Bull. Iran. Math. Soc. https://doi.org/10.1007/s41980-018-0017-x

Nontrivial Weak Solution for a Schrödinger–Kirchhoff-Type System Driven by a ( p1 , p2 )-Laplacian Operator D. Ma1 · L. Liu2,3 · Y. Wu3

Received: 20 March 2017 / Accepted: 26 September 2017 © Iranian Mathematical Society 2018

Abstract In this paper, we investigate the existence of nontrivial weak solution for a Schrödinger–Kirchhoff-type system driven by a ( p1 , p2 )-Laplacian operator under appropriate hypotheses. The proofs are based on the variational methods. Keywords Kirchhoff-type elliptic system · Nontrivial weak solution · Critical point theory Mathematics Subject Classification Primary 35J10; Secondary 35J50 · 35J92

1 Introduction Throughout the paper, suppose that p1 , p2 > 1,  ⊂ R N (N > 1) is a non-empty bounded open set with a smooth boundary ∂, F :  × R2 → R is a function, such that F(x, ·, ·) is C 1 in R2 for every x ∈  and F(·, u 1 , u 2 ) is continuous in  for all (u 1 , u 2 ) ∈ R2 , and Fs denotes the partial derivative of F with respect to s.

Communicated by Davod Khojasteh Salkuyeh.

B

D. Ma [email protected]; [email protected] L. Liu [email protected] Y. Wu [email protected]

1

Department of Mathematics, North China Electric Power University, Beijing 102206, China

2

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

3

Department of Mathematics and Statistics, Curtin University, Perth WA6845, Australia

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We study the following Schrödinger–Kirchhoff-type elliptic system: ⎧    M1  (|∇u 1 | p1 + a1 (x)|u 1 | p1 )dx − p1 (u 1 ) + a1 (x)|u 1 | p1 −2 u 1 ⎪ ⎪ ⎪ ⎪ ⎨ =Fu 1 (x, u 1 , u 2 ) in ,   M2  (|∇u 2 | p2 + a2 (x)|u 2 | p2 )dx − p2 (u 2 ) + a2 (x)|u 2 | p2 −2 u 2 ⎪ ⎪ = Fu 2 (x, u 1 , u 2 ) in , ⎪ ⎪ ⎩ u 1 = u 2 = 0 on ∂,

(1.1)

where Mi : [0, +∞) → R is a continuous function for i = 1, 2, ai ∈ L ∞ () with essinfx∈ ai (x) ≥ 0 for i = 1, 2 and s (u) = div(|∇u|s−2 ∇u) is the s-Laplacian operator. System (1.1) is related to the stationary analogue of a model presented by Kirchhoff [1]. More precisely, Kirchhoff extended the classical D’Alembert’s wave equation for free vibrations of elastic strings and proposed a model given by the equation:

∂ 2u ρ 2 − ∂t

E P0 + h 2L



L 0

 ∂u 2   dx ∂x



∂ 2u = 0, ∂x2

(1.2)

which takes into account the changes in length of string during the vibrations. The parameters ρ, P0 , h, E, L in (1.2) are constants which have accurate physical meanings. For more mathematical and physical background on Kirchhoff problems, we refer the reader to [1–4]. The stationary counterpart of (1.2) is the following Kirchhoff equation:

|∇u|2 dx 2 (u) = f (x, u), x ∈ , (1.3) − a+b 

which has been extensively studied by various authors, see some recent papers [5–17] and their references. Kirchhoff-type equation

|∇u| p dx  p (u) = f (x, u), x ∈  (1.4) −M 

and Schrödinger–Kirchhoff-type equation

 (|∇u| p + a(x)|u| p )dx − p (u) + a(x)|u| p−2 u = f (x, u), x ∈  M 

(1.5) generalize Eq. (1.3), where M : R+ → R is a given function. If we take p = 2 and M(t) = a +bt, then (1.4) coincidentally reduces to (1.3). Many results of the existence and multiplicity of solutions for (1.4) and (1.5) have been established by variational method, the reader may consult some recent papers [3,18–22] in case p = 2 for (1.4), [23–28] in case p = 2 for (1.4), and [29–31] for (1.5). From the variational point of view, one always made the assumption: M(t) ≥ m 0 > 0, ∀t > 0,

123

(1.6)

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where m 0 is a constant, see [18,23,26,27,30] for examples. In this paper, we will use the critical point theory to study the existence of a nontrivial weak solution for (1.1) in cases pi > N and pi < N . The condition (1.6) is weakened and we impose brief conditions on F, which are easily to be verified. To our knowledge, there are few results for Schrödinger–Kirchhoff-type system. We point out that in a recent work [32], using Nehari manifold methods and Mountain pass theorem, the authors obtained the existence of nontrivial and radially symmetric solution for the following Schrödinger–Kirchhoff-type system:    ⎧ a + c( R N (|∇u| p + b|u| p )dx)τ − p (u) + b|u| p−2 u ⎪ ⎪ ⎨ 1 q−2 N  = d Fu (u, v) +pλ|u| up in τR ,  a + c( R N (|∇v| + b|v| )dx) − p (v) + b|v| p−2 v ⎪ ⎪ ⎩ = d1 Fv (u, v) + λ|v|q−2 v in R N ,

(1.7)

where 1 < p < N . Comparing with (1.1), we have M1 (t) = M2 (t) = a + ct τ in (1.7).

2 Preliminary We first give a critical point theorem which was developed by Bonanno and D’Aguì [33] as our tool to obtain the main results. Lemma 2.1 [33] Let X be a reflexive real Banach space, ϕ : X → R be a sequentially weakly lower semicontinuous functional, and ψ : X → R be a sequentially weakly upper semicontinuous functional, such that ϕ − ψ is coercive. Assume that there exists a sequentially weakly continuous function I : X → R and r ∈ (inf X (ϕ + I ), sup(ψ + X

I )), such that (ψ + I )(y) − ρ(I, r ) :=

sup

sup

(ϕ+I )(x)≤r

(ψ + I )(x) > 1.

(ϕ + I )(y) − r

(ϕ+I )(y)>r

Then, the restriction of the function ϕ −ψ to (ϕ + I )−1 (r, +∞) has a global minimum. 1, pi

We denote W0

() the closure of C0∞ () with respect to the norm

u i ∗ =



|∇u i (x)| pi dx

1/ pi

.

1, p

Then, W0 i () is a reflexive real Banach space. 1, p For u i ∈ W0 i (), we define

u i  =



(|∇u i (x)| pi + ai (x)|u i (x)| pi )dx

1/ pi

.

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Obviously, u i ∗ ≤ u i . When pi > N , if we put   N −1/ pi N 1/N pi − 1 1−1/ pi ||1/N −1/ pi ,  1+ √ 2 pi − N π

mi =

where  denotes the Gamma function and || is the Lebesgue measure of the set . It is well known that 1, pi

max |u i (x)| ≤ m i u i ∗ , ∀u i ∈ W0

(),

(2.1)

pi > N .

(2.2)

x∈

and thus, one has p

u i  ≤ (1 + m i i ai ∞ ||1/ pi )u i ∗ ,

∗

1, p

When 1 < pi < N , we know that W0 i () is continuously embedded in L pi (), where pi∗ = NN−ppi i is the critical Sobolev exponent. Thus, there exists a constant li , such that

u i 

L

pi∗

=

pi∗



1/ p∗ i

|u i (x)| dx

1, pi

≤ li u i ∗ , ∀u i ∈ W0

(),

(2.3)

and thus, one has ∗

u i  ≤ (1 + li i ||1− pi / pi a∞ )1/ pi u i ∗ , p

pi < N ,

(2.4)

where a∞ = esssupx∈ |a(x)|. From (2.2) and (2.4), we know that u i  and u i ∗ are equivalent norms in 1, p 1, p W0 i () for pi = N . Hence, W0 i () is a reflexive real Banach space with norm u i  for pi = N . 1, p 1, p In what follows, X will denote the Sobolev space W0 1 () × W0 2 () equipped with the norm: (u 1 , u 2 ) = u 1  + u 2 . Then, X is a reflexive real Banach space. For each (u 1 , u 2 ) ∈ X , we define the functions ϕ, ψ, I : X → R by the following: 1 ˆ 1 ˆ M1 (u 1  p1 ) + M2 (u 2  p2 ); p1 p2

ψ(u 1 , u 2 ) = F(x, u 1 , u 2 )dx; ϕ(u 1 , u 2 ) =



p (1+α)

I (u 1 , u 2 ) = u 1 ∗ 1

123

p (1+α)

+ u 2 ∗ 2

,

(2.5) (2.6) (2.7)

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t where Mˆ i (t) = 0 Mi (s)ds, t ≥ 0 and α ≥ 0 is a parameter. Standard arguments show that ϕ and ψ are well defined and continuously Gâteaux differentiable whose Gâteaux derivatives at the point (u 1 , u 2 ) ∈ X are the functions ϕ  (u 1 , u 2 ) ∈ X ∗ and ψ  (u 1 , u 2 ) ∈ X ∗ given by the following: ϕ  (u 1 , u 2 )(v1 , v2 ) =

2 

Mi (u i  pi )

i=1

ψ  (u 1 , u 2 )(v1 , v2 ) =



 

 |∇u i | pi −2 ∇u i ∇vi + ai (x)|u i | pi −2 u i vi dx;

[Fu 1 (x, u 1 , u 2 )v1 + Fu 2 (x, u 1 , u 2 )v2 ]dx,

(2.8) (2.9)

for every (v1 , v2 ) ∈ X . Moreover, ϕ is sequentially weakly lower semicontinuous, ψ is sequentially weakly upper semicontinuous and I is sequentially weakly continuous. Definition 2.2 A pair of functions (u 1 , u 2 ) ∈ X is said to be a weak solution of (1.1) if for any (v1 , v2 ) ∈ X , one has 2 

Mi (u i  pi )

i=1

=





 |∇u i | pi −2 ∇u i ∇vi + ai (x)|u i | pi −2 u i vi dx

[Fu 1 (x, u 1 , u 2 )v1 + Fu 2 (x, u 1 , u 2 )v2 ]dx.

From (2.8), (2.9), and Definition 2.2, we know that a critical point (u ∗1 , u ∗2 ) of ϕ −ψ in X must be a weak solution of (1.1). Definition 2.3 A pair of functions (u 1 , u 2 ) ∈ X is said to be a nontrivial weak solution of (1.1) if (u 1 , u 2 ) is a weak solution of (1.1) satisfying (u 1 , u 2 ) = u 1 +u 2  > 0. Lemma 2.4 Suppose that m, n, C are three constants satisfying m, n > 1 and C ≥ 0. Then x m + y n − C(x + y) ≥ −δ(m, n, C), ∀(x, y) ∈ [0, ∞) × [0, ∞), m

(2.10)

n

where δ(m, n, C) = (m − 1)(C/m) m−1 + (n − 1)(C/n) n−1 . Proof Let f (x, y) = x m + y n − C(x + y), (x, y) ∈ [0, ∞) × [0, ∞). 1

1

Obviously, f attains its global minimum at ((C/m) m−1 , (C/n) n−1 ) on [0, ∞) × [0, ∞). Thus m

n

f min = −(m − 1)(C/m) m−1 − (n − 1)(C/n) n−1 , which means that (2.10) holds.

 

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3 Existence of Nontrivial Weak Solutions for (1.1) In this section, we will give our main results about the existence of nontrivial weak solutions for (1.1) in cases pi > N and 1 < pi < N for i = 1, 2. For convenience, we list the following conditions. (H0) There are constants α ≥ 0 and Ai > 0 for i = 1, 2, such that Mi (t) ≥ Ai t α , i = 1, 2. (H1) There exists ω = (ω1 , ω2 ) ∈ X , such that



F(x, ω1 (x), ω2 (x))dx >

1 ˆ 1 ˆ M1 (ω1  p1 ) + M2 (ω2  p2 ) > 0. p1 p2

(H2) F(x, 0, 0) = 0 for   any x ∈ . p (1+α) , si ∈ (0, pi (1+α)), bi (x) (H3) There exist ci ∈ 0, Ai / pi (1 + α)||m i i ∈ L 1 (, R+ ) for i = 1, 2, and k(x) ∈ L 1 (), such that F(x, u 1 , u 2 ) ≤

2 

ci |u i | pi (1+α) +

i=1

bi (x)|u i |si +k(x), (x, u 1 , u 2 ) ∈  × R2 .

i=1





(H4) There exist ci ∈ 0, Ai / α)) and bi (x) ∈ L F(x, u 1 , u 2 ) ≤

2 

2 

pi∗ pi∗ −si

1− p (1+α) pi (1 + α)li i ||

 , si ∈ (0, pi (1 +

(, R+ ) for i = 1, 2, such that

ci |u i | pi (1+α) +

i=1

(1+α) pi pi∗

2 

bi (x)|u i |si , (x, u 1 , u 2 ) ∈  × R2 .

i=1

Theorem 3.1 Let pi > N for i = 1, 2. Assume that (H0), (H1), (H2), and (H3) hold. Then, (1.1) has at least one nontrivial weak solution u ∗ = (u ∗1 , u ∗2 ) ∈ X . Proof We will apply Lemma 2.1 to prove the existence of a critical point for ϕ − ψ. For any (u 1 , u 2 ) ∈ X , from (H0), (H3), and (2.1), we get

1 ˆ 1 ˆ F(x, u 1 , u 2 )dx M1 (u 1  p1 ) + M2 (u 2  p2 ) − p1 p2 

2 2  Ai u i  pi (α+1)  − ≥ ci |u i (x)| pi (1+α) dx pi (1 + α) 

ϕ(u) − ψ(u) =

i=1 2



−

i=1

123

i=1



bi (x)|u i (x)|si dx −



k(x)dx

Bull. Iran. Math. Soc.

≥

2   i=1 2 

−

 Ai p (1+α) u i  pi (1+α) − ci ||m i i pi (1 + α) m isi

i=1





bi (x)dxu i si −



k(x)dx



  ≥ M u 1  p1 (1+α) + u 2  p2 (1+α) − N u 1 s1 + u 1 s2

 M u 1  p1 (1+α) + u 2  p2 (1+α) − k(x)dx = 2     2N   M u 1  p1 (1+α) + u 2  p2 (1+α) − u 1 s1 + u 1 s2 + 2 M

k(x)dx, (3.1) − 

where 

 A1 A2 p (1+α) p (1+α) > 0; − c1 ||m 1 1 − c2 ||m 2 2 , p1 (1 + α) p2 (1 + α)  



N : = max m s11 b1 (x)dx, m s22 b2 (x)dx .

M : = min





From Lemma 2.4, we have  p1 (1+α)   p2 (1+α)  u 1  p1 (1+α) + u 2  p2 (1+α) = u 1 s1 s1 + u 2 s2 s2  p1 (1 + α) p2 (1 + α) 2N 2N  s1 s2 u 1  + u 1  − δ ≥ , , M s1 s2 M

(3.2)

and u 1  p1 (1+α) + u 2  p2 (1+α) ≥ (u 1  + u 1 ) − δ( p1 (1 + α), p2 (1 + α), 1), (3.3) where δ(·, ·, ·) is the function defined in Lemma 2.4. We substitute (3.2) and (3.3) into (3.1) to get M ϕ(u) − ψ(u) ≥ M 2 (u 1  + u 2 ) − 2 δ( p1(1 + α), p2 (1 + α), 1) p1 (1+α) p2 (1+α) 2N −M , s2 , M − k(x)dx. 2 δ s1 

Thus, by M > 0, we have lim

u 1 +u 2 →+∞

(ϕ(u) − ψ(u)) = +∞,

which means that ϕ(u) − ψ(u) is coercive. Next, we will prove ρ(I, r ) > 1 for some r ∈ (inf E (ϕ + I ), sup E (ψ + I )).

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First, by (H1), we have 

(ψ − ϕ)(ω) =



F(x, ω1 (x), ω2 (x))dx −

 1 ˆ 1 ˆ M1 (ω1  p1 ) + M2 (ω2  p2 ) > 0 p1 p2 (3.4)

and  (ϕ + I )(ω) =

 1 ˆ 1 ˆ p (1+α) p1 p2 M1 (ω1  ) + M2 (ω2  ) + ω1 ∗ 1 p1 p2 p (1+α)

+ ω2 ∗ 2

> 0.

(3.5)

For r ≥ 0, we denote 

 p1 (1+α) p2 (1+α)  |t |t | | 1 2 K (r ) = (t1 , t2 ) ∈ R2  + ≤r . p1 (1 + α) p2 (1 + α)  Let c0 = min

lim

A1 + p1 (1+α) p (1+α)

m1 1

 (ψ+I )(ω)− 

,

A2 + p2 (1+α) p (1+α)

m2 2

max

(t1 ,t2 )∈K (r/c0 )

(ϕ+I )(ω)−r

r →0+

 . Then, by (3.4), (3.5), and (H2), we have

F(x,t1 ,t2 )dx−r

=

(ψ+I )(ω) (ϕ+I )(ω)

=1+

(ψ−ϕ)(ω) (ϕ+I )(ω)

> 1. (3.6)

Second, by (ϕ + I )(ω) > 0 and (3.6), we choose a constant r0 ∈ R satisfying 0 < r0 < (ϕ + I )(ω) and

(ψ + I )(ω) −



max  (t ,t )∈K (r0 /c0 ) 1 2

(3.7)

F(x, t1 , t2 )dx − r0

(ϕ + I )(ω) − r0

> 1.

(3.8)

For (u 1 , u 2 ) ∈ X , by (H0) and (2.1), we have 

 1 ˆ 1 ˆ p (1+α) p (1+α) p1 p2 + u 2 ∗ 2 M1 (u 1  ) + M2 (u 2  ) + u 1 ∗ 1 (ϕ + I )(u 1 , u 2 ) = p1 p2 A1 + p1 (1 + α) A2 + p2 (1 + α) p (1+α) p (1+α) u 1 ∗ 1 u 2 ∗ 2 ≥ + p1 (1 + α) p2 (1 + α) A1 + p1 (1 + α) max |u 1 (x)| p1 (1+α) ≥ p (1+α) 1 m1 p1 (1 + α) x∈ A2 + p2 (1 + α) max |u 2 (x)| p2 (1+α) + p (1+α) m2 2 p2 (1 + α) x∈ ⎧ ⎫ p (1+α) ⎪ max |u 2 (x)| p2 (1+α) ⎪ ⎨ max |u 1 (x)| 1 ⎬ x∈ x∈ ≥ c0 + . ⎪ ⎪ p1 (1 + α) p2 (1 + α) ⎩ ⎭

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Thus, we obtain    (u 1 , u 2 ) ∈ X (ϕ + I )(u 1 , u 2 ) ≤ r0   p2 (1+α)  |u 1 (x)| p1 (1+α) (x)| |u r 2 0 ⊆ (u 1 , u 2 ) ∈ X  + ) ≤ , ∀x ∈  , p1 (1 + α) p2 (1 + α) c0 and it follows that: sup

(ϕ+I )(u 1 ,u 2 )≤r0

(ψ + I )(u 1 , u 2 ) ≤ ≤

sup

(ϕ+I )(u 1 ,u 2 )≤r0

ψ(u 1 , u 2 ) +

sup

(t1 ,t2 )∈M(r0 /c0 ) 

sup

(ϕ+I )(u 1 ,u 2 )≤r0

F(x, t1 , t2 )dx + r0 .

I (u 1 , u 2 ) (3.9)

Therefore, by (3.7), (3.8), and (3.9), we get (ψ+I )(u 1 ,u 2 )−

ρ(I, r0 ) =

sup

(ϕ+I )(u 1 ,u 2 )≤r0

sup

(ϕ+I )(u 1 ,u 2 )−r0

(ϕ+I )(u 1 ,u 2 ))>r0 (ψ+I )(ω)−

sup

(ϕ+I )(u 1 ,u 2 )≤r0

≥

(ψ+I )(u 1 ,u 2 )

(ψ+I )(u 1 ,u 2 )

(ϕ+I )(ω)−r0

(ψ+I )(ω)−

sup

(t1 ,t2 )∈K (r0 /c0 )

≥

 

(ϕ+I )(ω)−r0

F(x,t1 ,t2 )dx−r0

> 1.

Lemma 2.1 guarantees that ϕ − ψ has a critical point u ∗ = (u ∗1 , u ∗2 ) ∈ (ϕ + 0 , +∞). Moreover,

I )−1 (r

r0 < (ϕ +

I )(u ∗1 , u ∗2 )

u ∗  p2 2 1 p (1+α) M1 (s)ds + M2 (s)ds + u ∗1 ∗ 1 p 2 0 0

(u ∗  p1 +u ∗  p2 ) 1 2 M1 (s) M2 (s) p (1+α) ds + u ∗2 ∗ 2 ≤ + p1 p2 0  1+α + u ∗1  p1 + u ∗2  p2 .

1 = p1



u ∗1  p1

Let h(t) =

t 0

M1 (s) M2 (s) ds + t 1+α , t ∈ R. + p1 p2

From h  (t) ≥ 0, h(0) = 0 and lim h(t) = +∞, we conclude that there exists a t→+∞

unique constant t ∗ > 0, such that h(t ∗ ) =

0

t∗



M1 (s) M2 (s) ds + t ∗ 1+α = r0 . + p1 p2

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Since h  (t) ≥ 0, we conclude that u ∗1  p1 + u ∗2  p2 ≥ t ∗ > 0.  



p1 p2 Theorem 3.2 Let pi < N for i = 1, 2 and α ≤ min N − p1 , N − p2 . Assume that (H0), (H1), and (H4) hold. Then, (1.1) has at least one nontrivial weak solution u ∗ = (u ∗1 , u ∗2 ) ∈ X .

Proof We will apply Lemma 2.1 to prove the existence of a critical point for ϕ − ψ. For any (u 1 , u 2 ) ∈ X , from (H0) and (H4), we get

1 ˆ 1 ˆ F(x, u 1 , u 2 )dx M1 (u 1  p1 ) + M2 (u 2  p2 ) − p1 p2 

2 2  Ai u i  pi (α+1)  − ≥ ci |u i (x)| pi (1+α) dx pi (1 + α) 

ϕ(u) − ψ(u) =

i=1 2



−

i=1

i=1



bi (x)|u i (x)|si dx.

(3.10)

 p1 p2 ∗ From α ≤ min N − p1 , N − p2 , we know pi (1 + α) ≤ pi , and then, we use the Hölder’s inequality and (2.3) to get



pi (1+α)

||

1−

pi (1+α) pi∗

pi (1+α)

||

1−

pi (1+α) pi∗

|u i (x)| pi (1+α) dx ≤ li ≤ li

p (1+α)

u i ∗ i

u i  pi (1+α) .

(3.11)

By si ∈ (0, pi (1 + α)) ⊆ (0, pi∗ ), we still use the Hölder’s inequality and (2.3) to get



bi (x)|u i (x)|si dx ≤ lisi bi 

L

pi∗ /( pi∗ −si )

u i s∗i

≤ lisi bi 

L

pi∗ /( pi∗ −si )

u i si .

(3.12)

We substitute (3.11) and (3.12) into (3.10) to get    ϕ(u) − ψ(u) ≥ M u 1  p1 (1+α) + u 2  p2 (1+α) − N u 1 s1 + u 2 s2 , (3.13) where  M := min

123

p (1+α) 1− 1 p∗ A1 p (1+α) 1 − c1 l1 1 || , p1 (1 + α)

Bull. Iran. Math. Soc.

 p (1+α) 1− 2 p∗ A2 p2 (1+α) 2 − c2 l2 × || > 0; p2 (1 + α)  N := max l1s1 b1  p1∗ /( p1∗ −s1 ) , l2s2 b2  p2∗ /( p2∗ −s2 ) . L

L

Following, we substitute (3.2) and (3.3) in Theorem 3.1 with M and N changing to M and N into (3.13) to obtain: ϕ(u) − ψ(u) ≥

M M δ( p1 (1 + α), p2 (1 + α), 1) (u 1  + u 2 ) − 2 2   p1 (1 + α) p2 (1 + α) 2N M − δ , , . 2 s1 s2 M

Thus, by M > 0, we have lim

u 1 +u 2 →+∞

(ϕ(u) − ψ(u)) = +∞,

which means that ϕ(u) − ψ(u) is coercive. Next, we will prove ρ(I, r ) > 1 for some r ∈ (inf E (ϕ + I ), sup E (ψ + I )). First, as in (3.4) and (3.5) in Theorem 3.1, we have (ψ − ϕ)(ω) > 0 and (ϕ + I )(ω) > 0.

(3.14)

We denote pi (1+α)

Fi = 1 + ci li Li =

||

lisi bi  pi∗ /( pi∗ −si ) , L

1−

pi (1+α) pi∗

, i = 1, 2;

i = 1, 2;

 A1 + p1 (1 + α) A2 + p2 (1 + α) , ; c0 = min p1 (1 + α) p2 (1 + α)    r p1 (1+α) p2 (1+α)  , r > 0. Br = (u 1 , u 2 ) ∈ X u 1 ∗ + u 2 ∗ ≤ c0 

Then, from (3.14), we have

lim

(ψ + I )(ω) − sup(u 1 ,u 2 )∈Br

! 2

p (1+α)

i i=1 Fi u i ∗

(ϕ + I )(ω) − r (ψ − ϕ)(ω) (ψ + I )(ω) =1+ > 1. = (ϕ + I )(ω) (ϕ + I )(ω)

+

!2

si i=1 Li u i ∗



r →0+

(3.15)

Second, by (ϕ + I )(ω) > 0 and (3.15), we choose a constant r0 ∈ R, such that 0 < r0 < (ϕ + I )(ω)

(3.16)

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and (ψ + I )(ω) − sup(u 1 ,u 2 )∈Br

! 2

pi (1+α) i=1 Fi u i ∗

0

+

!2

si i=1 Li u i ∗



(ϕ + I )(ω) − r0

> 1. (3.17)

For any (u 1 , u 2 ) ∈ X , by (H0), we have 

 1 ˆ 1 ˆ p (1+α) p (1+α) p1 p2 (ϕ + I )(u 1 , u 2 ) = + u 2 ∗ 2 M1 (u 1  ) + M2 (u 2  ) + u 1 ∗ 1 p1 p2 A1 + p1 (1 + α) A2 + p2 (1 + α) p (1+α) p (1+α) u 1 ∗ 1 u 2 ∗ 2 ≥ + p1 (1 + α) p2 (1 + α)  p (1+α) p (1+α) . ≥ c0 u 1 ∗ 1 + u 2 ∗ 2 Thus, we obtain    (u 1 , u 2 ) ∈ X (ϕ + I )(u 1 , u 2 ) ≤ r0    r0 p1 (1+α) p2 (1+α)  = Br0 . + u 2 ∗ ≤ ⊆ (u 1 , u 2 ) ∈ X u 1 ∗ c0 Then, by (H4), we use (3.11) and (3.12) to obtain sup

(ϕ+I )(u 1 ,u 2 )≤r0

≤ ≤

(ψ + I )(u 1 , u 2 ) ≤  2 

sup

(u 1 ,u 2 )∈Br0

sup

+

ci

 2  

(u 1 ,u 2 )∈Br0 2 

i=1

sup

(u 1 ,u 2 )∈Br0



|u i (x)|

pi (1+α)

(ψ + I )(u 1 , u 2 )

dx +

i=1

1− p (1+α) 1 + ci li i ||

i=1

pi (1+α) pi∗

"

=

i=1

bi (x)|u i (x)| dx + I (u 1 , u 2 ) si



p (1+α)

u i ∗ i



lisi bi  pi∗ /( pi∗ −si ) u i s∗i L



2



 2 

sup

(u 1 ,u 2 )∈Br0

p (1+α) Fi u i ∗ i

i=1

+

2 



Li u i s∗i

.

i=1

(3.18) Therefore, from (3.16), (3.17), and (3.18), we get (ψ+I )(u 1 ,u 2 )−

ρ(I, r0 ) = ≥ ≥

123

sup

(ϕ+I )(u 1 ,u 2 )≤r0

sup

(ϕ+I )(u 1 ,u 2 )>r0 (ψ+I )(ω)− sup

(ψ+I )(u 1 ,u 2 )

(ϕ+I )(u 1 ,u 2 )−r0

(ϕ+I )(u 1 ,u 2 )≤r0

(ψ+I )(u 1 ,u 2 )

(ϕ+I )(ω)−r0

!  pi (1+α) !2 s 2 (ψ+I )(ω)− sup + i=1 Li u i ∗i i=1 Fi u i ∗ (u 1 ,u 2 )∈Br0

(ϕ+I )(ω)−r0

> 1.

Bull. Iran. Math. Soc.

Lemma 2.1 guarantees that ϕ − ψ has a critical point u ∗ = (u ∗1 , u ∗2 ) ∈ (ϕ + 0 , +∞). Just as in Theorem 3.1, we conclude

I )−1 (r

u ∗1  p1 + u ∗2  p2 ≥ t ∗ > 0, where t ∗ > 0 is the unique solution of

t 0

M1 (s) M2 (s) ds + t 1+α = r0 . + p1 p2  

4 Examples In this section, we will give two examples to illustrate our main results. Example 4.1 Let  ⊂ R2 be a non-empty bounded open set with a smooth boundary ∂. Considering the following Schrödinger–Kirchhoff-type elliptic system: ⎧  −4 (u 1 )  |∇u 1 |4 dx = Fu 1 (x, u 1 , u 2 ) in , ⎪ ⎪ ⎪ ⎨  −4 (u 2 )  |∇u 2 |4 dx = Fu 2 (x, u 1 , u 2 ) in , ⎪ ⎪ ⎪ ⎩ on ∂, u1 = u2 = 0

(4.1)

where F :  × R2 → R is a function defined by the following: F(x, u 1 , u 2 ) = c(u 81 + u 82 ) + (u 41 + u 42 )

(4.2)

for some positive constant c. Comparing (4.1) with (1.1), we have p1 = p2 = 4 > N = 2, a1 (x) = a2 (x) = 0 for x ∈  and M1 (t) = M2 (t) = t for t ∈ [0, +∞). Following, we will verify that all conditions in Theorem 3.1 are satisfied for (4.1). First, (H0) is obvious, since we may take A1 = A2 = 1 and α = 1. Moreover, (H2) is obvious, since F(x, 0, 0) = 0 for any x ∈ . Next, choose x0 ∈  and R > 0, such that B(x0 , R) ⊆ , and put υε (x) =

⎧ ⎨0 ⎩

2ε R (R

ε

if x ∈ \B(x0 , R) − |x − x0 )| if x ∈ B(x0 , R)\B(x0 , R/2) if x ∈ B(x0 , R/2),

where ε > 0 is a parameter to be confirmed. Clearly, one has υε ∈ W01,4 () and

υε 4 = υε 4∗ =





|∇υε (x)|4 dx =

B(x0 ,R)\B(x0 ,R/2)

2ε R

4 dx =

12 4 πε . R2

123

Bull. Iran. Math. Soc.

Let ω1 (x) = ω2 (x) = υε (x), one has ω = (ω1 , ω2 ) ∈ X = W01,4 () × W01,4 () and 1 ˆ 1 ˆ 1 M1 (ω1  p1 ) + M2 (ω2  p2 ) = p1 p2 2 On the other hand, we have



F(x, ω1 (x), ω2 (x))dx ≥ 

B(x0 ,R/2)



υε 4 0

F(x, ε, ε)dx =

Taking a proper ε satisfying 21 (cε8 + ε4 )π R 2 > obviously by (4.3) and (4.4). Finally, for (4.1), we obtain

sds =

36 2 8 π ε . R4

1 8 (cε + ε4 )π R 2 . 2

(4.3)

(4.4)

36 2 8 π ε , we conclude that (H1) holds R4

  

 p (1+α) = 0, 1/(8||m 8 ) , 0, Ai / pi (1 + α)||m i i where m is the constant satisfying (2.1)# for any u ∈ W01,4 (). Therefore, when we let the constant c in (4.2) satisfying c ∈ 0, 1/(8||m 8 ) , it is easy to verify that (H3) holds. Above all, by Theorem 3.1, we conclude that (4.1) has at least one nontrivial weak 1,4 1,4 ∗ ∗ ∗ solution # u = (u 18, u2 ) ∈ W0 () × W0 () when the constant c in (4.2) satisfying c ∈ 0, 1/(8||m ) . Example 4.2 Let  ⊂ R3 be a non-empty bounded open set with a smooth boundary ∂. Considering the following Schrödinger–Kirchhoff-type elliptic system: ⎧  ⎨ −2 (u 1 )  |∇u 1 |2 dx = Fu 1 (x, u 1 , u 2 ) in , −2 (u 2 )  |∇u 2 |2 dx = Fu 2 (x, u 1 , u 2 ) in , ⎩ on ∂, u1 = u2 = 0

(4.5)

where F :  × R2 → R is a function defined by the following: F(x, u 1 , u 2 ) = c(u 41 + u 42 ) + (u 21 + u 22 )

(4.6)

for some positive constant c. Comparing (4.5) with (1.1), we have p1 = p2 = 2 < N = 3, a1 (x) = a2 (x) = 0 for x ∈  and M1 (t) = M2 (t) = t for t ∈ [0, +∞). Following, we will verify that all conditions in Theorem 3.2 are satisfied for (4.5). First, (H0) is obvious, since we may take A1 = A2 = 1 and α = 1. Next, choose x0 ∈  and R > 0, such that B(x0 , R) ⊆ , and put υε (x) =

123

⎧ ⎨0 ⎩

2ε R (R

ε

if x ∈ \B(x0 , R) − |x − x0 )| if x ∈ B(x0 , R)\B(x0 , R/2) if x ∈ B(x0 , R/2),

Bull. Iran. Math. Soc.

where ε > 0 is a parameter to be confirmed. Clearly, one has υε ∈ W01,2 () and

υε 2 = υε 2∗ =





|∇υε (x)|2 dx =

B(x0 ,R)\B(x0 ,R/2)

2ε R

2 dx =

14 π Rε2 . 3

Let ω1 (x) = ω2 (x) = υε (x), one has ω = (ω1 , ω2 ) ∈ X = W01,2 () × W01,2 () and 1 ˆ 1 ˆ M1 (ω1  p1 ) + M2 (ω2  p2 ) = p1 p2



υε 2

sds =

0

98 2 2 4 π R ε . 9

(4.7)

On the other hand, we have





F(x, ω1 (x), ω2 (x))dx ≥

B(x0 ,R/2)

F(x, ε, ε)dx =

Taking a proper ε satisfying 13 (cε4 + ε2 )π R 3 > holds obviously by (4.7) and (4.8). Finally, for (4.5), we obtain 



1− p (1+α) pi (1 + α)li i ||

0, Ai /

(1+α) pi pi∗

1 4 (cε + ε2 )π R 3 . 3

98 2 2 4 9 π R ε ,



(4.8)

we conclude that (H1)

  1 = 0, 1/(4l 4 || 3 ) ,

where l is the constant satisfying (2.3)for any u ∈ W01,2  (). Therefore, when we let 1 4 3 the constant c in (4.6) satisfying c ∈ 0, 1/(4l || ) , it is easy to verify that (H4) holds. Above all, by Theorem 3.2, we conclude that (4.5) has at least one nontrivial weak 1,2 1,2 ∗ ∗ ∗ solution  u = (u 1 , u2 ) ∈ W0 () × W0 () when the constant c in (4.6) satisfying 1

c ∈ 0, 1/(4l 4 || 3 ) .

Acknowledgements The authors would like to thank the referees for making good suggestions. The second author was supported financially by the National Natural Science Foundation of China (11371221).

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