一类藕合k-Hessian 系统非线性径向k-凸解的渐近行为 The Asymptotic Behavior of Nontrivial Radial k-Convex Solutions for a Class of Coupled k-Hessian System

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AdvancesinAppliedMathematics A^êÆ?Ð,2022,11(7),4979-4989

PublishedOnlineJuly2022inHans.//www.abtbus.com/journal/aam

https://doi.org/10.12677/aam.2022.117522

˜ aÍÜk-HessianXÚš‚5»•k-à)

ì C1•

 •••ÿÿÿ

∗

Ü “‰ŒÆ§êÆ†ÚOÆ§[‹=²

Â vFÏµ2022c625F¶¹^FÏµ2022c720F¶uÙFÏµ2022c727F

Á ‡

Ä uIþØÄ:½n§©Ì‡ïÄ˜aÍÜk-HessianXÚš‚5»•k-à)ìC1•"

' …c

Í Ük-HessianXÚ§š‚5»•k-à)§ìC1•§ØÄ:½n

TheAsymptoticBehaviorofNontrivial

Radial k-ConvexSolutionsforaClassof

Coupled k-HessianSystem

CunyanYue

∗

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Jun.25

,2021;accepted:Jul.20

,2022;published:Jul.27

,2022

Abstract

Basedontheﬁxed-pointtheoremincone,westudytheasymptoticbehaviorofnon-

trivialradial k-convexsolutionsforaclassofcoupled k-Hessiansystem.

∗ Email:yuecunyan@163.com

4979-4989.DOI:10.12677/aam.2022.117522

 •ÿ

Keywords

Coupled k-HessianSystem,NontrivialRadial k-ConvexSolution,Asymptotic

Behavior,Fixed-PointTheorem

 2022byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

1. 0

 ©Ì‡•Ä˜aÍÜk-HessianXÚ











( D

) = λ

( −u

)inΩ ,

( D

) = λ

( −u

)inΩ ,

( D

) = λ

( −u

)inΩ ,

= u

= ···= u

= 0on ∂Ω ,

(1 .1)

š ‚5»•k-à)•35ÚìC1•,Ù¥λ

´ ëê ,f

∈ C([0,+∞ ),[0,+∞ )),Ω={ x∈

: |x|≤1 }( N≥2), u

∈ C

( R

), D

´ ëY‡©¼ê u

 HessianÝ,S

( λ( D

))

´ 1kÐé¡õ‘ª ,´ HessianÝD

 ¤kk×kÌfª Ú,i= 1,2,···,n.

˜ „/,·‚½Â Xek-HessianŽf:

( λ( D

u )) =

1 ≤j

<...

≤ N

...λ

,k = 1, 2, ···,N.

A O/,k=1ž,k-HessianŽfò z•LaplaceŽfS

( λ( D

u ))=

i =1

=∆ u, •„[1,2];

 k=Nž,k-HessianŽfòz•Monge-Amp`ereŽfS

( λ( D

u ))=

i =1

=det( λ( D

u )),•

„ [3–7].

C c5,Laplace¯KÚMonge-Amp`ere¯K®2•A^uêÆ†A^êÆÌ‡©|.

k 'Laplace¯KÚMonge-Amp`ere¯K)•35!Ø•35!õ)5!•˜5ÚìC-

½ 5 ƒ'(J•„©z[1–7].AO/ ,32021c,¾{r[7]¥$^Iþ ØÄ:½ n

DOI:10.12677/aam.2022.1175224980 A^êÆ?Ð

 •ÿ

 Monge-Amp`ereXÚ











det( D

) = λ

( −u

)inΩ ,

det( D

) = λ

( −u

)inΩ ,

det( D

) = λ

( −u

)inΩ ,

= u

= ···= u

= 0on ∂Ω ,

š ‚5»•à)•35ÚìC1•.

k -Hessian•§´˜aš‚5‡©•§,3AÛÆ!6NåÆÚÙ¦A^Æ‰¥kX-

‡ A^.NõÆöÏLüNS“•{!þe)•{!C©•{!ØÄ:½n±9£Ä²¡•

{ Ãõk'k-Hessian• ¯K)•35!Ø•35!õ)5!•˜5ÚìC-½5`

D (J,•„©z[8–12].~X,32019c,¾{r[8]¥$^IþØÄ:½nk-Hessian

X Ú











( D

) = λ

( −u

)inΩ ,

( D

) = λ

( −u

)inΩ ,

= u

= 0on ∂Ω ,

š ‚5»•à)•35ÚìC1•.

É ©z[7]Ú[8]éu,©òÏLØÄ:½nïÄÍÜk-HessianXÚ(1.1)š‚5»•k-à

) •359ìC 1•.©Ì‡óŠ´é©z[7,8]í2.

2. ý•£

 !‰Ñ˜7 ‡ÚnÚÌ‡óä.

é ?¿ k= 1,2,···,N,½ Â8Ü

:= {ν∈R

: S

( ν) >0 ,1 ≤k≤N}⊂R

½ Â1.1.([13])Ω´R

¥ ˜‡k.m8,eé?¿x∈Ω,HessianÝ A •þ

,ν

, ···,ν

÷ v^‡(ν

,ν

, ···,ν

) ∈Γ

, K¡u( x) ∈C

(Ω) ´k- à¼ê.

Ú n2.1([14])v(r)∈C

[0 ,R) ´˜‡»•é¡¼ê…v

(0)= 0, K¼êu( |x|)= v( r) ∈

( B

), r= |x|, …

λ (D

u ) =











( v

( r) ,

( r)

,...,

( r)

,r ∈(0,R ),

( v

(0) ,v

(0) ,...,v

(0)) ,r= 0;

( λ( D

u )) =











k −1

N −1

( r)+



( r)



k −1

+ C

N −1



( r)



,r ∈(0,R ),

( v

(0))

,r = 0,

DOI:10.12677/aam.2022.1175224981 A^êÆ?Ð

 •ÿ

Ù ¥r=|x|=

i =1

, B

:= {x∈R

: |x|}, C

N !

k !(N −k )!

Ï LÚn3.1,·‚Œ±òk-HessianXÚ (1.1)=z•Xe~‡©>Š¯K













N −k



( r)





= λ

( C

k −1

N −1

)

− 1

N −1

( −u

( r)) ,0 1 ,



N −k



( r)





= λ

( C

k −1

N −1

)

− 1

N −1

( −u

( r)) ,0 1 ,



N −k



( r)





= λ

( C

k −1

N −1

)

− 1

N −1

( −u

( r)) ,0 1 ,

(0) = u

(0) = 0 ,i= 1 ,2 ,···,n.

(2 .1)

Š C†v

= −u

( i= 1 ,2 ,···,n), KŒò~‡©XÚ(2.1) =z•Xe~‡©XÚ













N −k



− v

( r)





= λ

( C

k −1

N −1

)

− 1

N −1

( v

( r)) ,0 1 ,



N −k



− v

( r)





= λ

( C

k −1

N −1

)

− 1

N −1

( v

( r)) ,0 1 ,



N −k



− v

( r)





= λ

( C

k −1

N −1

)

− 1

N −1

( v

( r)) ,0 1 ,

(0) = v

(0) = 0 ,i= 1 ,2 ,···,n.

(2 .2)

K (u

) = ( −v

, −v

) ´k-Hesian XÚ(1.1) »•)…=( v

) ´È©XÚ











( r) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ,

( r) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ,

( r) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ.

(2 .3)

 ˜‡).

½ Â¼ê˜mE=C[0,1],KU‰êkxk=max

0 ≤t≤1

| x(t)| ¤Banach˜m.½Â

P :=

x ∈E :x (t )≥0,t ∈[0, 1],x (t )≥θ kx k,t ∈[θ, 1−θ ]

⊂ E

´ Eþ˜‡I,Ù¥θ∈(0,

). w,, P´Eþ˜‡5I.

DOI:10.12677/aam.2022.1175224982 A^êÆ?Ð

 •ÿ

é ?¿ v∈P,·‚½ ÂŽfT

: P→E( i= 1 ,2 ,···,n) •

( T

v )(t ) =λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v( s)) ds



dτ,

( T

v )(t ) =λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v( s)) ds



dτ,

( T

v )(t ) =λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v( s)) ds



dτ.

(2 .4)

½ Â˜‡EÜŽf

= T

··· T

Ú n2.2([6])ŽfT

( i=1 ,2 ,,n) ´šKà¼ê, ÏddArzel` a½nŒ•T

: P→E( i=

1 ,2 ,···,n) ´ëYŽf. ?˜Ú/, dT½ÂŒ•T•´˜‡ëYŽf.

d ©z[5]Œ•,λ

= λ

= ···= λ

ž ,(v

, ···,v

) ∈C

[0 ,1] ×C

[0 ,1] ×···×C

[0 ,1]

|{z}

´ È©XÚ(2.3))…=(v

, ···,v

) ∈P\{0 }×P\{0 }×···×P\{0 }

|{z}

¿ …÷v

= T

, ···,v

= T

. ùL²ev

∈ P\{ 0} ´

 ˜‡ØÄ: ,@o ·‚½ Â

= T

, ···,v

= T

ž ,(v

, ···,v

) ∈C

[0 ,1] ×C

[0 ,1] ×···×C

[0 ,1]

{z}

´ È©XÚ(2.3)

 ˜‡).,˜•¡,e(v

, ···,v

) ∈C

[0 ,1] ×C

[0 ,1] ×···×C

[0 ,1]

|{z}

´ È©XÚ(2.3)˜

‡ ),Kv

´ ëYŽf

 ˜‡š"ØÄ: .

Ï d,‡ yÈ©XÚ(2.3)k˜ ‡),·‚•IyëYŽf

k ˜‡š"ØÄ:=Œ.

a q,·‚Œ±½ÂÙ¦EÜŽf,Xe:

= T

··· T

= T

··· T

= T

··· T

e ¡‰Ñ©Ì‡ïÄóä.

Ú n2.3([15])-Ω

Ú Ω

´ Banach˜mEþü‡k.m8,…0∈Ω,

Ω

⊂ Ω

, -

P :P ∩(

Ω

\ Ω

) →P´˜‡ëYŽf, Ù¥P´Eþ˜‡I. e

(i) kTxk≤kxk, ∀x∈P∩∂Ω

, …kTxk≥kxk, ∀x∈P∩∂Ω

;

(ii) kTxk≥kxk, ∀x∈P∩∂Ω

, …kTxk≤kxk, ∀x∈P∩∂Ω

K T3 P∩(

Ω

\ Ω

) ¥–k˜‡ØÄ:.

DOI:10.12677/aam.2022.1175224983 A^êÆ?Ð

 •ÿ

3. Ì‡(J

Ä k,éi= 1,2,···,n,·‚‰ÑXePÒ:

=lim

x →0

f (x )

∞

=lim

x →∞

f (x )

½ n3.1bf

∈ C([0,+∞ ),[0,+∞ )),…f

=0 ,f

∞

= ∞, Kéλ

> 0,k -HessianXÚ(1.1)

• 3˜‡š‚5»•à)u= (u

, ···,u

) ÷vlim

→ 0

k u

k =∞ ,Ù¥i= 1,2,··· ,n.

y ²:éi= 1,2,···,n,·‚•Iy²éλ

> 0,È©XÚ(2.3)•3˜‡)v = (v

, ···,v

)

÷ vlim

→ 0

k v

k =∞ =Œ.Ï•f

= 0, K•3˜‡~êr

> 0¦é?¿ε> 0,k

( v

) ≤εv

, ∀0≤v

≤ r

( v

) ≤εv

, ∀0≤v

≤ r

( v

) ≤εv

, ∀0≤v

≤ r

Ù ¥ε÷v

( λ

··· λ

)

≤ 1.

(3 .1)

Ï d,év

∈ P∩ ∂Ω

, i= 1 ,2 ,···,n,Ω

= {x∈R

: kxk≤r

} ,k

( T

)( t) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ

≤ λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ

≤ λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

εv

( s) ds



dτ

≤ λ



k −1

N −1



k v



k −N

N −1



dτ

≤ λ

k v

k ,t∈ [0,1],

Ó nŒ,

( T

)( t) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ

≤ λ

k v

k ,t∈ [0,1],

( T

)( t) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ

≤ λ

k v

k ,t∈ [0,1].

DOI:10.12677/aam.2022.1175224984 A^êÆ?Ð

 •ÿ

 d

½ ÂÚ(3.1)Œ•,

k =k T

··· T

≤ λ

k T

··· T

≤ (λ

)

k T

··· T

≤ (λ

··· λ

)

k v

≤k v

k ,v

∈ P∩ ∂Ω

(3 .2)

Ï •f

∞

= ∞, K•3˜‡~êR

) ¦é?¿~êη>0, k

( v

) ≥ηv

, ∀v

≥ R

( v

) ≥ηv

, ∀v

≥ R

( v

) ≥ηv

, ∀v

≥ R

Ù ¥η÷v

( λ

··· λ

)



ηk

k −1

N −1



(1 −θ)

n (k −N )

n (2k +N −1)

≥ 1.

(3 .3)

- R

> max{R

} ,Kév

∈ P∩ ∂Ω

, i= 1 ,2 ,···,n,Ω

= {x∈R

: kxk≤R

} ,k

( t) ≥θkv

k =θR

≥ R

,t ∈[θ, 1−θ ].

Ï d,év

∈ P∩ ∂Ω

, k

( T

)( t) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ

≥ λ

1 −θ

kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds

dτ

≥ λ

1 −θ

kτ

k −N

N −1

( C

k −1

N −1

)

− 1

ηv

( s) ds

dτ

≥ λ

1 −θ

k (1−θ )

k −N

N −1

( C

k −1

N −1

)

− 1

η (θ kv

k )

dτ



ηk

k −1

N −1



(1 −θ)

k −N

2 k+ N−1

k v

k ,t∈ [0,1],

DOI:10.12677/aam.2022.1175224985 A^êÆ?Ð

 •ÿ

Ó nŒ,

( T

)( t) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ

≥



ηk

k −1

N −1



(1 −θ)

k −N

2 k+ N−1

k v

k ,t∈ [0,1],

( T

)( t) = λ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ

≥



ηk

k −1

N −1



(1 −θ)

k −N

2 k+ N−1

k v

k ,t∈ [0,1].

 d

½ ÂÚ(3.3)Œ•,

k =k T

··· T

≥



ηk

k −1

N −1



(1 −θ)

k −N

2 k+ N−1

k T

··· T

≥ (λ

)



ηk

k −1

N −1



(1 −θ)

2( k−N)

2(2 k+ N−1)

k T

··· T

≥ (λ

··· λ

)



ηk

k −1

N −1



(1 −θ)

n (k −N )

n (2k +N −1)

k v

≥k v

k ,v

∈ P∩ ∂Ω

(3 .4)

( ÜÚn2.3Œ•,Žf

k ˜‡ØÄ:v

∈ P∩ (

Ω

\ Ω

). ½ÂT

= v

, ···,T

= v

, K

( v

, ···,v

Ó n,·‚•Œ±ÑŽf

k ˜‡ØÄ:v

∈ P∩ (

Ω

\ Ω

), ···, Žf

k ˜‡ØÄ:

∈ P∩ (

Ω

\ Ω

 e5,·‚y²λ

→ 0

ž ,kv

k→ +∞ ,i=1,2,··· ,n.b•3~êβ

> 0ÚS

→ 0

¦ 

k v

k≤ β

( m= 1 ,2 ,···) .

K S{kv

k} •3˜‡Âñu~êα

(0 ≤α

≤ β

) fS, •{Bå„, ·‚b{kv

Â ñuα

(i) eα

iλ

k} >

. -

= max {f

( v

) ,r

≤k v

k≤ R

} ,

= max {f

( v

) ,r

≤k v

k≤ R

} ,

= max {f

( v

) ,r

≤k v

k≤ R

} .

DOI:10.12677/aam.2022.1175224986 A^êÆ?Ð

 •ÿ

d T

= v

, ···,T

= v

Œ •

1 m



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ k

k v

1 λ

1 m

≤



kτ

k −N

N −1

( C

k −1

N −1

)

− 1



k v

1 λ

1 m

≤

2 k−N

k v

1 λ

1 m

≤

2 kF

| 2k− N| α

2 m



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ k

k v

2 λ

2 m

≤

2 kF

| 2k− N| α



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ k

k v

nλ

≤

2 kF

| 2k− N| α

(3 .5)

(3.5) L²λ

→ +∞ (m→ +∞ ),ù†λ

→ 0

g ñ,i= 1,2,···,n.

(ii) eα

iλ

k}→ 0.df

= 0 Œ•, é?¿δ>0,

• 3˜‡~êr

> 0,¦

( v

2 λ

2 m

) ≤δv

2 λ

2 m

, ∀0≤v

2 λ

2 m

≤ r

( v

3 λ

3 m

) ≤δv

3 λ

3 m

, ∀0≤v

3 λ

3 m

≤ r

( v

1 λ

1 m

) ≤δv

1 λ

1 m

, ∀0≤v

1 λ

1 m

≤ r

Ï d,év

iλ

∈ P∩ ∂Ω

, kv

iλ

k =r

, k

1 m



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ k

k v

1 λ

1 m

≤



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

δv

2 λ

2 m



k v

1 λ

1 m

≤

kδ

k v

| 2k− N| r

2 m



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ k

k v

2 λ

2 m

≤

kδ

k v

| 2k− N| r

(3 .6)

DOI:10.12677/aam.2022.1175224987 A^êÆ?Ð

 •ÿ



kτ

k −N

N −1

( C

k −1

N −1

)

− 1

( v

( s)) ds



dτ k

k v

nλ

≤

kδ

k v

| 2k− N| r

(3.6) L²λ

→ +∞ (m→ +∞ ),ù†λ

→ 0

g ñ,i= 1,2,···,n.

n þŒ•,λ

→ 0

ž ,kv

k→ +∞ ,i= 1,2,··· ,n.

a qu½n 3.1y²,·‚kXe½n .

½ n3.2bf

∈ C([0,+∞ ),[0,+∞ )),…f

= ∞, f

∞

=0, Ké¤kλ

> 0,k -

Hessian •§(1.1) •3˜‡š‚5»•à)u=( u

, ···,u

) ÷vlim

→ 0

k u

k =0,Ù¥

i = 1, 2, ···,n .

ë •©z

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