t ) ≤ sup t > 0 t 2 ϕ ( t ) Φ ( t ) ≕ m < min { N , l ∗ } , where Φ ( t ) ≔ ∫ 0 ∣ t ∣ s ϕ ( s ) d s , t ∈ R , l ∗ = l N N − l ;

  4. there exist positive constants c 1 and c 2 such that

    c 1 ∣ t ∣ l ≤ Φ ( t ) ≤ c 2 ∣ t ∣ l , ∀ ∣ t ∣ < 1 ;

Moreover, we introduce the following conditions on f , V , and M :

  1. f : R N × R → R is a C 1 function such that f ( x , 0 ) = 0 , x ∈ R N ;

  2. V ∈ C ( R N , R ) and inf R N V ( x ) > 1 ;

  3. there exist a constant c 3 > 0 such that

    lim ∣ z ∣ → ∞ meas { x ∈ R N : ∣ x − z ∣ ≤ c 3 , V ( x ) ≤ c 4 } = 0 for every c 4 > 0 ,

    where meas ( ⋅ ) denotes the Lebesgue measure in R N ;

  4. M ∈ C ( R + , R + ) and c 5 ≤ M ( t ) ≤ c 6 , ∀ t ≥ 0 for some c 5 , c 6 > 0 ;

  5. M ^ ( t ) ≔ ∫ 0 t M ( s ) d s ≥ M ( t ) t .

Similar to the results in Section 3, we can obtain the following results.

Theorem 4.1

Assume that ϕ and f satisfy ( ϕ 1 ) ′ – ( ϕ 4 ) ′ , ( F 0 ) ′ , ( M 0 ) ′ , ( M 1 ) ′ , ( V 0 ) ′ , ( V 1 ) ′ , and the following conditions:

  1. There exist a constant c 7 > 0 and a continuous function ψ : [ 0 , + ∞ ) → R such that

    ∣ f ( x , u ) ∣ ≤ c 7 ( ∣ u ∣ l − 1 + ψ ( ∣ u ∣ ) )

    for all ( x , u ) ∈ R N × R , where

    Ψ ( t ) ≔ ∫ 0 ∣ t ∣ ψ ( s ) d s , t ∈ R

    is an N-function satisfying

    m < l Ψ ≔ inf t > 0 t ψ ( t ) Ψ ( t ) ≤ sup t > 0 t ψ ( t ) Ψ ( t ) ≕ m Ψ < l ∗ ≔ l N N − l .

  2. There exists a constant c 8 ∈ [ 0 , 1 ) such that

    limsup u → 0 F ( x , u ) Φ ( u ) = c 8 uniformly i n x ∈ R N ,

    where F ( x , u ) = ∫ 0 u f ( x , s ) d s for all ( x , u ) ∈ R N × R .

  3. There exists a domain G ⊂ R N such that

    lim u → ∞ F ( x , u ) ∣ u ∣ m = + ∞ , for a . e . x ∈ G .

  4. There exists a continuous function γ ¯ : [ 0 , ∞ ) → R and constants σ ∈ l ( m Γ ¯ − 1 ) m Γ ¯ , min l , l ∗ ( l Γ ¯ − 1 ) l Γ ¯ , c 9 , r 2 > 0 such that

    (4.2) Γ ¯ F ( x , u ) ∣ u ∣ σ 1 ≤ c 9 F ¯ ( x , u ) , for a l l x ∈ R N and a l l u ∈ R with ∣ u ∣ ≥ r 2 ,

    where Γ ¯ ( t ) ≔ ∫ 0 ∣ t ∣ γ ¯ ( s ) d s , t ∈ R , is an N-function with

    (4.3) 1 < l Γ ¯ ≔ inf t > 0 t γ ¯ ( t ) Γ ¯ ( t ) ≤ sup t > 0 t γ ¯ ( t ) Γ ¯ ( t ) ≕ m Γ ¯ < + ∞

    and

    F ¯ ( x , u ) ≔ 1 m f ( x , u ) u − F ( x , u ) , ∀ ( x , u ) ∈ R N × R .

Then the equation (4.1) has a nontrivial weak solution.

Theorem 4.2

Assume that ( ϕ 1 ) ′ - ( ϕ 4 ) ′ , ( F 0 ) ′ , ( F 1 ) ′ , ( M 0 ) ′ , ( M 1 ) ′ , ( V 0 ) ′ , ( V 1 ) ′ , ( F 4 ) ′ , and the following conditions hold:

( F 3 ′ ) ′

lim t → + ∞ F ( x , u ) ∣ u ∣ m = + ∞ uniformly i n x ∈ R N .

( F 5 ) ′   F ( x , − u ) = F ( x , u ) , for all ( x , u ) ∈ R N × R .

Then the equation (4.1) possesses infinitely many weak solutions { u k } such that

I ( u k ) ≔ M ^ ∫ R N Φ ( ∣ ∇ u k ∣ ) d x + ∫ R N V ( x ) Φ ( ∣ u k ∣ ) d x − ∫ R N F ( x , u k ) d x → + ∞ , as k → ∞ .

Remark 4.3

  1. It is clear that the conditions ( F 3 ′ ) ′ and ( F 4 ) ′ extend the conditions ( F 3 ) and ( F 5 ) in [29].

  2. Comparing Theorem 1.1 in [23] and Theorem 1.5 in [5] with Theorem 4.1, it is easy to see that the condition ( F 3 ′ ) ′ is weaker than the following global (A-R) condition:

  3. (A-R) there exists a constant θ > m such that for all u ∈ R / { 0 } ,

    0 < F ( u ) ≔ ∫ 0 u f ( x , s ) d s ≤ 1 θ u f ( x , u ) .

  4. If we consider the system (4.1) on a bounded domain Ω with Dirichlet boundary condition, then it is natural that we restrict those assumptions of Theorem 4.1 on the bounded domain Ω . Then the condition ( F 4 ) ′ is different from the condition ( f 4 ) in Theorem 5.1 of [32] and ( f 2 ) in [9]. To exemplify this, let F ( x , t ) = ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) ( ∣ t ∣ 5 ln ( 1 + ∣ t ∣ ) ) . If we choose l Γ ¯ = 6 5 , then the condition ( F 4 ) holds. But in [9,32], the constant l Γ must satisfy l Γ > N l i , i.e., l Γ > 3 2 > 6 5 . So the condition ( f 4 ) in [32] and ( f 2 ) in [9] do not hold.

5 Example

(5.1) − M 1 ∫ R 6 ( ∣ ∇ u ∣ 4 + ∣ ∇ u ∣ 5 ) d x div [ ( 4 ∣ ∇ u ∣ 2 + 5 ∣ ∇ u ∣ 3 ) ∇ u ] + V 1 ( x ) ( 4 ∣ u ∣ 2 + 5 ∣ u ∣ 3 ) u = F u ( x , u , v ) , x ∈ R 6 , − M 2 ∫ R 6 ( ∣ ∇ v ∣ 4 ln ( e + ∣ ∇ v ∣ ) ) d x div 4 ∣ ∇ v ∣ 2 ln ( e + ∣ ∇ v ∣ ) + ∣ ∇ v ∣ 3 e + ∣ ∇ v ∣ ∇ v + V 1 ( x ) 4 ∣ v ∣ 2 ln ( e + ∣ v ∣ ) + ∣ v ∣ 3 e + ∣ v ∣ v = F v ( x , u , v ) , x ∈ R 6 ,

where

(5.2) M 1 ( t ) = 2 + 1 e + t , t ≥ 0 , M 2 ( t ) = 3 + 1 e t 2 3 + 3 t , t ≥ 0 ,

(5.3) F ( x , t , s ) = ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) ( ∣ t ∣ 5 ln ( 1 + ∣ t ∣ ) + ∣ s ∣ 5 ln ( 1 + ∣ s ∣ ) + ∣ t ∣ 3 ∣ s ∣ 3 ) .

Let N = 6 , ϕ 1 ( t ) = 4 ∣ t ∣ 2 + 5 ∣ t ∣ 3 and ϕ 2 ( t ) = 4 ∣ t ∣ 2 ln ( e + ∣ t ∣ ) + ∣ t ∣ 3 e + ∣ t ∣ . Then ϕ i ( i = 1 , 2 ) satisfy ( ϕ 1 ) – ( ϕ 4 ) , M i satisfy ( M 0 ) – ( M 1 ) by (5.2), l 1 = l 2 = 4 , m 1 = m 2 = 5 and Φ 1 ( t ) = ∣ t ∣ 4 + ∣ t ∣ 5 , Φ 2 ( t ) = ∣ t ∣ 4 ln ( e + ∣ t ∣ ) . So, l 1 ∗ = l 2 ∗ = 12 .

Let V 1 ( x ) = ∑ i = 1 6 x i 2 + 1 and V 2 ( x ) = ∑ i = 1 6 x i 4 + 2 for all ( x , t , s ) ∈ R N × R × R . Then it is obvious that V i , i = 1 , 2 satisfy (V0) and (V1).

By (5.3), we have

(5.4) F t ( x , t , s ) = ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 5 ∣ t ∣ 3 t ln ( 1 + ∣ t ∣ ) + ∣ t ∣ 4 t 1 + ∣ t ∣ + 3 ∣ t ∣ ∣ s ∣ 3 t ,

(5.5) F s ( x , t , s ) = ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 5 ∣ s ∣ 3 s ln ( 1 + ∣ s ∣ ) + ∣ s ∣ 4 s 1 + ∣ s ∣ + 3 ∣ t ∣ 3 ∣ s ∣ s .

Hence,

(5.6) F ¯ ( x , t , s ) = ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) ∣ t ∣ 6 5 ( 1 + ∣ t ∣ ) + ∣ s ∣ 6 5 ( 1 + ∣ s ∣ ) + 1 5 ∣ t ∣ 3 ∣ s ∣ 3 ≥ ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) ∣ t ∣ 6 5 ( 1 + ∣ t ∣ ) + ∣ s ∣ 6 5 ( 1 + ∣ s ∣ ) ≥ sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ 10 ( ∣ t ∣ 5 + ∣ s ∣ 5 ) , if ∣ t ∣ ≥ 1 , ∣ s ∣ ≥ 1 , sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ 10 ( ∣ t ∣ 6 + ∣ s ∣ 5 ) , if 0 ≤ ∣ t ∣ < 1 , ∣ s ∣ ≥ 1 , sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ 10 ( ∣ t ∣ 5 + ∣ s ∣ 6 ) , if ∣ t ∣ ≥ 1 , 0 ≤ ∣ s ∣ < 1 , sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ 10 ( ∣ t ∣ 6 + ∣ s ∣ 6 ) , if 0 ≤ ∣ t ∣ < 1 , 0 ≤ ∣ s ∣ < 1 .

It is easy to see that conditions ( F 0 ) and ( F 5 ) hold. Since

lim ∣ ( t , s ) ∣ → 0 F ( x , t , s ) ∣ t ∣ m 1 + ∣ s ∣ m 2 = 0 , and lim ∣ ( t , s ) ∣ → + ∞ F ( x , t , s ) ∣ t ∣ m 1 + ∣ s ∣ m 2 = + ∞ ,

by (2) of Lemma 2.3, we can see that ( F 2 ) and ( F 3 ) hold with G = ( 1 / 8 , 3 / 8 ) × R 5 . Choose Ψ 1 ( t ) = Ψ 2 ( t ) = t 6 , Γ ¯ ( t ) = ∣ t ∣ 6 5 , and σ 1 , σ 2 = 11 6 . Then

limsup ∣ ( u , v ) ∣ → ∞ ∣ F ( x , u , v ) ∣ ∣ u ∣ 11 6 + ∣ v ∣ 11 6 6 5 1 F ¯ ( x , u , v ) ≤ limsup ∣ ( u , v ) ∣ → ∞ 10 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 ( ∣ u ∣ 5 ln ( 1 + ∣ u ∣ ) + ∣ v ∣ 5 ln ( 1 + ∣ v ∣ ) + ∣ u ∣ 3 ∣ v ∣ 3 ) 6 5 ∣ u ∣ 11 6 + ∣ v ∣ 11 6 6 5 ( ∣ u ∣ 5 + ∣ v ∣ 5 ) , if ∣ u ∣ ≥ 1 , ∣ v ∣ ≥ 1 , limsup ∣ ( u , v ) ∣ → ∞ 10 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 ( ∣ u ∣ 5 ln ( 1 + ∣ u ∣ ) + ∣ v ∣ 5 ln ( 1 + ∣ v ∣ ) + ∣ u ∣ 3 ∣ v ∣ 3 ) 6 5 ∣ u ∣ 11 6 + ∣ v ∣ 11 6 6 5 ( ∣ u ∣ 6 + ∣ v ∣ 5 ) , if 0 ≤ ∣ u ∣ < 1 , ∣ v ∣ ≥ 1 , limsup ∣ ( u , v ) ∣ → ∞ 10 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 ( ∣ u ∣ 5 ln ( 1 + ∣ u ∣ ) + ∣ v ∣ 5 ln ( 1 + ∣ v ∣ ) + ∣ u ∣ 3 ∣ v ∣ 3 ) 6 5 ∣ u ∣ 11 6 + ∣ v ∣ 11 6 6 5 ( ∣ u ∣ 5 + ∣ v ∣ 6 ) , if ∣ u ∣ ≥ 1 , 0 ≤ ∣ v ∣ < 1

≤ 40 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 limsup ∣ ( u , v ) ∣ → ∞ ∣ u ∣ 6 ( ln ( 1 + ∣ u ∣ ) ) 6 5 + ∣ v ∣ 6 ( ln ( 1 + ∣ v ∣ ) ) 6 5 + ∣ u ∣ 36 5 + ∣ v ∣ 36 5 ∣ u ∣ 36 5 + ∣ v ∣ 36 5 , if ∣ u ∣ ≥ 1 , ∣ v ∣ ≥ 1 , 40 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 limsup ∣ ( u , v ) ∣ → ∞ ∣ u ∣ 6 ( ln ( 1 + ∣ u ∣ ) ) 6 5 + ∣ v ∣ 6 ( ln ( 1 + ∣ v ∣ ) ) 6 5 + ∣ u ∣ 36 5 + ∣ v ∣ 36 5 ∣ u ∣ 41 5 + ∣ v ∣ 36 5 , if 0 ≤ ∣ u ∣ < 1 , ∣ v ∣ ≥ 1 , 40 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 limsup ∣ ( u , v ) ∣ → ∞ ∣ u ∣ 6 ( ln ( 1 + ∣ u ∣ ) ) 6 5 + ∣ v ∣ 6 ( ln ( 1 + ∣ v ∣ ) ) 6 5 + ∣ u ∣ 36 5 + ∣ v ∣ 36 5 ∣ u ∣ 36 5 + ∣ v ∣ 41 5 , if ∣ u ∣ ≥ 1 , 0 ≤ ∣ v ∣ < 1

≤ 40 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 limsup ∣ ( u , v ) ∣ → ∞ ∣ u ∣ 6 ∣ u ∣ 6 5 + ∣ v ∣ 6 ∣ v ∣ 6 5 + ∣ u ∣ 36 5 + ∣ v ∣ 36 5 ∣ u ∣ 36 5 + ∣ v ∣ 36 5 , if ∣ u ∣ ≥ 1 , ∣ v ∣ ≥ 1 , 40 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 limsup ∣ ( u , v ) ∣ → ∞ ∣ u ∣ 6 ∣ u ∣ 6 5 + ∣ v ∣ 6 ∣ v ∣ 6 5 + ∣ u ∣ 36 5 + ∣ v ∣ 36 5 ∣ u ∣ 41 5 + ∣ v ∣ 36 5 , if 0 ≤ ∣ u ∣ < 1 , ∣ v ∣ ≥ 1 , 40 ( sin ( 2 π x 1 ) + ∣ sin ( 2 π x 1 ) ∣ ) 11 5 limsup ∣ ( u , v ) ∣ → ∞ ∣ u ∣ 6 ∣ u ∣ 6 5 + ∣ v ∣ 6 ∣ v ∣ 6 5 + ∣ u ∣ 36 5 + ∣ v ∣ 36 5 ∣ u ∣ 36 5 + ∣ v ∣ 41 5 , if ∣ u ∣ ≥ 1 , 0 ≤ ∣ v ∣ < 1 < + ∞ .

So the condition ( F 4 ) holds. Then by Theorem 3.1, system (5.1) has at least one nontrivial weak solution. If we let

(5.7) F ( x , t , s ) ≡ ∣ t ∣ 5 ln ( 1 + ∣ t ∣ ) + ∣ s ∣ 5 ln ( 1 + ∣ s ∣ ) + ∣ t ∣ 3 ∣ s ∣ 3 for all x ∈ R N and ( t , s ) ∈ R 2 ,

then by Theorem 3.9, system (5.1) has infinitely many nontrivial weak solutions of high energy.

6 Remark on the semi-trivial solutions of (1.1)

In Theorems 3.1 and 3.9, we do not exclude the possibility of semi-nontrivial solutions. Hence, it is possible that the solutions of system are ( u , v ) = ( 0 , v ) or ( u , v ) = ( u , 0 ) , which are called as semi-nontrivial solutions. In general, it is not a simple work to rule out the semi-nontrivial solutions and some extra assumptions have to be added. We refer readers to [10], [11], and [35] for the related work. If we make the extra assumption F ( x , u , v ) = F ( x , ∣ u ∣ , ∣ v ∣ ) , by using Corollary 2.6 in [35] and combing with the proofs of Theorem 3.9, it is easy to exclude the semi-trivial solutions, that is, system (1.1) possesses infinitely many non semi-trivial solutions. Especially, we can obtain that system (5.1) with F satisfying (5.7) has infinitely many nonsemi-trivial solutions.

Acknowledgments

The authors sincerely thank the reviewers for their valuable comments.

  1. Funding information: This project was supported by Yunnan Ten Thousand Talents Plan Young and Elite Talents Project and Candidate Talents Training Fund of Yunnan Province, China (No: 2017HB016).

  2. Author contributions: These authors contributed equally to this work.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Not available.

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Received: 2022-06-15
Revised: 2022-09-21
Accepted: 2022-09-26
Published Online: 2023-03-02

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
https://doi.org/10.1515/anona-2022-0289
Schlagwörter für diesen Artikel
Orlicz-Sobolev spaces; mountain pass theorem; symmetric mountain pass theorem; locally super-(m 1, m 2) condition
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Artikel in diesem Heft

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
  5. Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
  6. Double-phase parabolic equations with variable growth and nonlinear sources
  7. Logistic damping effect in chemotaxis models with density-suppressed motility
  8. Bifurcation diagrams of one-dimensional Kirchhoff-type equations
  9. Standing wave solution for the generalized Jackiw-Pi model
  10. Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
  11. Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
  12. Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
  13. Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
  18. On a strongly damped semilinear wave equation with time-varying source and singular dissipation
  19. Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
  20. Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
  21. Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
  22. Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
  23. Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
  24. Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
  25. Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
  26. Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
  27. On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
  28. Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
  29. Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
  30. Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
  31. Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
  32. Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
  33. Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
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