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

doi:10.12677/AAM.2022.117522

设为首页加入收藏期刊导航网站地图

期刊菜单

文章导航

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

PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam

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

˜aÍÜk-HessianXÚš‚5»•k-à)

ìC1•

•••ÿÿÿ

∗

Ü“‰ŒÆ§êÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c625F¶¹^FÏµ2022c720F¶uÙFÏµ2022c727F

Á‡

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

'…c

ÍÜk-HessianXÚ§š‚5»•k-à)§ìC1•§ØÄ:½n

TheAsymptoticBehaviorofNontrivial

Radialk-ConvexSolutionsforaClassof

Coupledk-HessianSystem

CunyanYue

∗

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Jun.25

,2021;accepted:Jul.20

,2022;published:Jul.27

,2022

Abstract

Basedontheﬁxed-pointtheoremincone,westudytheasymptoticbehaviorofnon-

trivialradialk-convexsolutionsforaclassofcoupledk-Hessiansystem.

∗Email:yuecunyan@163.com

©ÙÚ^:•ÿ.˜aÍÜk-HessianXÚš‚5»•k-à)ìC1•[J].A^êÆ?Ð,2022,11(7):

4979-4989.DOI:10.12677/aam.2022.117522

•ÿ

Keywords

Coupledk-HessianSystem,NontrivialRadialk-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











) = λ

(−u

)inΩ,

) = λ

(−u

)inΩ,

) = 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

),D

´ëY‡©¼êu

HessianÝ,S

(λ(D

))

´1kÐé¡õ‘ª,´HessianÝD

¤kk×kÌfªÚ,i= 1,2,···,n.

˜„/,·‚½ÂXek-HessianŽf:

(λ(D

u)) =

1≤j

<...<j

≤N

...λ

,k= 1,2,···,N.

AO/,k=1ž,k-HessianŽfòz•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].

Cc5,Laplace¯KÚMonge-Amp`ere¯K®2•A^uêÆ†A^êÆÌ‡©|.

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

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

DOI:10.12677/aam.2022.1175224980A^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‰¥k X-

‡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

XÚ











) = λ

(−u

)inΩ,

) = 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

)•359ìC1•.©Ì‡óŠ´é©z[7,8]í2.

2.ý•£

!‰Ñ˜7‡ÚnÚÌ‡óä.

é?¿k= 1,2,···,N,½Â8Ü

:= {ν∈R

: S

(ν) >0,1 ≤k≤N}⊂R

½Â1.1.([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.

Ún2.1([14])v(r)∈C

[0,R)´˜‡»•é¡¼ê…v

(0)= 0,K¼êu(|x|)= v(r)∈

),r= |x|<R,…

λ(D

u) =











(r),

(r)

,...,

(r)

,r∈(0,R),

(0),v

(0),...,v

(0)),r= 0;

(λ(D

u)) =











k−1

N−1

(r)+



(r)



k−1

N−1



(r)



,r∈(0,R),

(0))

,r= 0,

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

•ÿ

Ù¥r= |x|=

i=1

:= {x∈R

: |x|<R},C

k!(N−k)!

ÏLÚn3.1,·‚Œ±òk-HessianXÚ(1.1)=z•Xe~‡©>Š¯K













N−k



(r)





= λ

k−1

N−1

)

−1

N−1

(−u

(r)),0 <r<1,



N−k



(r)





= λ

k−1

N−1

)

−1

N−1

(−u

(r)),0 <r<1,



N−k



(r)





= λ

k−1

N−1

)

−1

N−1

(−u

(r)),0 <r<1,

(0) = u

(0) = 0,i= 1,2,···,n.

(2.1)

ŠC†v

= −u

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













N−k



−v

(r)





= λ

k−1

N−1

)

−1

N−1

(r)),0 <r<1,



N−k



−v

(r)





= λ

k−1

N−1

)

−1

N−1

(r)),0 <r<1,



N−k



−v

(r)





= λ

k−1

N−1

)

−1

N−1

(r)),0 <r<1,

(0) = v

(0) = 0,i= 1,2,···,n.

(2.2)

K(u

) = (−v

,−v

)´k-HesianXÚ(1.1)»•)…=(v

)´È©XÚ











(r) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ,

(r) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ,

(r) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ.

(2.3)

˜‡).

½Â¼ê˜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) ≥θkxk,t∈[θ,1−θ]

⊂E

´Eþ˜‡I,Ù¥θ∈(0,

).w,,P´Eþ˜‡5I.

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

•ÿ

é?¿v∈P,·‚½ÂŽfT

: P→E(i= 1,2,···,n)•

v)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(v(s))ds



dτ,

v)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(v(s))ds



dτ,

v)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(v(s))ds



dτ.

(2.4)

½Â˜‡EÜŽf

= T

···T

Ún2.2([6])ŽfT

(i=1,2,,n)´šK à¼ê,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

,···,v

.ùL²ev

∈P\{0}´

˜‡ØÄ:,@o·‚½Â

= 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:.

Ïd,‡yÈ©XÚ(2.3)k˜‡),·‚•IyëYŽf

k˜‡š"ØÄ:=Œ.

aq,·‚Œ±½ÂÙ¦EÜŽf,Xe:

= T

···T

= T

···T

= T

···T

e¡‰Ñ©Ì‡ïÄóä.

Ún2.3([15])-Ω

ÚΩ

´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

KT3P∩(

Ω

\Ω

)¥–k˜‡ØÄ:.

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

•ÿ

3.Ì‡(J

Äk,éi= 1,2,···,n,·‚‰ÑXePÒ:

=lim

x→0

f(x)

∞

=lim

x→∞

f(x)

½n3.1bf

∈C([0,+∞),[0,+∞)),…f

=0,f

∞

=∞,Kéλ

>0,k-HessianXÚ(1.1)

•3˜‡š‚5»•à)u= (u

,···,u

)÷vlim

→0

k= ∞,Ù¥i= 1,2,···,n.

y²:éi= 1,2,···,n,·‚•Iy²éλ

>0,È©XÚ(2.3)•3˜‡)v= (v

,···,v

)

÷vlim

→0

k= ∞=Œ.Ï•f

= 0,K•3˜‡~êr

>0¦é?¿ε>0,k

) ≤εv

,∀0 ≤v

≤r

) ≤εv

,∀0 ≤v

≤r

) ≤εv

,∀0 ≤v

≤r

Ù¥ε÷v

(λ

···λ

)

≤1.

(3.1)

Ïd,év

∈P∩∂Ω

,i= 1,2,···,n,Ω

= {x∈R

: kxk≤r

},k

)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ

≤λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ

≤λ



kτ

k−N

N−1

k−1

N−1

)

−1

εv

(s)ds



dτ

≤λ



k−1

N−1





k−N

N−1



dτ

≤λ

k,t∈[0,1],

ÓnŒ,

)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ

≤λ

k,t∈[0,1],

)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ

≤λ

k,t∈[0,1].

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

•ÿ

d

½ÂÚ(3.1)Œ•,

k= kT

···T

≤λ

···T

≤(λ

)

···T

≤(λ

···λ

)

≤kv

k,v

∈P∩∂Ω

(3.2)

Ï•f

∞

= ∞,K•3˜‡~êR

(0 <r

)¦é?¿~êη>0,k

) ≥ηv

,∀v

≥R

) ≥ηv

,∀v

≥R

) ≥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].

Ïd,év

∈P∩∂Ω

)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ

≥λ

1−θ

kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds

dτ

≥λ

1−θ

kτ

k−N

N−1

k−1

N−1

)

−1

ηv

(s)ds

dτ

≥λ

1−θ

k(1−θ)

k−N

N−1

k−1

N−1

)

−1

η(θkv

dτ



ηk

k−1

N−1



(1−θ)

k−N

2k+N−1

k,t∈[0,1],

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

•ÿ

ÓnŒ,

)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ

≥



ηk

k−1

N−1



(1−θ)

k−N

2k+N−1

k,t∈[0,1],

)(t) = λ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτ

≥



ηk

k−1

N−1



(1−θ)

k−N

2k+N−1

k,t∈[0,1].

d

½ÂÚ(3.3)Œ•,

k=kT

···T

≥



ηk

k−1

N−1



(1−θ)

k−N

2k+N−1

···T

≥(λ

)



ηk

k−1

N−1



(1−θ)

2(k−N)

2(2k+N−1)

···T

≥(λ

···λ

)



ηk

k−1

N−1



(1−θ)

n(k−N)

n(2k+N−1)

≥kv

k,v

∈P∩∂Ω

(3.4)

(ÜÚn2.3Œ•,Žf

k˜‡ØÄ:v

∈P∩(

Ω

\Ω

).½ÂT

= v

,···,T

= v

,···,v

Ón,·‚•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≤β

(m= 1,2,···).

KS{kv

k}•3˜‡Âñu~êα

(0≤α

≤β

)fS,•{Bå„,·‚b{kv

Âñuα

(i)eα

iλ

k}>

= max{f

),r

≤kv

k≤R

= max{f

),r

≤kv

k≤R

= max{f

),r

≤kv

k≤R

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

•ÿ

= v

,···,T

= v

Œ•



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτk

1λ

≤



kτ

k−N

N−1

k−1

N−1

)

−1



1λ

≤

2k−N

1λ

≤

2kF

|2k−N|α



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτk

2λ

≤

2kF

|2k−N|α



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτk

nλ

≤

2kF

|2k−N|α

(3.5)

(3.5)L²λ

→+∞(m→+∞),ù†λ

→0

gñ,i= 1,2,···,n.

(ii)eα

iλ

k}→0.df

= 0Œ•,é?¿δ>0,

•3˜‡~êr

>0,¦

2λ

) ≤δv

2λ

,∀0 ≤v

2λ

≤r

3λ

) ≤δv

3λ

,∀0 ≤v

3λ

≤r

1λ

) ≤δv

1λ

,∀0 ≤v

1λ

≤r

Ïd,év

iλ

∈P∩∂Ω

,kv

iλ

k= r



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτk

1λ

≤



kτ

k−N

N−1

k−1

N−1

)

−1

δv

2λ



1λ

≤

kδ

|2k−N|r



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτk

2λ

≤

kδ

|2k−N|r

(3.6)

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

•ÿ



kτ

k−N

N−1

k−1

N−1

)

−1

(s))ds



dτk

nλ

≤

kδ

|2k−N|r

(3.6)L²λ

→+∞(m→+∞),ù†λ

→0

gñ,i= 1,2,···,n.

nþŒ•,λ

→0

ž,kv

k→+∞,i= 1,2,···,n.

aqu½n3.1y²,·‚kXe½n.

½n3.2b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=0,Ù¥

i= 1,2,···,n.

ë•©z

[1]Trudinger,N.andWang,X.(2008)TheMonge-Amp`ereEquationsandItsGeometricAppli-

cations.HandbookofGeometricAnalysis,1,467-524.

[2]Shivaji,R.,Sim,I.andSon,B.(2017)AUniquenessResultforaSemipositonep-Laplacian

ProblemontheExteriorofaBall.JournalofMathematicalAnalysisandApplications,445,

459-475.https://doi.org/10.1016/j.jmaa.2016.07.029

[3]Zhang, Z. (2015)BoundaryBehavior of Large Solutionstothe Monge-Amp`ereEquations with

Weight.JournalofDiﬀerentialEquations,259,2080-2100.

https://doi.org/10.1016/j.jde.2015.03.040

[4]Zhang,Z.(2018)LargeSolutionstotheMonge-Amp`ereEquationswithNonlinearGradient

Terms:ExistenceandBoundaryBehavior.JournalofDiﬀerentialEquations,264,263-296.

https://doi.org/10.1016/j.jde.2017.09.010

[5]Zhang, Z.andQi, Z.(2015)OnaPower-TypeCoupledofMonge-Amp`ere Equations.Topolog-

icalMethodsinNonlinearAnalysis,46,717-729.

[6]Lazer,A.andMcKenna,P.(1996)OnSingularBoundaryValueProblemsfortheMonge

Amp`ereOperator.JournalofMathematicalAnalysisandApplications,197,342-362.

https://doi.org/10.1006/jmaa.1996.0024

[7]Feng,M.(2021)ConvexSolutionsofMonge-Amp`ereEquationsandSystems:Existence,U-

niquenessandAsymptoticBehavior.AdvancesinNonlinearAnalysis,10,371-399.

https://doi.org/10.1515/anona-2020-0139

[8]Feng,M.andZhang,X.(2021)ACoupledSystemofk-HessianEquations.Mathematical

MethodsintheAppliedSciences,44,7377-7394.https://doi.org/10.1002/mma.6053

[9]Gao, C.,He, X.andRan, M.(2021)On aPower-Type CoupledSystem ofk-HessianEquations.

QuaestionesMathematicae,44,1593-1612.https://doi.org/10.2989/16073606.2020.1816586

[10]Zhang, X. and Feng, M. (2019) TheExistence and Asymptotic Behavior of Boundary Blow-Up

Solutionstothek-HessianEquation.JournalofDiﬀerentialEquations,267,4626-4672.

https://doi.org/10.1016/j.jde.2019.05.004

DOI:10.12677/aam.2022.1175224988A^êÆ?Ð

•ÿ

[11]Wan,H.,Shi,Y.andQiao,X.(2021)EntireLargeSolutionstothek-HessianEquationswith

Weights:Existence,UniquenessandAsymptoticBehavior.JournalofMathematicalAnalysis

andApplications,503,ArticleID:125301.https://doi.org/10.1016/j.jmaa.2021.125301

[12]Sun, H.andFeng,M. (2018) BoundaryBehavior ofk-ConvexSolutionsforSingulark-Hessian

Equations.NonlinearAnalysis,176,141-156.https://doi.org/10.1016/j.na.2018.06.010

[13]Trudinger, N. (1995)On the Dirichlet Problem for HessianEquations. ActaMathematica, 175,

151-164.https://doi.org/10.1007/BF02393303

[14]Ji, X.and Bao, J.(2010) Necessaryand Suﬃcient Condition onSolvabilityforHessian Inequal-

ities.ProceedingsoftheAMS,138,175-188.https://doi.org/10.1090/S0002-9939-09-10032-1

[15]Guo,D.andLakshmikantham,V.(1988)NonlinearProblemsinAbstractCones.Academic

Press,Inc.,Cambridge,MA.

DOI:10.12677/aam.2022.1175224989A^êÆ?Ð