A critical p -biharmonic system with negative exponents

Computers and Mathematics with Applications xxx (xxxx) xxx

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A critical p-biharmonic system with negative exponents ∗

Yanbin Sang , Yan Ren Department of Mathematics, School of Science, North University of China, Taiyuan, Shanxi, 030051, China

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a b s t r a c t In this paper, we are devoted to the study of critical p-biharmonic system with a parameter λ, which involves strongly-coupled critical nonlinearities and negative exponents. We prove that there exists a positive constant λ∗ such that the above problem admits at least two solutions if λ ∈ (0, λ∗ ). © 2019 Elsevier Ltd. All rights reserved.

Article history: Received 2 September 2018 Received in revised form 28 August 2019 Accepted 31 August 2019 Available online xxxx Keywords: p-biharmonic system Negative exponents Sobolev critical constant Extremal function Variational method

1. Introduction In this paper, we are concerned with the following critical p-biharmonic system

⎧ 2 1 α1 −1 β1 2 ∆p u = λα u v + α α+β h(x)uα2 −1 v β2 , ⎪ p∗ ⎪ 2 2 ⎪ ⎨ 2 1 α1 β1 −1 2 ∆p v = λβ u v + α β+β h(x)uα2 v β2 −1 , p∗ 2 2 ⎪ ⎪ ⎪ ⎩ u > 0, v > 0, u = ∆u = 0, v = ∆v = 0,

x ∈ Ω, x ∈ Ω,

(1.1)

x ∈ Ω, x ∈ ∂Ω,

where ∆2p u = ∆(|∆u|p−2 ∆u), Ω ⊂ RN is a bounded domain with smooth boundary, 1 < p < ∞, N > 2p, λ > 0 is a Np parameter and α1 > 1, β1 > 1 satisfy α1 + β1 = p∗ , 0 < α2 < 1, 0 < β2 < 1 and α2 + β2 < p, where p∗ = N −2p denotes ∞ the Sobolev critical exponent, h(x) ∈ L (Ω ) and h(x) > 0. In recent years, quasi-linear biharmonic equations have been studied by some researchers, see [1–9] and references therein. This class of problems describe a beam from a material which conforms to a nonlinear Hook’s law rather than the linear one [10]. Further physical background information on the above problems can be found in [11–14]. In particular, much attention has been paid to biharmonic systems or equations with critical exponents [15–24]. In [16], H. Bueno et al. considered the following fourth-order p-biharmonic problems

{

∗ −2

∆2p u = λf (x)|u|q−2 u + |u|p u = ∂∂ un = 0,

,

in Ω , on ∂ Ω ,

(1.2)

,

in Ω , on ∂ Ω ,

(1.3)

and

{

∗ −2

∆2p u = λf (x)|u|q−2 u + |u|p u = ∆u = 0,

∗ Corresponding author. E-mail address: [email protected] (Y. Sang). https://doi.org/10.1016/j.camwa.2019.08.032 0898-1221/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

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Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

where 1 < p < ∞, N > 2p, 1 < q < p. The authors utilized Ljusternik–Schnirelmann methods to show the existence of infinitely many solutions for the problems (1.2) and (1.3) with both Navier and Dirichlet boundary conditions. In fact, the main theorems obtained in [16] generalize the biharmonic operator studied in [17] to the p-biharmonic operator. Shen and Zhang [18] proved the existence of at least two positive solutions of semilinear p-biharmonic system by means of a variational method which is similar to the fibering method of Pohozaev’s. Furthermore, Zhang [19] studied (1.3) with f (x) ≡ 1 and p = 2, and gave uniform estimates for the supremum of the set of λ. Liang et al. [21] showed the existence and multiplicity of semiclassical solutions of perturbed biharmonic equation with critical nonlinearity. Very recently, Deng et al. [22] established the multiplicity of G-invariant solutions for a singular biharmonic system involving multiple critical exponents. By studying the minimizers of a Rellich–Sobolev constant, Kang and Xu [23] obtained the mountain pass type solution to a biharmonic system with strongly coupled critical Rellich–Sobolev terms. We should point out that above problems only involve nonlinearities with lower order term |u|q−2 u (1 < q < p) and subcritical perturbation |u|q−2 u (p ≤ q < p∗ ). On the other hand, nonlinear biharmonic equations with negative exponents have been studied extensively [25–31]. Guerra [27] gave a complete description of entire radially symmetric solutions for the following biharmonic equation

∆2 u = −u−q ,

u > 0 in R3 ,

where q > 1. Moreover, Lai and Ye [30] proved that, if 0 < q ≤ 1, the following biharmonic equation

∆2 u = −u−q ,

u > 0 in RN

(1.4)

has no entire smooth solution. Very recently, Huang et al. [31] studied entire radial solutions to Eq. (1.4) for all q > 0 and N ≥ 3. We note that Kang et al. [32] considered the following elliptic system with singular nonlinearities and critical exponents

⎧ ηα1 α1 −1 β1 2 v + α α+β h(x)uα2 −1 v β2 , ⎪ 2 2 ⎨ −∆u = 2∗ u 1 α1 β1 −1 2 −∆v = ηβ u v + α β+β h(x)uα2 v β2 −1 , 2∗ 2 2 ⎪ ⎩ u > 0, v > 0, x ∈ Ω; u = v = 0,

x ∈ Ω, x ∈ Ω,

(1.5)

x ∈ ∂Ω,

where αi , βi > 0, i = 1, 2, α1 + β1 = 2∗ , α2 + β2 < 2 and 2∗ = N2N is the Sobolev critical constant, h(x) ∈ L∞ (Ω ). −2 They studied the infimum of the energy functional corresponding to (1.5) on the constraint sets and obtained the explicit expression of a critical constant. Furthermore, they showed the existence of positive solutions to the system (1.5) using Ekeland’s variational principle. To our knowledge, p-biharmonic system with the combinations of strongly-coupled critical nonlinearities and negative exponents has not been studied up to now. Motivated by above results, we deal with the problem (1.1). Owing to singular 1,p term, the energy functional corresponding to (1.1) may not be differentiable on E := (W 2,p (Ω ) ∩ W0 (Ω )) × (W 2,p (Ω ) ∩ 1,p W0 (Ω )). Therefore, previous tools and methods adopted in [15–24] cannot be applied to problem (1.1). We must find a new method to prove that there are two solutions to the problem (1.1). Our idea comes from Sun and Wu [33]. We work in the product space E with the standard norm

∫

∥(u, v )∥p =

Ω

|∆u|p dx +

∫ Ω

|∆v|p dx = ∥∆u∥p + ∥∆v∥p .

Associated with the problem (1.1), the energy functional is defined on E by J(u, v ) =

1 p

∥(u, v )∥ − p

λ

∫

α1

β1

|u| |v| dx −

p∗

Ω

∫

1

α2 + β2

h(x)|u|α2 |v|β2 dx,

Ω

then J ∈ C (E , R). A pair of functions (u, v ) ∈ E is said to be a solution of (1.1) if (u, v ) ̸ = (0, 0) and satisfies

∫ 0

= −

(

∫ Ω( Ω

α2 h(x)|u|α2 −2 u|v|β2 φ α2 +β2

+

β2 α2 +β2

∫ (

) α1 β1 −2 1 |u|α1 −2 u|v|β1 φ + λβ | u | |v| vϕ dx ∗ p Ω ) h(x)|u|α2 |v|β2 −2 vϕ dx, ∀ (φ, ϕ ) ∈ E .

) |∆u|p−2 ∆u∆φ + |∆v|p−2 ∆v ∆ϕ dx −

λα1 p∗

1

Below, D := D2,p (RN ) is the completion of C0∞ (RN ) with respect to the norm ( RN |∆u|p dx) p . For all α, β ∈ (0, p∗ ) satisfying α + β = p∗ , by using Sobolev inequality and Young inequality the following best constants are well defined (see [23,34,35])

∫

∫ S :=

RN

inf

u∈D\{0}

(∫

∫

|∆u|p dx p∗

)

p p∗

,

¯ α, β ) := S(

inf

(u,v )∈(D\{0})2

RN

( ) |∆u|p + |∆v|p dx

(∫

)

|u|α |v|β dx

|u| dx RN

p p∗

.

(1.6)

RN

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

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Working as in the proof of Kang and Xu (see [23], Lemma 3.2), we derive that

(( ) β α ) ( ) α+β α+β β α ¯ α, β ) = + S. S( β α

(1.7)

Define the following manifold

Λ := {t(u, v )(u, v )|(u, v ) ∈ E \ {(0, 0)}}, where t(u, v ) are the zeros of the map

, t v)

1 d J(tu ∗ t p −1 dt

t→

=

1 ∗ t p −1

( t

p−1

∥(u, v )∥ − λt p

p∗ −1

∫

α1

β1

|u| |v| dx − t

α2 +β2 −1

∫

α2

)

β2

h(x)|u| |v| dx

Ω

(1.8)

Ω

∫ ∫ α1 β1 p−p∗ p α2 +β2 −p∗ =t ∥(u, v )∥ − λ |u| |v| dx − t h(x)|u|α2 |v|β2 dx. Ω

Ω

We decompose Λ into the following subsets

∫

Λ := {(u, v ) ∈ Λ | (p − p )∥(u, v )∥ + (p − α2 − β2 ) +

p

∗

∗

Λ := {(u, v ) ∈ Λ | (p − p )∥(u, v )∥ + (p − α2 − β2 ) 0

p

∗

∗

Ω

∫

h(x)|u|α2 |v|β2 dx > 0},

h(x)|u|α2 |v|β2 dx = 0},

Ω

∫ Λ− := {(u, v ) ∈ Λ | (p − p∗ )∥(u, v )∥p + (p∗ − α2 − β2 ) h(x)|u|α2 |v|β2 dx < 0}. Ω

Let

λ∗

=S

⎛ ⎜ ⎜ ⎜ ⎝

(

p∗ −α2 −β2 p−α2 −β2

(

p−α2 −β2 p∗ −α2 −β2

)(

p∗ −p

p∗ −α2 −β2 p∗

2

(

2

p∗ −α2 −β2

2 ) α2 +β p ( ) β2 ( ) α2 α2 α2 +β2 β2 α2 +β2 + α2 β2

∥h∥∞ |Ω |

p ) p−αp∗ −−β

( ) β∗1 α1 β1

p

+

( ) α∗1 β1 α1

) pp∗

p

p ⎞ p−αp∗ −−β 2

⎟ ⎟ ⎟ ⎠

2

.

This article is summarized as follows. In Section 2, we will obtain the value λ∗ utilizing the relation between Λ and the fibering maps corresponding to the functional J. Moreover, some necessary lemmas are also given. In Section 3, based on these lemmas and some delicate estimates, the existence of two solutions (u0 , v0 ) ∈ Λ+ and (U0 , V0 ) ∈ Λ− for the problem (1.1) will be established by the Ekeland’s variational principle. 2. Preliminaries and lemmas Define φ (t , u, v ) : (0, +∞) × {E \{(0, 0)}} −→ R by (1.8). Lemma 2.1. Assume that λ ∈ (0, λ∗ ). Then for every (u, v ) ∈ E \ {(0, 0)}, there exist two and only two numbers t − (u, v ) and t + (u, v ) with 0 < t − (u, v ) < t + (u, v ) such that φ (t − (u, v ), u, v ) = φ (t + (u, v ), u, v ) = 0, t − (u, v )(u, v ) ∈ Λ+ , and t + (u, v )(u, v ) ∈ Λ− .

⎞ p−α−1−β

⎛ Proof. Let φ ′ (t , u, v ) = 0, which yields t := tmax

⎜ =⎜ ⎝

2

(p∗ − p)∥(u, v )∥p (p∗ − α2 − β2 )

( φ ′′ (tmax , u, v ) = ∥(u, v )∥p (p − p∗ )(p − α2 − β2 )

∫

h(x)|u|α2 |v|β2 dx

(p − p )∥(u, v )∥ (α2 + β2 −

, and

Ω

∗

p∗ )

2

⎟ ⎟ ⎠

∫

Ω

p∗ −2 ) αp−+β −p

p

α2

2

β2

h(x)|u| |v| dx

2

< 0.

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

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Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

Therefore φ (t , u, v ) achieves its maximum at tmax , and

φ (tmax , u, v )

( =

(p∗ − α2 − β2 )

(

2

Ω

(p∗ − α2 − β2 )

=

h(x)|u|α2 |v|β2 dx

Ω

2

Ω

∥(u, v )∥

2

p ∗ − α 2 − β2

∫

∥(u, v )∥p

−α2 −β2 ) pp∗−α ∫ 2 −β2

p ) p−αp∗ −−β

p∗ − p

(

∫

2

h(x)|u|α2 |v|β2 dx

∫

(p∗ − p)∥(u, v )∥p

−

−λ

p ) p−αp∗ −−β

(p∗ − p)∥(u, v )∥p

(∫

Ω

h(x)|u|α2 |v|β2 dx − λ

p∗ −α −β p p−α 2−β 2 2 2

2

|u|α1 |v|β1 dx Ω −α2 −β2 ) pp∗−α −β

p∗ − p

(

− p ) p−αp∗ −−β

h(x)|u|α2 |v|β2 dx

∫

2

p∗ − α2 − β2

2

∥(u, v )∥

2

(∫

Ω

p∗ −α −β p p−α 2−β 2 2 2

h(x)|u|α2 |v|β2 dx

p ) p−αp∗ −−β 2

2

|u|α1 |v|β1 dx Ω

p − α2 − β2

( =

p ∗ − α 2 − β2 p − α2 − β2

( ≥

p ) p−αp∗ −−β

p∗ − p

)(

2

p∗ − α2 − β2 p∗ − p

)(

p∗ − α2 − β2

∥(u, v )∥

2

(∫ )

Ω

p∗ −α −β p p−α 2−β 2 2 2

β2

α2

h(x)|u| |v| dx

)

p∗ −p p−α2 −β2

p ∗ −p p−α2 −β2

−λ

∫

|u|α1 |v|β1 dx Ω

∥(u, v )∥

p∗ − α2 − β2

p∗ −α −β p p−α 2−β 2 2 2

p ) p−αp∗ −−β ( p∗ −α2 −β2 α +β 2 2 p∗ α2 p∗ β2 − 2 p 2 ∗ α +β p ¯ S( α +β , α +β ) ∥(u, v )∥ 2 2 |Ω | 2 2 2 2

p∗ −p p−α2 −β2

∥h∥∞ ∗

−λ

∥(u, v )∥p ¯ α1 , β1 ) S(

p∗ p

( p − α2 − β2

( =

p ∗ − α 2 − β2

⎛

α1 − λ⎝ β1

(

) β∗1

2

α2 β2

2

2

+

β1 α1

) α∗1

⎞− pp∗ S

⎠

+

(

∗

p

2

− pp

2 ( ) α α+β

β2 α2

∥h∥∞ |Ω |

⎛ −λ⎝

(

α1 β1

) β∗1

(

)(

p∗ − α2 − β2

(

p

p − α2 − β2

+

β1 α1

) α∗1

⎞− pp∗

S

p

p∗ −p p−α2 −β2

p∗ −α2 −β2 p∗

p ) p−αp∗ −−β 2

∥(u, v )∥p

2

⎜ p ⎜ ) p−αp∗ −−β 2 2 ⎜ ⎜ ⎜ p∗ − α2 − β2 ⎜ ⎝ p∗ − p

p ⎞ p−αp∗ −−β 2 ) α2 +β 2 2 p β2 2 2 2 2 ⎟ + α ⎟ 2 ⎟ ⎟ ∗ ⎟ p −α2 −β2 ∗ ⎟ ∥h∥∞ |Ω | p ⎠

2 ( ) α β+β

α2 β2

2 ( ) α α+β

⎞ ∗

p

⎠

α2 +β2

2

∗

⎛( p∗ −p p−α2 −β2

p∗ −p p−α2 −β2

∥(u, v )∥p

⎛ ⎜ ⎜ ⎜ α +β ⎜ 2 2 p =⎜ ⎜S ⎜ ⎜ ⎝

2

2 ) α2 +β p

∗

p∗ − α2 − β2

(

p

p ) p−αp∗ −−β

p∗ − p

)(

2 ( ) α β+β

S

− pp

⎟ ⎟ ∥(u, v )∥p∗ ⎠

∗

:= D(λ)∥(u, v )∥p .

(2.1)

Note that D(λ) = 0, if and only if λ = λ∗ , where we have applied the following inequalities

∫

α1

β1

|u| |v| dx ≤ Ω

(

∥(u, v )∥p ¯ α1 , β1 ) S(

1 ) α1 +β p

∗

=

∥(u, v )∥p ¯ α1 , β1 ) S(

p∗ p

,

(2.2)

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

5

and

∫

h(x)|u|α2 |v|β2 dx ≤ ∥h∥∞

∫

|u|α2 |v|β2 dx Ω

Ω

2 (∫ ) p∗ −α∗2 −β2 ) α2 +β ∗ p∗ p ( α β ) α p+β 2 2 2 2 dx |u| |v| 1dx

(∫ ≤ ∥h∥∞ Ω

= ∥h∥∞ |Ω |

≤ ∥h∥∞ |Ω |

Ω

p∗ −α2 −β2 p∗

p∗ α2

(∫

p∗ β2

p

|u| α2 +β2 |v| α2 +β2 Ω

⎛

p∗ −α2 −β2 p∗

⎝

p∗ −α2 −β2 p∗

¯ S(

2 ⎞ α2 +β ∗ p

p∗

∥(u, v )∥ p∗ α2

¯ S( α

2 +β2

= ∥h∥∞ |Ω |

2 ) α2 +β ∗

p∗ α2

β2 , αp +β ) ∗

2

p∗ p

2

p ∗ β2

,

⎠ −

α2 + β2 α2 + β2

)

α2 +β2 p

∥(u, v )∥α2 +β2 ,

(2.3)

because of (1.6) and (1.7). By λ < λ∗ , we conclude that D(λ) > 0 and φ (tmax , u, v ) > 0. An simple computation gives lim φ = −∞,

lim φ = −λ

t →0+

∫

t →+∞

Ω

|u|α1 |v|β1 dx < 0.

Consequently φ (t , u, v ) has exactly two zeros t − (u, v ) and t + (u, v ) with 0 < t − (u, v ) < t + (u, v ). Owing to

φ (t , u, v ) = 0 ⇒ t p−p ∥(u, v )∥p − t α2 +β2 −p ∗

∗

∫ Ω

h(x)|u|α2 |v|β2 dx − λ

∫ Ω

|u|α1 |v|β1 dx = 0,

so

∥(tu, t v )∥p −

∫

∫

Ω

h(x)|tu|α2 |t v|β2 dx − λ

Ω

|tu|α1 |t v|β1 dx = 0.

Then t(u, v )(u, v ) ∈ Λ. Furthermore, ∗ −1

φ ′ (t , u, v ) > 0 ⇒ (p − p∗ )t p−p

∥(u, v )∥p − (α2 + β2 − p∗ )t α2 +β2 −p

∗ −1

∫ Ω

h(x)|u|α2 |v|β2 dx > 0,

thus

∫

(p − p∗ )∥(tu, t v )∥p − (α2 + β2 − p∗ )

Ω

h(x)|tu|α2 |t v|β2 dx > 0.

Then t − (u, v )(u, v ) ∈ Λ+ . Similarly, we get t + (u, v )(u, v ) ∈ Λ− . This completes the proof. Lemma 2.2. For λ ∈ (0, λ∗ ), we have Λ0 = ∅. Proof. Assume that Λ0 ̸ = ∅ for every λ > 0. If (u, v ) ∈ Λ0 , then

∥(u, v )∥p = λ

∫

|u|α1 |v|β1 dx + Ω

∫ Ω

h(x)|u|α2 |v|β2 dx,

(2.4)

and p∥(u, v )∥p = λp∗

∫ Ω

|u|α1 |v|β1 dx + (α2 + β2 )

∫ Ω

h(x)|u|α2 |v|β2 dx.

(2.5)

It follows from (2.2)–(2.5) that

∥(u, v )∥p

= ≤

α2 +β2 −p∗ p−p∗

∫

h(x)|u|α2 |v|β2 dx

Ω p∗ −α2 −β2 α2 +β2 −p∗ p∗ ∥ h ∥ | Ω | ∞ ∗ p−p

p∗ α2

¯ S( α

2 +β2

β2 − , αp +β ) ∗

2

α2 +β2

2

p

(2.6)

∥(u, v )∥α2 +β2 ,

and

∥(u, v )∥ = λ p

p∗ − α2 − β2 p − α 2 − β2

∫ Ω

|u|α1 |v|β1 dx ≤ λ

p∗ − α2 − β2 p − α 2 − β2

∗

¯ α1 , β1 ) S(

− pp

∗

∥(u, v )∥p .

(2.7)

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

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Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

By (2.6) and (2.7), we have

λ

≥ ≥

p∗

¯ α , β1 ) p ∥(u, v )∥p−p∗ ⎛

p−α2 −β2 S( 1 p∗ −α2 −β2

¯ α , β1 )

p−α2 −β2 S( 1 p∗ −α2 −β2

=S

p∗ −α2 −β2 p−α2 −β2

⎛

( (

⎜ ⎜ ×⎜ ⎝

α2 β2

(

) β2

p∗ p

p−α2 −β2 p∗ −α2 −β2

α2 +β2 + β2 α2

(

∥h∥∞ |Ω |

⎝

p∗ α2

α2 +β2

p∗ β2

p ¯ S( α2 +β2 , α2 +β2 ) p∗ −p p∗ −α2 −β2 p∗ −α2 −β2 p∗ ∥h∥∞ |Ω |

)(

p∗ −p p∗ −α2 −β2

) α2

α2 +β2

2 ) α2 +β p

2

α1 β1

2

2

⎠

( ( ) β∗1

p ) p−αp∗ −−β 2

p ⎞ p−αp∗ −−β

p

+

( ) α∗1 β1 α1

) pp∗

p

p ⎞ p−αp∗ −−β 2

2

⎟ ⎟ ⎟ ⎠

p∗ −α2 −β2 p∗

= λ∗ , which contradicts with λ ∈ (0, λ∗ ). This completes the proof of Lemma 2.2. Lemma 2.3. Assume that λ ∈ (0, λ∗ ). Then there exists the following gap structure in Λ:

∥(u, v )∥ < H0 ,

∀(u, v ) ∈ Λ+ ;

∥(u, v )∥ > H(λ) > H0 ,

∀(u, v ) ∈ Λ− ,

where

⎛

⎛

⎜ p − α2 − β2 ⎝ α1 H(λ) := ⎜ ⎝ λ(p∗ − α2 − β2 ) β1 (

) β∗1

(

p

+

) α∗1

β1 α1

⎞ pp∗

⎞

p

⎠

S

p∗ p

1 p∗ −p

,

⎟ ⎟ ⎠

and

⎛ ⎜ p − α 2 − β2 ∥h∥∞ |Ω | p∗ − p ∗

H0 := ⎝

p∗ −α2 −β2 p∗

⎛

2 ) α β+β

⎝ α2 β2 (

2

2

( +

β2 α2

2 ) α α+β 2

⎞ p−α21−β2

2 ⎞− α2 +β p

2

S

⎠

−

α2 +β2 p

⎟ ⎠

.

Proof. For (U , V ) ∈ Λ− (⊂ Λ), we have

∫

(p − α2 − β2 )∥(U , V )∥p − λ(p∗ − α2 − β2 )

|U |α1 |V |β1 dx Ω (

∫

α2

β2

= (p − α2 − β2 )∥(U , V )∥ − (p − α2 − β2 ) ∥(U , V )∥ − h(x)|U | |V | dx Ω) ( ∫ = − (p∗ − p)∥(U , V )∥p − (p∗ − α2 − β2 ) h(x)|U |α2 |V |β2 dx < 0, p

∗

p

)

Ω

which along with (2.2) yields (p − α2 − β2 )∥(U , V )∥

p

< λ(p − α2 − β2 ) ∗

∫

|U |α1 |V |β1 dx

Ω

p∗

¯ α1 , β1 )− p ∥(U , V )∥p∗ ≤ λ(p∗ − α2 − β2 )S( ( ) p∗ ( ) β∗1 ( ) α∗1 − p ∗ ∗ p −p α1 p = λ(p∗ − α2 − β2 ) + αβ1 S p ∥(U , V )∥p , β 1

1

therefore

⎛

⎛

⎜ p − α 2 − β2 ⎝ α1 ∥(U , V )∥ > ⎜ ⎝ λ(p∗ − α2 − β2 ) β1 (

) β∗1

(

p

+

β1 α1

) α∗1

⎞ pp∗

⎞

p

⎠

S

p∗ p

⎟ ⎟ ⎠

1 p∗ −p

:= H(λ).

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

7

If (u, v ) ∈ Λ+ , then (p − p)∥(u, v )∥ − (p − α2 − β2 ) p

∗

∗

∫ Ω

h(x)|u|α2 |v|β2 dx < 0.

Using (2.3), we get

∥(u, v )∥p

<

p∗ −α2 −β2 p∗ −p

≤

p∗ −α2 −β2 p∗ −p

∫

h(x)|u|α2 |v|β2 dx

Ω

∥h∥∞ |Ω |

p∗ −α2 −β2 p∗

(

2 ( ) α β+β

α2 β2

2

2

+

2 )− α2 +β p

2 ( ) α α+β

β2 α2

2

2

S

−

α2 +β2 p

∥(u, v )∥α2 +β2 ,

thus

⎛ ∥(u, v )∥

⎜ ∗ 2 −β2 < ⎝ p −α ∥h∥∞ |Ω | p∗ −p

p∗ −α2 −β2 p∗

(

2 ( ) α β+β

α2 β2

2

2

β2 α2

+

2

⎞ p−α 1−β

2 )− α2 +β p

2 ( ) α α+β

2

2

S

−

α2 +β2 p

2

⎟ ⎠

:= H0 . In addition, it is easy to verify that H(λ) = H0 ,

if and only if λ = λ∗ .

For all λ ∈ (0, λ∗ ), we deduce that

∥(U , V )∥ > H(λ) > H0 > ∥(u, v )∥,

∀(u, v ) ∈ Λ+ , ∀(U , V ) ∈ Λ− .

(2.8)

This completes the proof of Lemma 2.3. Lemma 2.4. Assume that λ ∈ (0, λ∗ ). Then Λ− is a closed set in E-topology. Proof. Suppose that {(un , vn )} ⊂ Λ− and (un , vn ) → (u, v ) strongly in E. In the following, we show that (u, v ) ∈ Λ− . For {(un , vn )} ⊂ Λ− , we get (p − p∗ )∥(un , vn )∥p + (p∗ − α2 − β2 )

∫ Ω

h(x)|un |α2 |vn |β2 dx < 0,

and thus (p − p∗ )∥(u, v )∥p + (p∗ − α2 − β2 )

∫ Ω

h(x)|u|α2 |v|β2 dx ≤ 0.

Therefore (u, v ) ∈ Λ− ∪ Λ0 . Let us prove that (u, v ) ∈ / Λ0 by contradiction. Otherwise, if (u, v ) ∈ Λ0 , the definition of Λ0 gives

∫ ∫ ∥(u, v )∥p − λ |u|α1 |v|β1 dx − h(x)|u|α2 |v|β2 dx = 0, Ω

(2.9)

Ω

(p∗ − p)∥(u, v )∥p − (p∗ − α2 − β2 )

∫ Ω

h(x)|u|α2 |v|β2 dx = 0.

(2.10)

It follows from (2.1), (2.9) and (2.10) that ∗

D(λ)∥(u, v )∥p

<

(

p−α2 −β2 p∗ −α −β 2

)(

p∗ −p

p ) p−αp∗ −−β 2

p∗ −α2 −β2

2

∥(u, v )∥

2

(∫

p∗ −α −β p p−α 2−β 2 2 2

α2

p ) p−αp∗ −−β

β2

2

h(x)|u| |v| dx Ω

=

−

(

(

p−α2 −β2 p∗ −α2 −β2

p−α2 −β2 p∗ −α2 −β2

)(

p∗ −p

p ) p−αp∗ −−β

p∗ −α2 −β2

)

2

(∫

Ω

p ∗ −p p p−α −β

∥(u, v )∥

2

2

∫ − λ |u|α1 |v|β1 dx

2

2

∥(u, v )∥p p ) p−αp∗ −−β

h(x)|u|α2 |v|β2 dx

2

2

Ω

∥(u, v )∥p = 0

which contradicts with D(λ) > 0 for all λ ∈ (0, λ∗ ). This completes the proof of Lemma 2.4. Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

8

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

Lemma 2.5. Given (u, v ) ∈ Λ± , then for every (ϕ, ψ ) ∈ E, there exist a number ε > 0 and a continuous function f : [0, +∞) −→ (0, +∞) such that f (0) = 1,

f (t)(u + t ϕ, v + t ψ ) ∈ Λ± ,

∀t ∈ R, |t | < ε.

Proof. We define F : E × R+ −→ R as follows: F (t , r) = r

p−p∗

∥(u + t ϕ, v + t ψ )∥ − λ p

∫

α1

Ω

β1

|u + t ϕ| |v + t ψ| dx − r

α2 +β2 −p∗

∫ Ω

h(x)|u + t ϕ|α2 |v + t ψ|β2 dx.

For (u, v ) ∈ Λ+ (⊂ Λ), we have F (0, 1) = ∥(u, v )∥p − λ

∫

|u|α1 |v|β1 dx −

∫

Ω

Ω

h(x)|u|α2 |v|β2 dx = 0,

and Fr (0, 1) = (p − p )∥(u, v )∥ − (α2 + β2 − p ) ∗

p

∗

∫ Ω

h(x)|u|α2 |v|β2 dx > 0.

We deduce that there exists ε¯ > 0 such that F (t , r) = 0 has a unique continuous solution r = f (t) > 0 for |t | < ε¯ , t ∈ R by using the implicit function theorem at the point (0, 1). Since F (0, 1) = 0, then f (0) = 1. By F (t , f (t)) = 0 for |t | < ε¯ , t ∈ R, we obtain 0

∫ ∗ = (f (t))p−p ∥(u + t ϕ, v + t ψ )∥p − λ |u + t ϕ|α1 |v + t ψ|β1 ∫ Ω α2 α2 +β2 −p∗ − (f (t)) h(x)|u + t ϕ| |v + t ψ|β2 ∫ Ω ∫ |f (t)(u+t ϕ )|α1 |f (t)(v+t ψ )|β1 −

∥f (t)(u+t ϕ,v+t ψ )∥p −λ Ω

=

h(x)|f (t)(u+t ϕ )|α2 |f (t)(v+t ψ )|β2

Ω

∗ f p (t)

,

that is f (t)(u + t ϕ, v + t ψ ) ∈ Λ,

∀t ∈ R, |t | < ε¯ .

Since Fr (0, 1) > 0 and Fr (t , f (t))

∗ −1

= (p − p∗ )(f (t))p−p

∥(u + t ϕ, v∫+ t ψ )∥p

− (α2 + β2 − p∗ )(f (t))α2 +β2 −p

∗ −1

∫Ω

h(x)|u + t ϕ|α2 |v + t ψ|β2 h(x)|f (t)(u+t ϕ )|α2 |f (t)(v+t ψ )|β2

(p−p∗ )∥f (t)(u+t ϕ,v+t ψ )∥p −(α2 +β2 −p∗ )

Ω

=

,

∗ (f (t))p +1

we can choose ε > 0 sufficiently small (ε < ε¯ ) such that for every t ∈ R with |t | < ε , (p − p∗ )∥f (t)(u + t ϕ, v + t ψ )∥p − (α2 + β2 − p∗ )

∫ Ω

h(x)|f (t)(u + t ϕ )|α2 |f (t)(v + t ψ )|β2 dx > 0,

that is f (t)(u + t ϕ, v + t ψ ) ∈ Λ+ ,

∀ t ∈ R, |t | < ε.

This completes the proof of Lemma 2.5. 3. Solution of (1.1) for all λ ∈ (0, λ∗ ) Theorem 3.1. Assume that λ ∈ (0, λ∗ ). Then the problem (1.1) admits a positive solution (u0 , v0 ) ∈ E with J(u0 , v0 ) < 0 and ∥(u0 , v0 )∥ < H0 . Proof. For (u, v ) ∈ Λ we obtain J(u, v )

∫ ∫ 1 = 1p ∥(u, v )∥p − pλ∗ |u|α1 |v|β1 dx − α +β h(x)|u|α2 |v|β2 dx 2 2 Ω ∫ Ω ( ) ( ) 1 = 1p − p1∗ ∥(u, v )∥p − α +β − p1∗ h(x)|u|α2 |v|β2 dx 2 2 Ω ( ) ( ) p∗ −α2 −β2 α +β ∗ ∗ 1 ¯ p α2 , p β2 )− 2 p 2 ∥(u, v )∥α2 +β2 , ≥ 1p − p1∗ ∥(u, v )∥p − α +β − p1∗ ∥h∥∞ |Ω | p∗ S( α +β α +β 2

2

2

2

2

2

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

9

thus J(u, v ) is coercive and bounded below in Λ. Considering the following minimization problems

⎧ ⎨ d¯ := inf J(u, v ), (u,v )∈Λ

⎩ d¯ ± :=

inf

(u,v )∈Λ±

(3.1)

J(u, v ).

Note that J(tu, t v ) is increasing in [t − (u, v ), t + (u, v )] for each (u, v ) ∈ E \{(0, 0)}. For (u, v ) ∈ Λ− , we have d¯ ≤ d¯ + ≤ J(t − (u, v )u, t − (u, v )v ) ≤ J(t + (u, v )u, t + (u, v )v ) ≤ J(u, v ). So d¯ ≤ d¯ + ≤ d¯ − . By Lemma 2.2, we know that Λ0 = ∅, then d¯ = d¯ + . So for (3.1), we only need to consider the following problems

{

d¯ + := inf(u,v )∈Λ+ J(u, v ), d¯ − := inf(u,v )∈Λ− J(u, v ).

We know that Λ+ ∪ {0} and Λ− are both closed in E for λ ∈ (0, λ∗ ) in terms of Lemma 2.4. Using Ekeland’s variational principle (see [36]), there exists a minimizing sequence {(un , vn )} ⊂ Λ+ ∪ {0} such that (i) J(un , vn ) ≤ infΛ+ ∪{0} J(u, v ) + 1n ; (ii) J(u, v ) ≥ J(un , vn ) − 1n ∥(u − un , v − vn )∥, ∀(u, v ) ∈ Λ+ ∪ {0}. Since J(u, v ) = J(|u|, |v|), we may assume that un , vn ≥ 0. Observe that J(u, v ) <

(

1 pp∗

(p∗ − p)∥(u, v )∥p − (p∗ − α2 − β2 )

)

∫

h(x)|u|α2 |v|β2 dx Ω

< 0,

for all (u, v ) ∈ Λ+ ,

thanks to α2 + β2 < p. Hence infΛ+ ∪{0} J = d¯ + < 0. By Lemmas 2.1 and 2.2, we have (un , vn ) ∈ Λ+ . Clearly, {(un , vn )} is bounded in E, passing to a subsequence if necessary, there exists (u0 , v0 ) with u0 ≥ 0, and v0 ≥ 0 such that

{

(un , vn ) ⇀ (u0 , v0 ), (un (x), vn (x)) → (u0 (x), v0 (x)),

w eakly a.e.

in E , in Ω ,

as n → ∞. Therefore, we can establish the properties of (u0 , v0 ) by studying sequence further)prove (√ the √ ) {(un ,(v√n )} and √

wn , p βp∗1 wn ⇀ 0 weakly in√E. In ) order to show(√ that (u0 ,√v0 ) ̸ ≡) (0, 0), we assume for a contradiction that (u0 , v0 ) ≡ (0, 0), then (√ + p α1 w , p β1 w p α1 w , p β1 w n n ∈ Λ with J n n −→ infΛ+ ∪{0} J. That is p∗ p∗ p∗ p∗

that (u0 , v0 ) is a solution to the problem (1.1). Write (un , vn ) = (u0 , v0 ) +

p

α1 p∗

wn ,

p

β1 p∗

wn with

p

α1 p∗

√ (√ ⏐β2 )p ⏐√ ⏐ ⏐√ ∫   ⏐ α1 ⏐α2 ⏐⏐ p β1 ⏐⏐ β1  p α1  p ∗ h(x) ⏐⏐ p ∗ wn ⏐⏐ ⏐ (p − p )  wn , wn  + (p − α2 − β2 ) wn ⏐ dx > 0,   ⏐ p∗ ⏐ p∗ p∗ p Ω ∗

and

( √  α

p∗ − p   pp∗ 

p

1

p∗

√ wn ,

p

⏐β2 )p ⏐ ⏐√ ⏐√ ∫  ⏐ α1 ⏐α2 ⏐⏐ p β1 ⏐⏐ p∗ − α2 − β2  p ⏐ ⏐ wn  − ∗ h(x) ⏐ wn ⏐ ⏐ wn ⏐ dx = inf J + o(1), ⏐ p∗ ⏐  p∗ p (α 2 + β 2 ) Ω p∗ Λ+ ∪{0}

β1

which yields 0

( √ √ )p  p α1  wn , p βp∗1 wn  − p∗ (α2 + β2 ) infΛ+ ∪{0} J + o(1)  p∗ < −p∗ (α2 + β2 ) inf J + o(1) < 0.

<

p2 −pp∗ +(p∗ −p)(α2 +β2 ) p

Λ+ ∪{0}

Consequently, (u0 , v0 ) ̸ ≡ (0, 0). For the convenience of description, define the following notations: Gi (u, v, ϕ, ψ ) = αi |u|αi −2 |v|βi uϕ + βi |u|αi |v|βi −2 vψ,

i = 1, 2,

H(u, v, ϕ, ψ ) = |∆u|p−2 ∆u∆ϕ + |∆v|p−2 ∆v ∆ψ, F (t , u, v, ϕ, ψ ) =

1 t

h(x) |u + t ϕ|α2 |v + t ψ|β2 − |u|α2 |v|β2 .

(

)

Claim 1. For every (ϕ, ψ ) ∈ E with ϕ ≥ 0, ψ ≥ 0, there holds h(x)G2 (u0 , v0 , ϕ, ψ )

∫ Ω

α2 + β2

∫ dx ≤ Ω

H(u0 , v0 , ϕ, ψ )dx −

λ p∗

∫ Ω

G1 (u0 , v0 , ϕ, ψ )dx.

(3.2)

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

10

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

For {(un , vn )} ⊂ Λ+ (⊂ Λ), we have

∫

lim inf (p∗ − p)∥(un , vn )∥p < (p∗ − α2 − β2 )

)

(

n→∞

Ω

h(x)|u0 |α2 |v0 |β2 dx.

(3.3)

Otherwise, assume that lim inf (p − p)∥(un , vn )∥ ∗

(

p

)

n→∞

∫

= (p − α2 − β2 ) ∗

Ω

h(x)|u0 |α2 |v0 |β2 dx.

If (un , vn ) ∈ Λ+ , then

∫

lim inf (p∗ − p)∥(un , vn )∥p ≤ lim sup (p∗ − p)∥(un , vn )∥p ≤ (p∗ − α2 − β2 )

(

)

)

(

n→∞

n→∞

Ω

h(x)|u0 |α2 |v0 |β2 dx.

Furthermore lim ∥(un , vn )∥ = p

p∗ − α2 − β2

∫

p∗ − p

n→∞

Ω

h(x)|u0 |α2 |v0 |β2 dx.

(3.4)

Therefore, limn→∞

) ( ∫ = limn→∞ ∥(un , vn )∥p − h(x)|un |α2 |vn |β2 dx

) ( ∫ λ |un |α1 |vn |β1 dx

Ω

Ω

p−α2 −β2 p∗ −p

=

∫ Ω

(3.5)

h(x)|u0 |α2 |v0 |β2 dx.

Since D(λ) > 0 for λ ∈ (0, λ∗ ). We derive from (3.4), (3.5) that 0

∗

< D(λ)∥(un , vn )∥p <

(

p ) p−αp∗ −−β (

p∗ −p

2

2

p∗ −α2 −β2

p−α2 −β2 p∗ −α −β 2

∥(un , vn )∥

)

2

(∫

→

p∗ −p p∗ −α2 −β2

p ) p−αp∗ −−β ( 2

2

p−α2 −β2 p∗ −α2 −β2

)

∗ 2 −β2 ⎝ p −α p∗ −p

⎛∫ ⎝

2 −β2 − p−α p∗ −p

∫

= 0,

Ω

h(x)|u0 |α2 |v0 |β2 dx

p ) p−αp∗ −−β 2

h(x)|un | |vn | dx

⎛

(

β2

α2

Ω

p∗ −α −β p p−α 2−β 2 2 2

2

Ω

∗ ⎞ p −α2 −β2

∫ Ω

h(x)|u0 |α2 |v0 |β2 dx⎠

⎞ Ω

∫ − λ |un |α1 |vn |β1 dx

h(x)|u0 |α2 |v0 |β2 dx⎠

p−α2 −β2

p∗ −p p−α2 −β2

(n → ∞)

which is clearly impossible. So (un , vn ) → (u0 , v0 ) strongly in E when J(un , vn ) → infΛ+ J(u, v ) < 0. By (3.3), we can choose a subsequence such that p − p ∥(un , vn )∥ − p − α2 − β2

(

∗

p

)

(

∗

)

∫ Ω

h(x)|un |α2 |vn |β2 dx ≤ −C ,

(3.6)

for suitable constant C > 0. Let (ϕ, ψ ) ∈ E with ϕ (x), ψ (x) ≥ 0. Applying Lemma 2.5, we deduce that there exists a continuous function fn (t) > 0 such that fn (t)(un + t ϕ, vn + t ψ ) ∈ Λ+ (⊂ Λ) for all small enough t > 0. Thus fn (0) = 1, moreover 0

= (fn (t))p ∥(un + t ϕ, vn + t ψ )∥p − (fn (t))α2 +β2 ∫ ∗ − λ (fn (t))p |un + t ϕ|α1 |vn + t ψ|β1 dx,

∫ Ω

h(x)|un + t ϕ|α2 |vn + t ψ|β2 dx

Ω

and 0 = ∥(un , vn )∥ − p

∫

α2

Ω

β2

h(x)|un | |vn | dx − λ

∫ Ω

|un |α1 |vn |β1 dx,

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

11

for t > 0 small enough, namely,

( ) = fnp (t) − 1 ∥(un + t ϕ, vn + t ψ )∥p + (∥(un + t ϕ, vn + t ψ )∥p − ∥(un , vn )∥p ) (∫ ( )∫ α +β h(x)|un + t ϕ|α2 |vn + t ψ|β2 dx − fn 2 2 (t) − 1 h(x)|un + t ϕ|α2 |vn + t ψ|β2 dx − Ω Ω ) ∫ ( ∗ )∫ − h(x)|un |α2 |vn |β2 dx − λ fnp (t) − 1 |un + t ϕ|α1 |vn + t ψ|β1 dx Ω Ω (∫ ) ∫ −λ |un + t ϕ|α1 |vn + t ψ|β1 dx − |un |α1 |vn |β1 dx

0

Ω

Ω

p fn (t)

− 1 ∥(un + t ϕ, vn + t ψ )∥p + (∥(un + t ϕ, vn + t ψ )∥p − ∥(un , vn )∥p ) )∫ ( ∗ )∫ α +β p − fn 2 2 (t) − 1 h(x)|un + t ϕ|α2 |vn + t ψ|β2 dx − λ fn (t) − 1 |un + t ϕ|α1 |vn + t ψ|β1 dx, ≤

(

)

(

Ω

Ω

dividing by t > 0 and taking the limit for t → 0, we get

∫

( ) ≤ pfn (0)∥(un , vn )∥ + p |∆un |p−2 ∆un ∆ϕ + |∆vn |p−2 ∆vn ∆ψ dx Ω ∫ ∫ β2 α2 ∗ ′ ′ −(α2 + β2 )fn (0) h(x)|un | |vn | dx − λp fn (0) |un |α1 |vn |β1 dx Ω Ω ( ∫ α2 ′ p = fn (0) p∥(un , vn )∥ − (α2 + β2 ) h(x)|un | |vn |β2 dx − p∗ (∥(un , vn )∥p )) ∫ Ω ∫ ( ) β2 α2 +p |∆un |p−2 ∆un ∆ϕ + |∆vn |p−2 ∆vn ∆ψ dx − h(x)|un | |vn | dx Ω Ω ( ) ∫ ′ ∗ p = fn (0) (p − p )∥(un , vn )∥ + (p∗ − α2 − β2 ) h(x)|un |α2 |vn |β2 dx Ω ∫ ( ) p−2 p−2 +p |∆un | ∆un ∆ϕ + |∆vn | ∆vn ∆ψ dx. p

′

0

Ω

Since (un , vn ) ∈ Λ+ , fn′ (0) ̸ = −∞. By (3.6), we have fn′ (0) is uniformly bounded from below. In the following, applying property (ii), we get J(un , vn ) ≤ J(fn (t)(un + t ϕ ), fn (t)(vn + t ψ )) +

1 n

∥(fn (t)(un + t ϕ ) − un , fn (t)(vn + t ψ ) − vn )∥,

(3.7)

for t > 0 small enough, that is 1 n

(|fn (t) − 1|∥(un , vn )∥ + tfn (t)∥(ϕ, ψ )∥)

≥ 1n ∥fn (t)[(un , vn ) + t(ϕ, ψ )] − (un , vn )∥ ≥ J(un , vn ) − J(fn (t)(un + t ϕ ), fn (t)(vn + t ψ )) ) ( )∫ ( 1 1 1 1 p |un |α1 |vn |β1 dx = p − α +β ∥(un , vn )∥ + λ α +β − p∗ 2 2 2 2 Ω ) ( 1 (fn (t))p ∥(un + t ϕ, vn + t ψ )∥p − 1p − α +β 2 2 ∫ ( ) ∗ 1 − λ α +β − p1∗ (fn (t))p |un + t ϕ|α1 |vn + t ψ|β1 dx 2 2 Ω ) ( 1 [(fn (t))p − 1]∥(un + t ϕ, vn + t ψ )∥p = − 1p − α +β 2 2 ∫ ( ) ∗ 1 − λ α +β − p1∗ [(fn (t))p − 1] |un + t ϕ|α1 |vn + t ψ|β1 dx 2 2 Ω ( ) ( ) 1 1 1 ∥(un , vn )∥p − p − α +β ∥(un + t ϕ, vn + t ψ )∥p + 1p − α +β 2 2 2 2 ∫ ( ) ( )∫ 1 1 1 1 α1 β1 − λ α +β − p∗ |un + t ϕ| |vn + t ψ| dx + λ α +β − p∗ |un |α1 |vn |β1 dx, 2

2

Ω

2

2

Ω

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

12

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

dividing by t > 0 and passing to the limit as t → 0, we derive that

( ′ ) |fn (0)|∥(un , vn )∥ + ∥(ϕ, ψ )∥

1 n

≥

p−α2 −β2

α2 +β2

−λ

fn′ (0)∥(un , vn )∥p +

p∗ −α2 −β2

′

∫

α1

fn (0)

p−α2 −β2

α2 +β2

∫

( Ω

) |∆un |p−2 ∆un ∆ϕ + |∆vn |p−2 ∆vn ∆ψ dx

β1

|un | |vn | dx Ω ( ) α1 |un |α1 −2 un |vn |β1 ϕ + β1 |un |α1 |vn |β1 −2 vn ψ dx −λ 2 2 ( Ω ) ∫ fn′ (0) α2 β2 ∗ p ∗ = α +β (p − p )∥(un , vn )∥ + (p − α2 − β2 ) h(x)|un | |vn | dx 2 2 Ω ∫ ( ) p−α2 −β2 |∆un |p−2 ∆un ∆ϕ + |∆vn |p−2 ∆vn ∆ψ dx + α +β 2 2 Ω∫ ( ) p∗ −α2 −β2 − λ p∗ (α +β ) α1 |un |α1 −2 un |vn |β1 ϕ + β1 |un |α1 |vn |β1 −2 vn ψ dx. 2 2 α2 +β2

p∗ −α2 −β2 p∗ (α +β )

∫

(3.8)

Ω

However, from (3.6), for n sufficiently large

−

(

1

α2 + β2

(p − p)∥(un , vn )∥ − (p − α2 − β2 ) p

∗

∗

∫

α2

Ω

β2

h(x)|un | |vn | dx

) −

∥(un , vn )∥ n

≥C

(3.9)

with C > 0 a suitable constant. Combining (3.8) and (3.9), we deduce that fn′ (0) is uniformly bounded from above. Consequently, fn′ (0) is uniformly bounded in n.

(3.10)

Using (3.7) again, 1 n

(|fn (t) − 1|∥(un , vn )∥ + tfn (t)∥(ϕ, ψ )∥)

≥ J(un , vn ) − J (fn (t)(u ∫ n + t ϕ ), fn (t)(vn + t ψ ))∫ λ

1 h(x)|un |α2 |vn |β2 dx |un |α1 |vn |β1 dx − α +β 2 2 Ω ∫ p p∗ − fnp(t) ∥(un + t ϕ, vn + t ψ )∥p + λfnp∗ (t) |un + t ϕ|α1 |vn + t ψ|β1 dx ∫ Ω α2 +β2 + fnα +β (t) h(x)|un + t ϕ|α2 |vn + t ψ|β2 dx 2 2 ( p Ω ) 1 = − p fn (t) − 1 ∥∫(un + t ϕ, vn + t ψ )∥p − 1p (∥(un + t ϕ, vn + t ψ )∥p − ∥(un , vn )∥p ) ( ∗ ) |un + t ϕ|α1 |vn + t ψ|β1 dx + pλ∗ fnp (t) − 1 Ω (∫ ) ∫ λ α1 β1 α1 β1 + p∗ |un + t ϕ| |vn + t ψ| dx − |un | |vn | dx Ω Ω ( )∫ α +β 1 fn 2 2 (t) − 1 h(x)|un + t ϕ|α2 |vn + t ψ|β2 dx + α +β 2 2 Ω (∫ ) ∫ 1 + α +β h(x)|un + t ϕ|α2 |vn + t ψ|β2 dx − h(x)|un |α2 |vn |β2 dx ,

= 1p ∥(un , vn )∥p −

2

2

Ω

p∗

Ω

Ω

dividing by t → 0 and passing to the limit as t → 0, we get

( ′ ) |fn (0)|∥(un , vn )∥ + ∥∫(ϕ, ψ )∥ ∫ ( ) ≥ −fn′ (0)∥(un , vn )∥p − |∆un |p−2 ∆un ∆ϕ + |∆vn |p−2 ∆vn ∆ψ dx + λfn′ (0) |un |α1 |vn |β1 dx Ω Ω (∫ ) ∫ + pλ∗ α1 |un |α1 −1 |vn |β1 ϕ + β1 |un |α1 |vn |β1 −1 ψ dx + fn′ (0) h(x)|un |α2 |vn |β2 dx Ω ) (∫ ∫ Ω 1 1 α2 β2 α2 β2 h(x)|un | |vn | dx h(x)|un + t ϕ| |vn + t ψ| dx − + lim inf t →0 Ω Ω ∫ ∫ t α2 + β2 ∫ λ F (t , un , vn , ϕ, ψ ) := − H(un , vn , ϕ, ψ )dx + ∗ G1 (un , vn , ϕ, ψ )dx + lim inf dx, t →0 p Ω α 2 + β2 Ω Ω 1 n

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

13

which yields

∫ lim inft →0

∫ ≤ Ω

F (t ,un ,vn ,ϕ,ψ )

α2 +β2

Ω

dx

H(un , vn , ϕ, ψ )dx −

∫

λ

p∗

Ω

G1 (un , vn , ϕ, ψ )dx +

|fn′ (0)|∥(un ,vn )∥+∥(ϕ,ψ )∥ n

.

For F (t , un , vn , ϕ, ψ ) ≥ 0 in Ω , Fatou’s Lemma implies that lim inft →0 F (t , un , vn , ϕ, ψ ) is integrable, and F (t , un , vn , ϕ, ψ )

∫ lim inf Ω∫

α2 + β2

t →0

≤ Ω

H(un , vn , ϕ, ψ )dx −

dx

∫

λ

p∗

Ω

G1 (un , vn , ϕ, ψ )dx +

|fn′ (0)|∥(un ,vn )∥+∥(ϕ,ψ )∥ n

.

(3.11)

Since F (t , un , vn , ϕ, ψ )

=

α2 + β2

|un + t ϕ|α2 |vn + t ψ|β2 − |un |α2 |vn |β2

h(x)

α2 + ⎧ β2 ⎪ ⎪ ⎪ −−−−→ ⎨ (t → 0) ⎪ ⎪ ⎪ ⎩

t un (x) = vn (x) = 0; ϕ (x), ψ (x) ̸ = 0, α2 + β2 > 1, ϕ α2 ψ β2 , un (x) = vn (x) = 0; ϕ (x), ψ (x) ̸ = 0, α2 + β2 = 1, ∞, un (x) = vn (x) = 0; ϕ (x), ψ (x) ̸ = 0, α2 + β2 < 1, ∞, un (x) = 0 or vn (x) = 0; ϕ (x), ψ (x) ̸ = 0, h(x) G (u , v , ϕ, ψ ), un (x), vn (x) > 0; ϕ (x), ψ (x) ̸ = 0. 2 n n α +β 0,

2

2

We take ϕ = ψ = e1 as a test-function in (3.11), where e1 is the first eigenfunction corresponding to the following problem

∆2p e1 + λ1 |e1 |p−2 e1 = 0, x ∈ Ω , e1 |∂ Ω = 0.

{

Since un (x), vn (x) > 0, then

∫

h(x)G2 (un ,vn ,ϕ,ψ )

α2 +β2

Ω∫

≤ Ω

dx

H(un , vn , ϕ, ψ )dx −

λ

∫

p∗

Ω

G1 (un , vn , ϕ, ψ )dx +

|fn′ (0)|∥(un ,vn )∥+∥(ϕ,ψ )∥ n

,

according to (3.10), we deduce that u0 (x), v0 (x) > 0 a.e. in Ω , and (3.2) holds. This completes the proof of Claim 1. For a ∈ Ω , let k ∈ C0∞ (Ω ) be a cut-off function such that 0 ≤ k(x) ≤ 1 in Ω and k(x) = 1 when x ∈ Br (a) ⊂ Ω for a suitable positive number r. Take Uε,a (x) = k(x)ε easy to check that

∥∆Uε,a ∥p = ∗

∥Uε,a ∥pp∗ =

∫ Ω

∫

|∆Uε,a |p dx = B + O(ε

U ∗ ( x−ε a ), where U ∗ (·) is a radially symmetric minimizer of S. It is

p

), B

N

∗

Ω

N −2p p−1

−(N −2p)

|Uε,a |p dx = A + O(ε p−1 ), S =

p

A p∗

,

where

∫

∫

∗

A= RN

(U ∗ )p dx,

B= RN

|∆U ∗ |p dx.

¯ α1 , β1 ) can be attained by By [32,34] we know that S(

(√ p

α1 p∗

Uε (x),

√ p

β1 p∗

)

Uε (x) , and

√ ⏐⏐(√ )⏐⏐p ⏐⏐ ⏐⏐ ( N −2p ) β1 ⏐⏐ p α1 ⏐⏐ p U (x) , U (x) ⏐⏐ = B + O ε p−1 , ⏐⏐ ε,a ε,a ∗ ∗ ⏐⏐ ⏐⏐ p p ⏐β1 ⏐α1 ⏐⏐√ ∫ ⏐√ ⏐ ( N ) ⏐ α1 ⏐ ⏐ p β1 ⏐ ⏐ p Uε,a (x)⏐ ⏐ p−1 ¯ , ⏐ p∗ ⏐ ⏐ p∗ Uε,a (x)⏐⏐ dx = A + O ε Ω

where A¯ =

(√ )α1 (√ )β1 p

α1 p∗

p

β1 p∗

¯ α1 , β1 ) = A. Thus S(

B

p

.

∗ A¯ p

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

14

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

Claim 2. If λ ∈ (0, λ∗ ), then (u0 , v0 ) ∈ Λ. Denote a0 = ∥(u0 , v0 )∥p − λ

∫ Ω

|u0 |α1 |v0 |β1 dx −

∫ Ω

h(x)|u0 |α2 |v0 |β2 dx.

Let ϕ = u0 , ψ = v0 in (3.2), we know that a0 ≥ 0. In order to show that a0 = 0, let us assume that a0 > 0. Under this assumption, there exists a unique c0 > 0 such that 1

(

¯ α1 , β1 ) c0 A¯ p∗ S(

)

( )p∗ 1 − λ c0 A¯ p∗ = −a0 .

On the other hand, J(un , vn ) → d¯ + ,

(un , vn ) ∈ Λ+ ,

in view of the Brezis–Lieb Lemma [37], we get

(

)∫

(

β2

α2

)∫

= J(un , vn ) = − α +β h(x)|un | |vn | dx + λ − |un |α1 |vn |β1 dx 2 2 Ω Ω ( ( )∫ )∫ 1 = 1p − α +β h(x)|u0 |α2 |v0 |β2 dx + λ 1p − p1∗ |u0 |α1 |v0 |β1 dx 2 2 Ω Ω ∫ ( ) (√ )α1 (√ )β1 p β1 p α1 w + λ 1p − p1∗ wn dx + o(1), n p∗ p∗

d¯ + + o(1)

1 p

1

1 p

1 p∗

Ω

and

∫ ∫ = ∥(un , vn )∥p − h(x)|un |α2 |vn |β2 dx − λ |un |α1 |vn |β1 dx Ω√ ∫ ( √ Ω )α ( √ ⏐⏐(√ )⏐⏐p )β1 1 ⏐⏐ ⏐⏐ p β1 p α1 w = a0 + ⏐⏐ p αp∗1 wn , p βp∗1 wn ⏐⏐ − λ w dx + o(1) n n ∗ ∗ p p Ω p (∫ (√ ) ∫ (√ )α1 (√ )β1 )α1 (√ )β1 p∗ p β1 p β1 p α1 w p α1 w ¯ α1 , β1 ) −λ ≥ a0 + S( wn dx wn dx + o(1), n n p∗ p∗ p∗ p∗

0

Ω

Ω

which yields

∫ (√ p

lim

n→∞

α1 p∗

Ω

wn

)α1 (√

β1

p

p∗

)β1 wn

dx,

exists and

⎛

∫ (√

lim ⎝

n→∞

p

α1 p∗

Ω

wn

)α1 (√ p

β1 p∗

)β1 ⎞ p1∗ wn

⎠

1

dx ≥ c0 A¯ p∗ .

Therefore, u0 and v0 satisfy d¯ + ≥

(

1 p

−

1

)∫

α2 + β2

Ω

h(x)|u0 |α2 |v0 |β2 dx + λ

(

1 p

−

1

)∫

p∗

Furthermore, for every (u, v ) ∈ E \{(0, 0)} with au,v = ∥(u, v )∥p − Ru,v > 0 satisfying ∥(u, v )∥p −

∫ Ω

h(x)|u|α2 |v|β2 dx − λ

∫ Ω

Ω

|u0 |α1 |v0 |β1 dx + λ

∫ Ω

(

1 p

h(x)|u|α2 |v|β2 dx − λ ∗

−

∫ Ω

1

)

p∗

∗

p c0 A¯ .

(3.12)

|u|α1 |v|β1 dx > 0. We can take

|u|α1 |v|β1 dx + Rpu,v B − λRpu,v A¯ < 0, hence

⏐ ⏐α2 ⏐ ⏐β2 ⏐⏐( √ √ )⏐⏐p ∫ √ √ ⏐⏐ ⏐ ⏐ ⏐ ⏐⏐ ⏐ α β α β ⏐⏐ u + Ru,v p p∗1 Uε,a , v + Ru,v p p∗1 Uε,a ⏐⏐ − h(x) ⏐u + Ru,v p p∗1 Uε,a ⏐ ⏐v + Ru,v p p∗1 Uε,a ⏐ dx Ω ∫ ⏐ ⏐α1 ⏐ ⏐β1 √ √ ⏐ ⏐ ⏐ ⏐ − λ ⏐u + Ru,v p αp∗1 Uε,a ⏐ ⏐v + Ru,v p βp∗1 Uε,a ⏐ dx Ω ⏐⏐(√ √ )⏐⏐p ⏐⏐ ⏐⏐ = ∥(u, v )∥p + Rpu,v ⏐⏐ p αp∗1 Uε,a , p βp∗1 Uε,a ⏐⏐ ∫ (√ √ ) p β1 p α1 |∆u|p−2 ∆u∆U + pRu,v |∆v|p−2 ∆v ∆Uε,a dx ε,a + p∗ p∗ Ω

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Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

∫

⏐ ⏐

h(x) ⏐u + Ru,v

−

Ω(∫

−λ

Ω

√ p

α1 p∗

⏐α2 ⏐ ⏐β2 √ ⏐ ⏐ ⏐ β ⏐v + Ru,v p p∗1 Uε,a ⏐ dx ∫ (√ )α1 (√ )β1 ∗

15

Uε,a ⏐

|u|α1 |v|β1 dx + Rpu,v

p

Ω

∗

α1

p

U p∗ ε,a

)

β1

dx + o(1)

U p∗ ε,a

= au,v + Rpu,v B − λRpu,v A¯ + o(1) < 0, for ε > 0 small. We can choose cε,u,v with 0 < cε,u,v < Ru,v satisfying ⏐⏐( ⏐α2 ⏐ ⏐β2 ⏐ √ √ )⏐⏐p ∫ √ √ ⏐⏐ ⏐⏐ ⏐ ⏐ ⏐ ⏐ α β α β ⏐⏐ u + cε,u,v p p∗1 Uε,a , v + cε,u,v p p∗1 Uε,a ⏐⏐ − h(x) ⏐u + cε,u,v p p∗1 Uε,a ⏐ ⏐v + cε,u,v p p∗1 Uε,a ⏐ dx Ω ∫ ⏐ ⏐α1 ⏐ ⏐β1 √ √ ⏐ ⏐ ⏐ ⏐ − λ ⏐u + cε,u,v p αp∗1 Uε,a ⏐ ⏐v + cε,u,v p βp∗1 Uε,a ⏐ dx = 0,

(3.13)

Ω

which implies that

(

√ p

u + cε,u,v

α1 p∗

√ Uε,a , v + cε,u,v

p

β1 p∗

) ∈ Λ.

Uε,a

Note that au,v > 0, let cu,v > 0 be the unique such that ∗

cup,v B − λcup,v A¯ = −au,v . Next, obviously cu,v >

( ) B λA¯

1 p∗ −p

p∗

p

. By (3.13), we obtain 0 = au,v + cε,u,v B − λcε,u,v A¯ + o(1), and thus

cε,u,v → cu,v as ε → 0,

(3.14)

which gives

⏐⏐( √ )⏐⏐p √ ⏐⏐ ⏐⏐ p α β ⏐⏐ u + cε,u,v p p∗1 Uε,a , v + cε,u,v p p∗1 Uε,a ⏐⏐ = ∥(u, v )∥p + cu,v B + o(1) ( ) ∗p p ( ) ∗p p −p ¯ α1 , β1 ) p∗ −p . > cup,v B > λBA¯ B = λ1 p −p S( Consequently,

⏐⏐( √ )⏐⏐ ( ) 1 √ p ⏐⏐ ⏐⏐ α β ∗ ¯ α1 , β1 ) p(p∗ −p) ⏐⏐ u + cε,u,v p p∗1 Uε,a , v + cε,u,v p p∗1 Uε,a ⏐⏐ > λ1 p −p S( (( ) ) p∗∗ p(p −p) ) ∗1 ( ( ) β∗1 ( ) α∗1 p − p p p α1 2 −β2 = H(λ). > λ(pp−α + αβ1 S ∗ −α −β ) β 2 2 1 1 (

(2.8) leads to u + cε,u,v

√ p

α1 p∗

Uε,a , v + cε,u,v

√ p

β1 p∗

Uε,a

)

∈ Λ− . Since d¯ = d¯ + , we get

√ ( ) √ ≤ J u + cε,u,v p αp∗1 Uε,a , v + cε,u,v p βp∗1 Uε,a ⏐α2 ⏐ ⏐β2 ⏐ √ ( )∫ √ ⏐ ⏐ ⏐ ⏐ α β 1 = 1p − α +β h(x) ⏐u + cε,u,v p p∗1 Uε,a ⏐ ⏐v + cε,u,v p p∗1 Uε,a ⏐ dx 2 2 Ω ⏐α1 ⏐ ⏐β1 √ ( )∫ ⏐ √ ⏐ ⏐ ⏐ ⏐ α β + λ 1p − p1∗ ⏐u + cε,u,v p p∗1 Uε,a ⏐ ⏐v + cε,u,v p p∗1 Uε,a ⏐ dx ( )Ω∫ ( )∫ ( 1 = 1p − α +β h(x)|u|α2 |v|β2 dx + λ 1p − p1∗ |u|α1 |v|β1 dx + λ 1p −

d¯ +

2

2

Ω

Ω

)

1 p∗

p∗

cε,u,v A¯ + o(1),

that is d¯ + ≤

(

1 p

−

)∫

1

α2 + β2

α2

Ω

β2

h(x)|u| |v| dx + λ

(

1

−

p

)∫

1 p∗

Ω

α1

β1

|u| |v| dx + λ

(

1 p

−

1

)

p∗

∗

cup,v A¯ .

(3.15)

It follows from (3.12) and (3.15) that d¯ + =

(

1 p

−

)∫

1

α2 + β2

α2

Ω

β2

h(x)|u0 | |v0 | dx + λ

(

1 p

−

1 p∗

)∫

α1

Ω

β1

|u0 | |v0 | dx + λ

(

1 p

−

1 p∗

)

∗

p c0 A¯ .

(3.16)

It can be derived that (u0 , v0 ) is the local extremal function pair of the following functional:

(

1 p

−

1

α 2 + β2

)∫ Ω

h(x)|u|α2 |v|β2 dx + λ

(

1 p

−

1 p∗

)∫ Ω

|u|α1 |v|β1 dx + λ

(

1 p

−

1 p∗

)

∗

cup,v A¯ .

(3.17)

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

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Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

For the functional cu,v , let (ϕ, ψ ) ∈ E and take the estimated function of cu,v as in a small neighborhood of t = 0,

g(t) := c(u0 +t ϕ,v0 +t ψ ) that is

∗ (g(t))(p B − λ (g(t))p A¯

= − ∥(u0 + t ϕ, v0 + t ψ )∥ − p

∫

α2

Ω

β2

h(x)|u0 + t ϕ| |v0 + t ψ| dx − λ

∫

α1

Ω

)

β1

|u0 + t ϕ| |v0 + t ψ| dx .

a0 > 0 gives g(t) exists and g(0) = c0 . By Claim 1 and Lebesgue Dominated Convergence Theorem,

∫ lim

Ω

h(x)|u0 + t ϕ|α2 |v0 + t ψ|β2 dx −

∫

Ω

h(x)|u0 |α2 |v0 |β2 dx

t

t →0

∫

h(x) α2 |u0 |α2 −2 |v0 |β2 u0 ϕ + β2 |u0 |α2 |v0 |β2 −2 v0 ψ dx,

(

= Ω

)

and thus ∗ −1 g(t)−g(0) t

limt →0 {B g p−1 (t) + g p−1 (0) − p∗ λA¯ [g(0) + θ (g(t) − g(0))]p ∗ ∗ (g(t))p B − λ (g(t))p A¯ − [g(0)]p B + λ[g(0)]p A¯ = limt →0 t ( ∫

(

)

}

= limt →0 − 1t ∥(u0 + t ϕ, v0 + t ψ )∥p − h(x)|u0 + t ϕ|α2 |v0 + t ψ|β2 dx Ω ) ∫ ∫ ∫ α1 β1 α2 β2 α1 β1 p − λ |u0 + t ϕ| |v0 + t ψ| dx − ∥(u0 , v0 )∥ + h(x)|u0 | |v0 | dx + λ |u0 | |v0 | dx Ω ∫ Ω Ω ( ( ) =− p |∆u0 |p−2 ∆u0 ∆ϕ + |∆v0 |p−2 ∆v0 ∆ψ dx Ω ∫ ( ) − h(x) α2 |u0 |α2 −2 u0 |v0 |β2 ϕ + β2 |u0 |α2 |v0 |β2 −2 v0 ψ dx Ω∫ ) ( ) α1 −2 β1 α1 β1 −2 −λ α1 |u0 | u0 |v0 | ϕ + β1 |u0 | |v0 | v0 ψ dx Ω∫ ∫ ∫ := −p H(u0 , v0 , ϕ, ψ )dx + h(x)G2 (u0 , v0 , ϕ, ψ )dx + λ G1 (u0 , v0 , ϕ, ψ )dx, Ω

Ω

Ω

′

which indicates that g (0) exists and g ′ (0)

=

−1 p−1 p∗ −1 ¯ pc0 B−λp∗ c0 A

∫

( ∫ p Ω

H(u0 , v0 , ϕ, ψ )dx −

)

∫ Ω

h(x)G2 (u0 , v0 , ϕ, ψ )dx

G1 (u0 , v0 , ϕ, ψ )dx .

−λ Ω

In view of (3.17) we obtain d

((

dt (

+λ

1 p

1 p

− −

)∫

1

α2 +β2 1 p∗

)

α2

Ω

β2

h(x)|u0 + t ϕ| |v0 + t ψ| dx + λ

(

1 p

−

)

∗ (g(t))p A¯ |t =0 = 0,

1 p∗

)∫ Ω

|u0 + t ϕ|α1 |v0 + t ψ|β1 dx

that is

(

1 p

−

α2 +β2

( ·

)∫

1

Ω

−1

h(x)G2 (u0 , v0 , ϕ, ψ )dx + λ

( ∫ p

p−1 p∗ −1 ¯ pc0 B−λp∗ c0 A

Ω

(

H(u0 , v0 , ϕ, ψ )dx −

1 p

−

∫ Ω

1 p∗

)∫ Ω

G1 (u0 , v0 , ϕ, ψ )dx + +λ

h(x)G2 (u0 , v0 , ϕ, ψ )dx − λ

∫ Ω

(

1 p

−

1 p∗

)

p∗ −1

p∗ c0

A¯

))

G1 (u0 , v0 , ϕ, ψ )dx

(3.18)

= 0.

Denote Gi (u0 , v0 ) =

H(u0 , v0 ) =

√ p

α1 p∗

√ p

α1 p∗

√ αi −2

αi |u0 |

βi

u0 |v0 | +

p

β1 p∗

√ | ∆ u0 |

p−2

∆u0 ∆Uε,a +

p

βi |u0 |αi |v0 |βi −2 v0 , i = 1, 2,

β1 p∗

|∆v0 |p−2 ∆v0 ∆Uε,a ,

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

17

and c¯ε := cε,u0 ,v0 = c0 + δε . From (3.14), δε → 0. According to (3.13), we get

⏐⏐( ⏐⏐

0 = ⏐⏐ u0 + c¯ε

√ p

α1 p∗

Uε,a , v0 + c¯ε

√ p

β1 p∗

)⏐⏐p ⏐⏐

∫

⏐ ⏐

Uε,a ⏐⏐ −

Ω ∫ ⏐ ⏐α1 ⏐ ⏐β1 √ √ ⏐ ⏐ ⏐ ⏐ α β − λ ⏐u0 + c¯ε p p∗1 Uε,a ⏐ ⏐v0 + c¯ε p p∗1 Uε,a ⏐ dx

h(x) ⏐u0 + c¯ε

√ p

α1 p∗

⏐α2 ⏐ ⏐β2 √ ⏐ ⏐ ⏐ β ⏐v0 + c¯ε p p∗1 Uε,a ⏐ dx

Uε,a ⏐

Ω

⏐⏐p ⏐⏐ √ √ ⏐⏐ ⏐⏐ = ∥(u0 , v0 )∥p + c¯εp ⏐⏐( p αp∗1 Uε,a , p βp∗1 Uε,a )⏐⏐ ∫ ∫ (√ √ ) p−2 p−2 p β1 p α1 ∆U | ∆ u | ∆ u + ∆ U | ∆ v | ∆ v dx − h(x)|u0 |α2 |v0 |β2 dx + pc¯ε ε, a 0 ε, a 0 0 0 ∗ ∗ p p Ω Ω ) (∫ ∫ N −2p h(x)c0 G2 (u0 , v0 )k(x)U ∗ (x − a)dx − λ |u0 |α1 |v0 |β1 dx −ε p Ω Ω ∫ ( ∫ (√ √ )α1 (√ )β1 ) √ α1 −1 β1 α1 β1 −1 α1 p∗ p β1 p β1 p α1 U p dx − λ α u c − λ¯cε U U v + β u v c U dx 1 0 ε, a ε, a ε, a 1 0 ε, a ∗ ∗ ∗ ∗ 0 0 0 0 p p p p Ω ∫ (Ω (√ )α1 −1 (√ )β1 α −1 α −1 β β β p α1 −λ α1 u0 c0 1 Uε,1a c0 1 p p∗1 Uε,1a p∗ Ω ) (√ )β1 −1 ( N −2p ) (√ )α1 α1 β1 −1 β1 −1 α1 p β1 p α1 U β v c U dx + o ε p + c0 1 0 ∗ ∗ ε,a ε,a 0 p p ∫ ∫ ( ) p p∗ ¯ p p∗ ¯ = − c0 − λc0 A + c¯ε B − λ¯cε A + pc0 H(u0 , v0 )dx − λc0 G1 (u0 , v0 )Uε,a dx Ω Ω ) (∫ N −2p ∗ h(x)c0 G2 (u0 , v0 )k(x)U (x − a)dx −ε p Ω ( ∫ (√ )α1 −1 (√ )β1 (√ )α1 (√ )β1 −1 ) ( N −2p ) p∗ −1 p∗ −1 α1 p β1 p β1 p p α1 α1 u0 − λc0 Uε,a + β v dx + o ε p 1 0 ∗ ∗ ∗ ∗ p p p p Ω

which yields

(

p−1

pc0

p∗ −1

B − λp∗ c0

∫

)

A¯ + o(1) (−δε )

H(u0 , v0 )dx − c0

= pc0

Ω∫ p∗ −1

− λc0

Ω

p∗ −1 Uε,a

(

α1 u0

∫

h(x)G2 (u0 , v0 )Uε,a dx − λc0

Ω (√ )α1 −1 (√ )β1

α1

p

p

p∗

β1

+ β1

p∗

∫

G1 (u0 , v0 )Uε,a dx

(√Ω )α1 p

α1 p∗

v0

(√ )β1 −1 ) p

β1 p∗

(

dx + o ε

N −2p p

)

.

By (3.18), we have

( ∫

c0

(−δε ) =

p−1

pc0

p∗ −1

B − λp∗ c0

p

A¯

Ω

p∗ −1

c0

−

p−1

pc0

p∗ −1

B − λp∗ c0

A¯

λ

∫ Ω

H(u0 , v0 )dx −

⎛ p∗ −1 Uε, a

⎝α1 u0

∫

(√ p

Ω

h(x)G2 (u0 , v0 )Uε,a dx − λ

α1 p∗

)α1 −1 (√ p

β1 p∗

)β1 + β1

∫ Ω

G1 (u0 , v0 )Uε,a dx

(√ p

α1 p∗

)α1

(√ v0

p

)

β1 p∗

)β1 −1 ⎞ ⎠ dx

( N −2p ) +o ε p ( )∫ ( )∫ 1 1 1 1 − h(x)G (u , v )U dx + λ − G1 (u0 , v0 )Uε,a dx 2 0 0 ε, a ∗ p α2 +β2 p p Ω Ω ( ) = c0 ∗ λ 1p − p1∗ p∗ c0p −1 A¯ ⎛ ⎞ (√ )α1 −1 (√ )β1 (√ )α1 (√ )β1 −1 ∫ p∗ −1 c0 α β α β ∗ 1 1 1 1 p ⎠ dx − p−1 λ Uε,p a−1 ⎝α1 u0 p ∗ + β1 p ∗ v0 p ∗ p∗ −1 ¯ p p∗ p p pc0 B − λp∗ c0 A Ω ( N −2p ) +o ε p . (

Also, δε = O ε

N −2p p

(3.19)

)

.

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

18

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

a0 > 0 leads to p−1

pc0

p∗ −1

B − λp∗ c0

(

Note that u0 + c¯ε

√ p

(

J u0 + c¯ε

α1 p∗

√ p

(

p

A¯ =

c0

p

c0 − λ

Uε,a , v0 + c¯ε

α1 p∗

√ p

Uε,a , v0 + c¯ε

β1 p∗

√ p

p∗ p

p∗

Uε,a

β1 p∗

)

c0 A¯

)

Uε,a

<

p ( c0

p

p∗

)

c0 − λc0 A¯ = −

p c0

a0 < 0.

∈ Λ, it follows from (3.16) and (3.19) that

)

⏐α2 ⏐ ⏐β2 √ ⏐ ⏐ ⏐ β ⏐v0 + c¯ε p p∗1 Uε,a ⏐ dx Ω ⏐α1 ⏐ ⏐β1 √ ( )∫ ⏐ √ ⏐ ⏐ ⏐ ⏐ α β 1 1 + λ p − p∗ ⏐u0 + c¯ε p p∗1 Uε,a ⏐ ⏐v0 + c¯ε p p∗1 Uε,a ⏐ dx Ω ( )∫ )∫ ( N −2p 1 1 1 1 − = p − α +β h(x)|u0 |α2 |v0 |β2 dx + ε p h(x)c0 G2 (u0 , v0 )k(x)U ∗ (x − a)dx p α2 +β2 2 2 Ω Ω ( ( )∫ ) ∗ |u0 |α1 |v0 |β1 dx + λ 1p − p1∗ c¯εp A¯ + λ 1p − p1∗ Ω ( ) ∫ 1 1 + λ p − p∗ c¯ε G1 (u0 , v0 )Uε,a dx Ω ( ) ∗ ∫ ( (√ )α1 −1 (√ )β1 (√ )α1 (√ )β1 −1 ) ∗ p β1 p β1 p α1 + λ 1p − p1∗ c¯εp −1 α1 p αp∗1 u + β v0 Uε,p a−1 dx 0 1 p∗ p∗ p∗ Ω ( )∫ ) ∫ ( 1 1 1 h(x)|u0 |α2 |v0 |β2 dx + 1p − α +β c0 h(x)G2 (u0 , v0 )Uε,a dx = p − α +β 2 2 2 2 Ω Ω ∫ ( ( )( ∗ ) ( ) )∫ ∗ p 1 1 1 1 α ∗ p −1 |u0 | 1 |v0 |β1 dx + c0 λ 1p − p1∗ + λ p − p∗ c0 + p c0 δε A¯ + λ p − p∗ G1 (u0 , v0 )Uε,a dx Ω Ω ) ( ∫ ) ∗ (√ )α1 −1 (√ )β1 (√ )α1 (√ )β1 −1 ( ∗ α p β1 p β1 α1 p αp∗1 u0 + β1 p p∗1 v0 Uε,p a−1 dx + λ 1p − p1∗ c0p −1 p∗ p∗ =

(

1 p

−

)∫

1

⏐ ⏐

h(x) ⏐u0 + c¯ε

α2 +β2

√ p

α1 p∗

Uε,a ⏐

Ω

( N −2p ) +o ε p ( = d¯ + + 1p −

) ∫

1

c0

α2 +β2

Ω

h(x)G2 (u0 , v0 )Uε,a dx

( )∫ δε A¯ + c0 λ 1p − p1∗ G1 (u0 , v0 )Uε,a dx Ω ( ∫ ( ) ∗ (√ )α1 −1 (√ )β1 (√ )α1 (√ )β1 −1 ) ∗ p −1 α1 α 1 1 p β1 p β1 p + λ p − p∗ c0 α1 u0 + β1 p p∗1 v0 Uε,p a−1 dx p∗ p∗ p∗

+λ

(

1 p

−

1 p∗

)

p∗ −1

p∗ c0

Ω

( N −2p ) +o ε p p∗ −1 ( ) ( ) ∗ ∗ c0 1 1 1 1 ∗ p −1 ¯ ∗ p −1 ¯ ¯ = d+ + λ p − p∗ p c0 δε A + λ p − p∗ p c0 A p−1 p∗ −1 ¯ pc0 B − λp∗ c0 A ) ∫ ( (√ )α1 −1 (√ )β1 (√ )α1 (√ )β1 −1 ∗ p − 1 p β1 ·λ α1 u0 p αp∗1 + β1 p αp∗1 Uε,a dx v0 p βp∗1 p∗ Ω

+λ

(

)

p∗ −1

¯ −δε ) A( ( (√ ) ∫ ( ) ∗ (√ )α1 (√ )β1 −1 ) ∗ α1 −1 (√ )β1 p −1 α1 α 1 1 p β1 p β1 p + λ p − p∗ c0 α1 u0 + β1 p p∗1 v0 Uε,p a−1 dx p∗ p∗ p∗ 1 p

−

1 p∗

p∗ c0

Ω

( N −2p ) +o ε p p−1 ( ) ∗ pc B = d¯ + + λ 1p − p1∗ c0p −1 p−1 0 p∗ −1 ¯ pc0 B − λp∗ c0 A ∫ ( (√ )α1 −1 (√ )β1 (√ )α1 (√ )β1 −1 ) ∗ ( N −2p ) p −1 α β α p 1 · α1 u0 p p∗1 + β1 p p∗1 v0 p βp∗1 Uε,a dx + o ε p < d¯ + , p∗ Ω

which is a contradiction. This concludes the proof of Claim 2. Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

19

Claim 3. (u0 , v0 ) is a solution of (1.1). For (ϕ, ψ ) ∈ E, ε > 0, define

Φ = (u0 + εϕ )+ ,

Ψ = (v0 + εψ )+ ,

Ω1 = {x|u0 + εϕ > 0},

(Φ , Ψ ) ∈ E ,

Ω2 = {x|v0 + εψ > 0},

Ω − = Ω − ( Ω1 ∩ Ω2 ) ,

so Φ (x) = (u0 + εϕ )|Ω1 , Ψ (x) = (v0 + εψ )|Ω2 . Applying Claim 2 and replacing (ϕ, ψ ) with (Φ , Ψ ) in (3.2), we get

∫ ( α −1 β α β −1 |∆u0 |p−2 ∆u0 ∆Φ + |∆v0 |p−2 ∆v0 ∆Ψ − pλ∗ α1 u0 1 v0 1 Φ − pλ∗ β1 u0 1 v0 1 Ψ Ω ) α2 −1 β2 α2 β2 −1 h(x) Φ − Ψ dx v v α u β u − α h(x) 2 2 0 0 0 0 α2 +β2 ∫2 +β2 ( β α −1 = |∆u0 |p−2 ∆u0 ∆(u0 + εϕ ) + |∆v0 |p−1 ∆v0 ∆(v0 + εψ ) − pλ∗ α1 u0 1 (u0 + εϕ )v0 1 Ω1 ∩Ω2 ) α −1 β α β −1 α β −1 − pλ∗ β1 u0 1 v0 1 (v0 + εψ ) − α h(x) α u 2 (u0 + εϕ )v0 2 − α h(x) β u 2 v 2 (v0 + εψ ) dx +β2 2 0 +β2 2 0 0 2 2 (∫ ∫ )( α −1 β = − |∆u0 |p−2 ∆u0 ∆(u0 + εϕ ) + |∆v0 |p−2 ∆v0 ∆(v0 + εψ ) − pλ∗ α1 u0 1 (u0 + εϕ )v0 1 − Ω Ω ) α β −1 α −1 β α β −1 − pλ∗ β1 u0 1 v0 1 (v0 + εψ ) − α h(x) α2 u0 2 (u0 + εϕ )v0 2 − α h(x) β2 u0 2 v0 2 (v0 + εψ ) dx 2 +β2 2 +β2 ∫ ∫ ∫ ( = ∥(u0 , v0 )∥p − h(x)|u0 |α2 |v0 |β2 dx − λ |u0 |α1 |v0 |β1 dx + ε |∆u0 |p−2 ∆u0 ∆ϕ

0

≤

Ω Ω Ω ) α −1 β α β −1 α2 −1 β2 α2 β2 −1 h(x) α u ϕv − β u v ψ dx + |∆v0 |p−2 ∆v0 ∆ψ − pλ∗ α1 u0 1 ϕv0 1 − pλ∗ β1 u0 1 v0 1 ψ − α h(x) 2 2 0 0 0 0 α2 +β2 2 +β2 ∫ ( α −1 β − |∆u0 |p−2 ∆u0 ∆(u0 + εϕ ) + |∆v0 |p−2 ∆v0 ∆(v0 + εψ ) − pλ∗ α1 u0 1 (u0 + εϕ )v0 1 Ω− ) α β −1 α2 −1 β2 α2 β2 −1 h(x) − v ( v + εψ ) dx − pλ∗ β1 u0 1 v0 1 (v0 + εψ ) − α h(x) α u (u + εϕ ) v β u 0 2 0 2 0 0 0 0 α2 +β2 2 +β2 ∫ ( ) h(x) := ε H(u0 , v0 , ϕ, ψ ) − pλ∗ G1 (u0 , v0 , ϕ, ψ ) − α +β G2 (u0 , v0 , ϕ, ψ ) dx 2 2 Ω ∫ ( α −1 β − |∆u0 |p−2 ∆u0 ∆(u0 + εϕ ) + |∆v0 |p−2 ∆v0 ∆(v0 + εψ ) − pλ∗ α1 u0 1 (u0 + εϕ )v0 1 Ω− ) α β −1 α −1 β α β −1 − pλ∗ β1 u0 1 v0 1 (v0 + εψ ) − α h(x) α2 u0 2 (u0 + εϕ )v0 2 − α h(x) β2 u0 2 v0 2 (v0 + εψ ) dx +β +β 2 2 2 2 ∫ ( ∫ ) h(x) λ ≤ε H(u0 , v0 , ϕ, ψ ) − p∗ G1 (u0 , v0 , ϕ, ψ ) − α +β G2 (u0 , v0 , ϕ, ψ ) dx − ε H(u0 , v0 , ϕ, ψ )dx. 2

Ω

2

Ω−

Obviously, limε→0 meas(Ω ) = limε→0 meas{x|u0 + εϕ ≤ 0, v0 + εψ ≤ 0, x ∈ Ω } = 0, and thus −

∫

( Ω−

) |∆u0 |p−2 ∆u0 ∆ϕ + |∆v0 |p−2 ∆v0 ∆ψ dx → 0.

Dividing by ε and letting ε → 0, we get

∫ ( Ω

H(u0 , v0 , ϕ, ψ ) −

λ

h(x)

p

α2 + β2

G (u , v , ϕ, ψ ) − ∗ 1 0 0

)

G2 (u0 , v0 , ϕ, ψ ) dx ≥ 0.

Consequently, (u0 , v0 ) is a solution of the problem (1.1). Note that (un , vn ) ⇀ (u0 , v0 ) weakly in E, the weak lower semicontinuity of ∥ · ∥ gives ∥(u0 , v0 )∥ ≤ lim infn→∞ ∥(un , vn )∥ ≤ H0 . Furthermore, the gap structure of Λ and Claim 2 lead to (u0 , v0 ) ∈ Λ+ . In addition, by J(un , vn ) → inf(u,v )∈Λ+ J(u, v ), we deduce that inf

(u,v )∈Λ+

J(u, v ) ≥

(

1 p

−

1 p∗

)

∥(u0 , v0 )∥p −

(

1

α 2 + β2

−

1 p∗

)∫ Ω

h(x)|u0 |α2 |v0 |β2 dx = J(u0 , v0 ),

which implies that J(u0 , v0 ) = inf(u,v )∈Λ+ J(u, v ). This completes the proof of Theorem 3.1. Theorem 3.2. Assume that λ ∈ (0, λ∗ ). Then the problem (1.1) has a solution (U0 , V0 ) ∈ E satisfying ∥(U0 , V0 )∥ > H(λ) > H0 . Proof. We apply the Ekeland’s variational principle to say that there exists a minimizing sequence {(Un , Vn )} ⊂ Λ− such that (i) J(Un , Vn ) < infΛ− J(u, v ) + 1n ; (ii) J(u, v ) ≥ J(Un , Vn ) − 1n ∥(u − Un , v − Vn )∥, ∀(u, v ) ∈ Λ− . Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

20

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

Clearly, {(Un , Vn )} is bounded in E, passing to a subsequence if necessary, there exists (U0 , V0 ) such that

{

(Un , Vn ) ⇀ (U0 , V0 ), (Un , Vn ) → (U0 , V0 ), (Un (x), Vn (x)) → (U0 (x), V0 (x)),

w eakly strongly a.e.

(√

as n → ∞. Write (Un , Vn ) = (U0 , V0 ) +

p

α1 p∗

√

Wn ,

p

β1 p∗

in E , in Lα2 (Ω ) × Lβ2 (Ω ), in Ω ,

Wn

)

with Wn ⇀ 0 weakly in E.

In the following, we will prove that lim inf (p∗ − p)∥(Un , Vn )∥p > (p∗ − α2 − β2 )

)

(

∫

n→∞

Ω

h(x)|U0 |α2 |V0 |β2 dx.

Otherwise, we assume that lim inf (p − p)∥(Un , Vn )∥ ∗

(

p

n→∞

)

= (p − α2 − β2 ) ∗

∫ Ω

h(x)|U0 |α2 |V0 |β2 dx.

∫

Since (Un , Vn ) ∈ Λ− and ∥(U , V )∥ > H(λ) > H0 , ∀(U , V ) ∈ Λ− , then

⎛ (p − p)∥(Un , Vn )∥

⎜

n→∞

h(x)|U0 |α2 |V0 |β2 dx ̸ = 0 and thus

⎞ ∗

lim inf ⎜ ⎝

Ω

(p∗ − α2 − β2 )

∫ Ω

p

h(x)|Un |α2 |Vn |β2 dx

⎟ ⎟ = 1, ⎠

which implies that there exists a subsequence {Unk , Vnk } of {Un , Vn } such that (p∗ − p)∥(Unk , Vnk )∥p (p∗ − α2 − β2 )

∫ Ω

h(x)|Unk |α2 |Vnk |β2 dx

→ 1,

k → ∞.

Hence,

∥(Unk , Vnk )∥p →

p∗ − α2 − β2 p∗ − p

∫ Ω

h(x)|U0 |α2 |V0 |β2 dx,

∫ ∫ ∫ p − α2 − β2 h(x)|U0 |α2 |V0 |β2 dx. λ |Unk |α1 |Vnk |β1 dx = ∥(Unk , Vnk )∥p − h(x)|Unk |α2 |Vnk |β2 dx → p∗ − p Ω Ω Ω For all λ ∈ (0, λ∗ ), we obtain 0

∗

∗

< D(λ)H p (λ) < D(λ)∥(Unk , Vnk )∥p p ( ∗ ) p−αp∗ −−β ( ) p−α2 −β2 p −p 2 2 < p∗ −α −β ∗ p −α −β (∫ 2

2

2

→ 0,

∥(Unk , Vnk )∥

p∗ −α −β p p−α 2−β 2 2 2

−λ p ) p−αp∗ −−β

2

Ω

(k → ∞),

h(x)|Unk |α2 |Vnk |β2 dx

2

2

∫ Ω

|Unk |α1 |Vnk |β1 dx

this is a contradiction. Therefore, according to the proof of Theorem 3.1, we have 1

∫

( ) h(x) α2 |U0 |α2 −2 U0 |V0 |β2 ϕ + β2 |U0 |α2 |V0 |β2 −2 V0 ψ dx α 2 + β2 Ω ∫ ∫ ( ) ( ) ≤ |∆U0 |p−2 ∆U0 ∆ϕ + |∆V0 |p−2 ∆V0 ∆ψ dx − pλ∗ α1 |U0 |α1 −2 U0 |V0 |β1 ϕ + β1 |U0 |α1 |V0 |β1 −2 V0 ψ dx, Ω

Ω

for every (ϕ, ψ ) ∈ E, ϕ ≥ 0, ψ ≥ 0. By letting ϕ = U0 , ψ = V0 we have that ∥(U0 , V0 )∥ − p

λ

∫ Ω

∫ Ω

h(x)|U0 |α2 |V0 |β2 dx −

|U0 |α1 |V0 |β1 dx ≥ 0. The remainder is to prove that (U0 , V0 ) ∈ Λ. That is to show that a¯ 0 = ∥(U0 , V0 )∥ − p

∫

∫ h(x)|U0 | |V0 | dx − λ |U0 |α1 |V0 |β1 dx = 0. α2

Ω

β2

Ω

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

21 p∗

p

We assume for a contradiction that a¯ 0 > 0. Thus there exists a unique number c¯0 > 0 such that c¯0 B − λ¯c0 A¯ = −¯a0 . For J(Un , Vn ) → d¯ − with (Un , Vn ) ∈ Λ− (⊂ Λ), we get

( )∫ ( )∫ 1 = J(Un , Vn ) = 1p − α +β h(x)|Un |α2 |Vn |β2 dx + λ 1p − p1∗ |Un |α1 |Vn |β1 dx 2 2 Ω Ω ( ( )∫ )∫ 1 = 1p − α +β h(x)|U0 |α2 |V0 |β2 dx + λ 1p − p1∗ |U0 |α1 |V0 |β1 dx 2 2 Ω Ω ∫ ( ) (√ )α1 (√ )β1 p β1 p α1 W + λ 1p − p1∗ dx + o(1), Wn n p∗ p∗

d¯ − + o(1)

Ω

and

∫

∫ = ∥(Un , Vn )∥ − h(x)|Un | |Vn | dx − λ |Un |α1 |Vn |β1 dx Ω√ ∫ (√ Ω ⏐⏐(√ )⏐⏐p )α1 (√ )β1 ⏐⏐ ⏐⏐ p β1 p α1 W dx + o(1) W = a¯ 0 + ⏐⏐ p αp∗1 Wn , p βp∗1 Wn ⏐⏐ − λ n n ∗ ∗ p p Ω p (∫ (√ ) ∫ )α1 (√ )β1 (√ )α1 (√ )β1 p∗ p β1 p β1 p α1 W p α1 W ¯ α1 , β1 ) ≥ a¯ 0 + S( dx dx + o(1), Wn −λ Wn n n p∗ p∗ p∗ p∗ p

0

α2

β2

Ω

Ω

which gives

∫ (√ p

lim

n→∞

α1 p∗

Ω

)α1 (√ p

Wn

β1 p∗

)β1 dx,

Wn

exists and

⎛

∫ (√ p

lim ⎝

n→∞

α1 p∗

Ω

)α1 (√ p

Wn

)β1 ⎞ p1∗ 1 ⎠ dx ≥ c¯0 A¯ p∗ , Wn ∗

β1 p

and thus d¯ − ≥

(

1 p

−

)∫

1

α2 + β2

α2

Ω

β2

h(x)|U0 | |V0 | dx + λ

(

1

−

p

1

)∫

p∗

α1

Ω

β1

|U0 | |V0 | dx + λ

(

1 p

−

)

1 p∗

∗

p c¯0 A¯ .

(3.20)

Following the proof of (3.15) in Theorem 3.1, we have d¯ − ≤

(

1 p

−

)∫

1

α2 + β2

α2

Ω

β2

h(x)|u| |v| dx + λ

(

1 p

−

1

)∫

p∗

α1

Ω

β1

|u| |v| dx + λ

(

1 p

−

)

1

∗

p∗

cup,v A¯ .

1

1

(3.21)

Now, combining (3.20) and (3.21), we derive that d¯ − =

(

1 p

−

)∫

1

α2 + β2

α2

Ω

β2

h(x)|U0 | |V0 | dx + λ

(

1 p

−

1

)∫

p∗

α1

Ω

β1

|U0 | |V0 | dx + λ

(

p

−

p∗

)

∗

p c¯0 A¯ ,

and for every (ϕ, ψ ) ∈ E, d

((

dt (

+λ

1 p

1 p

− −

1

)∫

α2 +β2 1 p∗

)

Ω

h(x)|U0 + t ϕ|α2 |V0 + t ψ|β2 dx + λ

(

1 p

−

1 p∗

)

∗ (G(t))p A¯ |t =0 = 0,

)∫ Ω

|U0 + t ϕ|α1 |V0 + t ψ|β1 dx

where ∗ (G(t))(p B − λ (G(t))p A¯

= − ∥(U0 + t ϕ, V0 + t ψ )∥ − p

∫ Ω

) ∫ α1 β1 h(x)|U0 + t ϕ| |V0 + t ψ| dx − λ |U0 + t ϕ| |V0 + t ψ| dx . α2

β2

Ω

In the following, we show that (U0 , V0 ) ∈ Λ− . Claim 4. There exists ε1 > 0 independent of R such that √ ⏐β1 ⏐ ⏐α ⏐⏐ √α ⏐ β ⏐ ⏐ 1 ⏐u0 (x)+R p p∗1 Uε,a (x)⏐ ⏐⏐v0 (x)+R p p∗1 Uε,a (x)⏐⏐ ⎜ ⎝ ⏐⏐⏐⏐( √ )⏐⏐p∗ √ ⏐⏐ Ω ⏐⏐ u0 (x)+R p α∗1 Uε,a (x),v0 (x)+R p β∗1 Uε,a (x) ⏐⏐ ⏐⏐ ⏐⏐ p p

⎛

⎞

∫

⎟ ⎠ dx ≥ min

⎧∫ ⎪ ⎨

|u0 |α1 |v0 |β1 dx Ω

∗

p ⎪ ⎩ (2∥(u0 ,v0 )∥p +1) p

,

A¯ p∗ 2(2(B+1)) p

⎫ ⎪ ⎬

,

(3.22)

⎪ ⎭

for every ε ∈ (0, ε1 ), ∀R ≥ 1. Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

22

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

The following two cases will be considered. Firstly, for R ≥ 1 with Rp B ≤ ∥(u0 , v0 )∥p , we get

∫ ⏐ ⏐α1 ⏐ ⏐β1 √ √ ⏐ ⏐ ⏐ ⏐ α β ⏐u0 + R p p∗1 Uε,a ⏐ ⏐v0 + R p p∗1 Uε,a ⏐ dx Ω∫ ∫ ( ∫ (√ )α (√ )β √ 1 1 p∗ α1 α1 β1 p∗ p β1 p Uε,a dx + α1 |u0 |α1 −1 |v0 |β1 R p αp∗1 Uε,a = |u0 | |v0 | dx + R p∗ p∗ Ω Ω ∫ ( √ ) ( √ )αΩ1 −1 ( √ )β1 β + β1 |u0 |α1 |v0 |β1 −1 R p βp∗1 Uε,a dx + α1 u0 R p αp∗1 Uε,a R p p∗1 Uε,a ( √ )α1 ( √ )β1Ω−1 ) ( N −2p ) α β p 1 1 + β1 R p p∗ Uε,a v0 R p∗ Uε,a dx + o ε p ∫ ( ( N −2p )) ∫ > |u0 |α1 |v0 |β1 dx, = |u0 |α1 |v0 |β1 dx + Rp A¯ + O ε p Ω

Ω

and

⏐⏐( √ )⏐⏐p √ ⏐⏐ ⏐⏐ α β ⏐⏐ u0 + R p p∗1 Uε,a , v0 + R p p∗1 Uε,a ⏐⏐ ⏐⏐(√ √ )⏐⏐p ⏐⏐ ⏐⏐ = ∥(u0 , v0 )∥p + Rp ⏐⏐ p αp∗1 Uε,a , p βp∗1 Uε,a ⏐⏐ ∫ ( √ ) √ +p |∆u0 |p−2 ∆u0 R p αp∗1 ∆Uε,a + |∆v0 |p−2 ∆v0 R p βp∗1 ∆Uε,a dx ) ∫Ω ( ( √ )p−1 ( ( √ )p−1 ( ( N −2p ) )p−1 )p−1 α1 p β1 p +p ∆u0 R p∗ + ∆v0 R p∗ dx + o ε p ∆Uε,a ∆Uε,a Ω ( N −2p ) = ∥(u0 , v0 )∥p + Rp B + O ε p < 2∥(u0 , v0 )∥p + 1, for ε > 0 small, hence

∫

∫ ⎛ ⏐ ⏐α1 ⏐ ⏐β1 ⎞ √ √ ⏐ ⏐ ⏐ ⏐ α1 p β1 p |u0 |α1 |v0 |β1 dx ⏐u (x) + R p∗ Uε,a (x)⏐ ⏐v0 (x) + R p∗ Uε,a (x)⏐ ⎟ ⎜ 0 Ω . ⎝ ⏐⏐( √ )⏐⏐p∗ ⎠ dx > √ p∗ ⏐⏐ ⏐⏐ p + 1) p Ω p β1 p α1 U (2 ∥ (u , v ) ∥ 0 0 u (x) + R (x) , v (x) + R U (x) ⏐⏐ 0 ⏐⏐ 0 p∗ ε,a p∗ ε,a

Furthermore, for R ≥ 1 with Rp B > ∥(u0 , v0 )∥p , we get

∫ ⏐ ⏐α1 ⏐ ⏐β1 √ √ ⏐ ⏐ ⏐ ⏐ α β ⏐u0 + R p p∗1 Uε,a ⏐ ⏐v0 + R p p∗1 Uε,a ⏐ dx Ω∫ ∫ ( ∫ (√ )α (√ )β √ 1 1 ∗ p∗ p β1 p α1 U dx + = |u0 |α1 |v0 |β1 dx + Rp α1 |u0 |α1 −1 R p αp∗1 Uε,a |v0 |β1 ∗ ∗ ε, a p p Ω Ω ∫ ( √ ) ( √ )αΩ1 −1 ( √ )β1 α1 β β1 −1 p β1 α1 p R p∗ Uε,a dx + + β1 |u0 | |v0 | α1 u0 R p∗ Uε,a R p p∗1 Uε,a ( √ )α1 ( √ )β1Ω−1 ) ( N −2p ) + β1 R p αp∗1 Uε,a v0 R p βp∗1 Uε,a dx + Rγ o ε p (γ ∈ (0, p∗ )) ∫ ( ( )) N −2p ∗ ∗ ∗ ¯ = |u0 |α1 |v0 |β1 dx + 12 Rp A¯ + Rp A2 + O ε p > 12 Rp A¯ , Ω

and

√ ⏐⏐( )⏐⏐p √ ⏐⏐ ⏐⏐ α β ⏐⏐ ⏐⏐ 1 1 p ⏐⏐ u0 + R p ∗ Uε,a , v0 + R ∗ Uε,a ⏐⏐ < 2Rp (B + 1) ⏐⏐ ⏐⏐ p p for ε > 0 small, and consequently,

∫

⎛ ⏐ ⏐α1 ⏐ ⏐β1 ⎞ √ √ ⏐ ⏐ ⏐ ⏐ α β 1 p∗ ¯ ⏐u0 (x) + R p p∗1 Uε,a (x)⏐ ⏐v0 (x) + R p p∗1 Uε,a (x)⏐ R A 1 A¯ ⎜ ⎟ 2 dx > = . ⎝ ⏐⏐( √ )⏐⏐p∗ ⎠ √ p∗ p∗ 2 ⏐⏐ ⏐⏐ α1 p (B + 1)) p Ω p p β1 p (2R (2(B + 1)) ⏐⏐ u0 (x) + R p∗ Uε,a (x), v0 (x) + R p∗ Uε,a (x) ⏐⏐

Hence for ε1 sufficiently small we conclude that (3.22) holds true for every ε ∈ (0, ε1 ) and R ≥ 1. Therefore there exist constants C > 0 and ε1 > 0 such that for all ε ∈ (0, ε1 ) and R ≥ 1,

∫

⎛ ⏐ ⏐α1 ⏐ ⏐β1 ⎞ √ √ ⏐ ⏐ ⏐ ⏐ α β ⏐u0 + R p p∗1 Uε,a ⏐ ⏐v0 + R p p∗1 Uε,a ⏐ ⎜ ⎟ ⎝ ⏐⏐( √ )⏐⏐p∗ ⎠ dx > C . √ ⏐⏐ ⏐⏐ α1 Ω p β1 p ⏐⏐ u0 + R p∗ Uε,a , v0 + R p∗ Uε,a ⏐⏐

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

23

Define

{

Σ1 = (u, v ) ∈ E \{(0, 0)} : ∥(u, v )∥ < t

+

(u, v )

(

)}

∥(u, v )∥

,

{ ( )} (u, v ) Σ2 = (u, v ) ∈ E \{(0, 0)} : ∥(u, v )∥ > t + . ∥(u, v )∥ It is easy to check that

{

Λ ⊂ Σ1 , Λ = (u, v ) ∈ E \{(0, 0)} : ∥(u, v )∥ = t +

−

(u, v )

(

+

)}

∥(u, v )∥

, E − Λ− = Σ1 ∪ Σ2 .

Claim 5. There exist ε2 with 0 < ε2 < ε1 and R0 > 1 such that

(

√ u0 + R 0

p

α1 p∗

√ Uε,a , v0 + R0

⎛

(

The definition of t + ⎝ ⏐⏐⏐⏐(

p

)

β1

Uε,a

p∗

∈ Σ2 , ∀ε ∈ (0, ε2 ).

√

α β u0 +R p ∗1 Uε,a ,v0 +R p ∗1 Uε,a p p

√

⎞

)

√ )⏐⏐ √ ⏐⏐ ⏐⏐ u0 +R p α∗1 Uε,a ,v0 +R p β∗1 Uε,a ⏐⏐ ⏐⏐ ⏐⏐ p p

λ

⎛

∫

⎞α1 ⎛

α u0 +R p ∗1 Uε,a p

√

√ )⏐⏐ ⎠ √ ⏐⏐ ⏐⏐ ⏐⏐ u0 +R p α∗1 Uε,a ,v0 +R p β∗1 Uε,a ⏐⏐ ⏐⏐ ⏐⏐ p p

⎝ ⏐⏐(

Ω

⎛

⎛

(

= ⎝t + ⎝ ⏐⏐⏐⏐(

⎠ leads to

√

α β u0 +R p ∗1 Uε,a ,v0 +R p ∗1 Uε,a p p

√

)

⎞β1 √ β v0 +R p p∗1 Uε,a ⎝ ⏐⏐( √ )⏐⏐ ⎠ √ ⏐⏐ ⏐⏐ ⏐⏐ u0 +R p α∗1 Uε,a ,v0 +R p β∗1 Uε,a ⏐⏐ ⏐⏐ ⏐⏐ p p

⎞⎞p−p∗

⎛ Ω

√

√ )⏐⏐ √ ⏐⏐ ⏐⏐ u0 +R p α∗1 Uε,a ,v0 +R p β∗1 Uε,a ⏐⏐ ⏐⏐ ⏐⏐ p p

⎛

(

By Claim 4, we have that t + ⎝ ⏐⏐⏐⏐( u0 + R

√ p

α1 p∗

⎠ √

√

√ p

√

⎞β2 √ β v0 +R p p∗1 Uε,a ⎝ ⏐⏐( √ )⏐⏐ ⎠ √ ⏐⏐ ⏐⏐ ⏐⏐ u0 +R p α∗1 Uε,a ,v0 +R p β∗1 Uε,a ⏐⏐ ⏐⏐ ⏐⏐ p p

α β u0 +R p ∗1 Uε,a ,v0 +R p ∗1 Uε,a p p

Uε,a , v0 + R

√

α β u0 +R p ∗1 Uε,a ,v0 +R p ∗1 Uε,a p p

)

⎞⎞α2 +β2 −p∗

√ )⏐⏐ ⎠⎠ √ ⏐⏐ ⏐⏐ u0 +R p α∗1 Uε,a ,v0 +R p β∗1 Uε,a ⏐⏐ ⏐⏐ ⏐⏐ p p

β1 p∗

Uε,a

dx.

⎞

)

√ )⏐⏐ √ ⏐⏐ ⏐⏐ u0 +R p α∗1 Uε,a ,v0 +R p β∗1 Uε,a ⏐⏐ ⏐⏐ ⏐⏐ p p

⎛ (

(

⎞α2 ⎛

α u0 +R p ∗1 Uε,a p

h(x) ⎝ ⏐⏐⏐⏐(

⎛

− ⎝t + ⎝ ⏐⏐⏐⏐(

√ )⏐⏐ ⎠⎠ √ ⏐⏐ ⏐⏐ u0 +R p α∗1 Uε,a ,v0 +R p β∗1 Uε,a ⏐⏐ ⏐⏐ ⏐⏐ p p

∫

⎛

dx

⎠ is uniformly bounded from above, namely

) ⎞

√ t + ⎝ ⏐⏐( )⏐⏐ ⎠ ≤ C , √ ⏐⏐ ⏐⏐ α β ⏐⏐ u0 + R p p∗1 Uε,a , v0 + R p p∗1 Uε,a ⏐⏐ ⎜

⎟

∀ε ∈ (0, ε1 ), ∀ R ≥ 1.

Moreover, for small enough ε > 0 and large enough R, we have

⏐⏐( √ )⏐⏐p ( ( N −2p )) √ ⏐⏐ ⏐⏐ α β > 12 Rp B > C p . ⏐⏐ u0 + R p p∗1 Uε,a , v0 + R p p∗1 Uε,a ⏐⏐ = ∥(u0 , v0 )∥p + 12 Rp B + Rp 2B + Rp o ε p Therefore there exist R0 and ε2 with R0 > 1 and 0 < ε2 < ε1 such that

⎛ ( √ ) ⎞ √ √ ⏐⏐( )⏐⏐ α β √ ⏐⏐ ⏐⏐ u0 + R p p∗1 Uε,a , v0 + R p p∗1 Uε,a α β ⏐⏐ ⏐⏐ ⎜ ⎟ 1 1 √ )⏐⏐ ⎠ , ⏐⏐ u0 + R p ∗ Uε,a , v0 + R p ∗ Uε,a ⏐⏐ > t + ⎝ ⏐⏐( √ ⏐⏐ ⏐⏐ ⏐⏐ ⏐⏐ α1 p p p β1 p ⏐⏐ u0 + R p∗ Uε,a , v0 + R p∗ Uε,a ⏐⏐ (

for all ε ∈ (0, ε2 ) and R ≥ R0 , which implies that u0 + R

√ p

α1 p∗

Uε,a , v0 + R

√ p

β1 p∗

Uε,a

)

∈ Σ2 , ∀ε ∈ (0, ε2 ), ∀R ≥ R0 .

Claim 6. There exists ε3 > 0 such that

( J

√ u0 + tR0

p

α1 p∗

√ Uε,a , v0 + tR0

p

β1 p∗

) Uε,a

< J(u0 , v0 ) +

2 N

N

S¯ 2p

( ) N −2p2p 1

λ

,

∀ t ∈ [0, 1], ∀ε ∈ (0, ε3 ).

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

24

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

Since (u0 , v0 ) is a solution (1.1), we have

(

J u0 + tR0

√ p

α1 p∗

Uε,a , v0 + tR0

√ p

β1 p∗

Uε,a

)

∫ ⏐ ⏐⏐( ⏐α1 √ )⏐⏐p √ √ ⏐ ⏐⏐ ⏐⏐ ⏐ α = 1p ⏐⏐ u0 + tR0 p αp∗1 Uε,a , v0 + tR0 p βp∗1 Uε,a ⏐⏐ − pλ∗ ⏐u0 + tR0 p p∗1 Uε,a ⏐ Ω ∫ ⏐ ⏐ ⏐β1 ⏐α2 ⏐ ⏐β2 √ √ √ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ β α β 1 h(x) ⏐u0 + tR0 p p∗1 Uε,a ⏐ ⏐v0 + tR0 p p∗1 Uε,a ⏐ dx ⏐v0 + tR0 p p∗1 Uε,a ⏐ dx − α2 +β 2 ⏐⏐(√ Ω √ )⏐⏐p ∫ ( √ ⏐⏐ ⏐⏐ = 1p ∥(u0 , v0 )∥p + 1p (tR0 )p ⏐⏐ p αp∗1 Uε,a , p βp∗1 Uε,a ⏐⏐ + |∆u0 |p−2 ∆u0 tR0 p αp∗1 ∆Uε,a Ω √ (√ )p−1 (√ )p−1 ) α1 p−2 p−1 p−1 p β1 p β1 p + |∆v0 | ∆v0 tR0 p∗ ∆Uε,a + ∆u0 (tR0 ) ∆Uε,a + ∆v0 (tR0 ) ∆Uε,a dx p∗ p∗ ∫ ∫ ( ∫ (√ )α (√ )β √ 1 1 ∗ α −1 β p∗ p β1 p α1 Uε,a dx − pλ∗ α1 u0 1 tR0 p αp∗1 Uε,a v0 1 − pλ∗ |u0 |α1 |v0 |β1 dx − pλ∗ (tR0 )p p∗ p∗ Ω √ (Ω √ )α1 −1 ( √ )β1 Ω α β −1 β + β1 u0 1 v0 1 tR0 p βp∗1 Uε,a + α1 u0 tR0 p αp∗1 Uε,a tR0 p p∗1 Uε,a ∫ ( √ )α1 ( √ )β1 −1 ) 1 + β1 tR0 p αp∗1 Uε,a v0 tR0 p βp∗1 Uε,a dx − α +β h(x)|u0 |α2 |v0 |β2 dx 2 2 Ω ∫ (√ )α2 (√ )β2 ( N −2p ) α +β α 1 p β1 − α +β h(x)(tR0 )α2 +β2 p p∗1 Uε,2a 2 dx + o ε p ∗ p 2 2 Ω ( ∫ )β1 ( √ )α1 −1 ( √ ∗ ∗ β = J(u0 , v0 ) + 1p (tR0 )p B − pλ∗ (tR0 )p A¯ − pλ∗ Uε,p a−1 α1 u0 tR0 p αp∗1 tR0 p p∗1 Uε,a ( √ )α1 ( √ )β1 −1Ω) ( N −2p ) α1 p β1 p + β1 v0 tR0 p∗ Uε,a tR0 p∗ Uε,a dx + o ε p (√ )α1 −1 (√ )β1 N −2p ∗ ∗ p β1 u0 (a)E ε p = J(u0 , v0 ) + 1p (tR0 )p B − pλ∗ (tR0 )p A¯ − pλ∗ (tR0 )p −1 α1 p αp∗1 p∗ (√ )α1 (√ )β1 −1 ( N −2p ) N −2p ∗ p β1 − pλ∗ (tR0 )p −1 β1 p αp∗1 v0 (a)E ε p + o ε p , p∗ ∫

∗ −1

(U ∗ )p

where E =

dx.

RN

Define

χ (s)

∗ λ Asp p∗ N −2p

¯ (

= Bp sp − − pλ∗ E ε

p

α1

(√ )α1 −1 (√ )β1 p

We can estimate χ (tR0 ). Set s0 =

α1

p

p∗

( ) B λA¯

1 p∗ −p

β1 p∗

u0 (a) + β1

(√ )α1 (√ )β1 −1 p

α1

p

p∗

β1 p∗

) ∗ v0 (a) sp −1 , ∀ s ≥ 0.

and sε > 0 satisfying χ (sε ) = maxs≥0 χ (s). Obviously, sε → s0 as ε → 0. Write

sε = s0 + lε with lε → 0. Necessarily, χ ′ (sε ) = 0, that is ∗ −1

¯ 0 + lε )p B(s0 + lε )p−1(− λA(s =

λ

E p∗

ε

N −2p p

α1

(√ )α1 −1 (√ )β1 p

α1 p∗

p

β1 p∗

u0 (a) + β1

(√ )α1 (√ )β1 −1 p

α1 p∗

p

β1 p∗

)

∗ −2

v0 (a) (p∗ − 1)(s0 + lε )p

,

which leads to

) ¯ 0 − λAs ( (√ ) ) (√ )α1 (√ )β1 −1 α1 −1 (√ )β1 α1 α1 p β1 p β1 p p α1 u0 (a) + β1 v0 (a) (p∗ − 1), p∗ p∗ p∗ p∗ ∗

(

p+1−p∗

¯ 0 + lε ) − Bs B(s0 + lε )p+1−p − λA(s 0

(

=

(

so lε = O ε

N −2p p

E p∗

λ

ε

N −2p p

)

)

. Hence

χ (tR0 ) ¯ p∗ ≤ χ (sε ) = Bp spε − pλ∗ As ) ( (√ ε) (√ )α1 (√ )β1 −1 α1 −1 (√ )β1 N −2p ∗ p β1 p β1 p α1 − pλ∗ E ε p u (a) + β v (a) spε −1 α1 p αp∗1 0 1 0 ∗ ∗ ∗ p p p ( ( N −2p )) ( ∗ ( N −2p )) ∗ ¯ = Bp sp0 + ps0 lε + o ε p − λp∗A sp0 + p∗ sp0 −1 lε + o ε p Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

25

) ( (√ ) (√ )α1 (√ )β1 −1 α1 −1 (√ )β1 N −2p p β1 p β1 p α1 u (a) + β v (a) − pλ∗ E ε p α1 p αp∗1 0 1 0 ∗ ∗ ∗ p p p ( ∗ ( N −2p )) p −1 p∗ −2 ∗ · s0 + (p − 1)s0 lε + o ε p N ( ) N −2p ¯ α1 , β1 ) 2p 1 2p = N2 S( ( (λ√ ) ) (√ )α1 (√ )β1 −1 ( N −2p ) α1 −1 (√ )β1 N −2p p∗ −1 p β1 p β1 p α1 − pλ∗ E ε p u (a) + β +o ε p α1 p αp∗1 v (a) s0 . 0 1 0 p∗ p∗ p∗ Therefore, for all t ∈ [0, 1], we get

(

J u0 + tR0

√ p

α1 p∗

Uε,a , v0 + tR0

√ p

β1

U p∗ ε,a N −2p

)

N ( ) 2 ¯ ≤ J(u0 , v0 ) + α1 , β1 ) 2p λ1 2p ( N S( ) ( N −2p ) (√ )α1 −1 (√ )β1 (√ )α1 (√ )β1 −1 N −2p p∗ −1 p β1 p β1 p α1 +o ε p − pλ∗ E ε p α1 p αp∗1 u (a) + β v (a) s0 . 0 1 0 p∗ p∗ p∗

Consequently, there exists ε3 with 0 < ε3 < ε2 such that

(

J u0 + tR0

√ p

α1 p∗

Uε,a , v0 + tR0

√ p

β1

Uε,a

p∗

)

N ( ) N −2p ¯ α1 , β1 ) 2p 1 2p < J(u0 , v0 ) + N2 S( ( ( √ ) λ ( √ )β ) (√ )α1 (√ )β1 −1 α1 −1 N −2p 1 p∗ −1 1 λ p β1 p β1 p α1 p − p p∗ E ε u (a) + β v (a) s0 α1 p αp∗1 0 1 0 ∗ ∗ ∗ p p p

for every ε ∈ (0, ε3 ) and ∀ t ∈ [0, 1]. In view of Claim 5, We know that

( (u0 , v0 ) ∈ Λ ⊂ Σ1 +

and

√ u0 + R0

p

α1 p∗

√ Uε,a , v0 + R0

p

β1 p∗

) Uε,a

∈ Σ2 .

By the proof of Theorem 3.1, there exists tε ∈ (0, 1) such that

(

√ u0 + t ε R 0

p

α1 p∗

√ Uε,a , v0 + tε R0

p

β1 p∗

) Uε,a

∈ Λ− .

Furthermore, Claim 6 leads to inf J < J(u0 , v0 ) + Λ−

2 N

N

¯ α1 , β1 ) 2p S(

( ) N −2p2p 1

λ

.

(3.23)

In addition, the fact that (Un , Vn ), (U0 , V0 ) ∈ Λ also gives 0

∫ ∫ = ∥(U0 , V0 )∥p − h(x)|U0 |α2 |V0 |β2 dx − λ |U0 |α1 |V0 |β1 dx ∫ (√ ⏐⏐(√ √ Ω )⏐⏐p )Ωα1 (√ )β1 ⏐⏐ ⏐⏐ p β1 p α1 W + ⏐⏐ p αp∗1 Wn , p βp∗1 Wn ⏐⏐ − λ W dx + o(1) n n ∗ ∗ p p ∫Ω (√ ⏐⏐(√ √ )⏐⏐p )α1 (√ )β1 ⏐⏐ ⏐⏐ p β1 p α1 W = ⏐⏐ p αp∗1 Wn , p βp∗1 Wn ⏐⏐ − λ Wn dx + o(1). n p∗ p∗

(3.24)

Ω

Lastly, we prove ⏐⏐(that )⏐⏐0p, V0 ) strongly in E by contradiction. Assuming that there exists a subsequence √ (Un , V√n ) → (U ⏐⏐ p α1 ⏐⏐ p β1 (Unk , Vnk ) with ⏐⏐ Wnk , p∗ Wnk ⏐⏐ ≥ C > 0 such that p∗

⏐β1 ⏐α1 ⏐⏐√ (¯ ) p∗p−∗ p ∫ ⏐√ ⏐ ⏐ α1 ⏐ ⏐ p Wn ⏐ ⏐⏐ p β1 Wn ⏐⏐ dx ≥ S(α1 , β1 ) + o(1) , ⏐ p∗ k ⏐ ⏐ p∗ k ⏐ λ Ω

and

√ ⏐⏐(√ )⏐⏐p (¯ )N ⏐⏐ ⏐⏐ β1 S(α1 , β1 ) 2p ⏐⏐ p α1 ⏐⏐ p Wnk , Wnk ⏐⏐ ≥ λ + o(1), ⏐⏐ ⏐⏐ ⏐⏐ p∗ p∗ λ

(3.25)

Please cite this article as: Y. Sang and Y. Ren, A critical p-biharmonic system with negative exponents, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.08.032.

26

Y. Sang and Y. Ren / Computers and Mathematics with Applications xxx (xxxx) xxx

thanks to (3.24). By (3.23) and (3.25), we derive that N ( ) N −2p ¯ α , β1 ) 2p 1 2p ) ⏐λ⏐( )∫ ( )⏐⏐p ( 1 1 1 1 ⏐ ⏐ ⏐ ⏐ Unk , Vnk − α +β − p∗ h(x)|Unk |α2 |Vnk |β2 dx > J(Unk , Vnk ) = p − p∗ 2 2 Ω √ ( ) ⏐⏐(√ )⏐⏐p ⏐⏐ ⏐⏐ = J(U0 , V0 ) + 1p − p1∗ ⏐⏐ p αp∗1 Wnk , p βp∗1 Wnk ⏐⏐ + o(1) √ ) ⏐⏐(√ )⏐⏐p ( ⏐⏐ ⏐⏐ ≥ infΛ J + 1p − p1∗ ⏐⏐ p αp∗1 Wnk , p βp∗1 Wnk ⏐⏐ + o(1) √ ) ⏐⏐(√ )⏐⏐p ( ⏐⏐ ⏐⏐ = J(U0 , V0 ) + 1p − p1∗ ⏐⏐ p αp∗1 Wnk , p βp∗1 Wnk ⏐⏐ + o(1) N ) (¯ ) 2p ( N ( ) N −2p ¯ α1 , β1 ) 2p 1 2p , + o(1) = J(U0 , V0 ) + N2 S( ≥ J(U0 , V0 ) + 1p − p1∗ λ S(α1λ,β1 ) λ

J(U0 , V0 ) +

2 S( 1 N

this is a contradiction. Lemma 2.3 leads to (U0 , V0 ) ∈ Λ− . It follows that infΛ− J

≤ J(U0 , V0 ) =

(

1 p

−

1 p∗

)

∥(U0 , V0 )∥ − p

(

1

α2 +β2

−

1 p∗

)∫

≤ lim infn→∞ J(Un , Vn ) = limn→∞ J(Un , Vn ) = infΛ− J ,

Ω

h(x)|U0 |α2 |V0 |β2 dx

and thus inf J = J(U0 , V0 ) = lim J(Un , Vn ), Λ−

n→∞

which yields

∥(U0 , V0 )∥p = lim ∥(Un , Vn )∥p . n→∞

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