一类k-Hessian 方程正径向解的存在性 The Existence of Positive Radial Solutions for a Class of k-Hessian Equation

doi:10.12677/PM.2021.116115

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PureMathematicsnØêÆ,2021,11(6),1010-1019

PublishedOnlineJune2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.116115

˜ak-Hessian•§»•)•35

•••ÿÿÿ

Ü“‰ŒÆêÆ†ÚOÆ§[‹=²

ÂvFÏµ2021c428F¶¹^FÏµ2021c531F¶uÙFÏµ2021c68F

Á‡

ÄuüNS“•{,ÏLE˜‡üNS“S,©Ì‡¼˜ak-Hessian•§»•)

•35.

'…c

k-Hessian•§§»•)§üNS“•{

TheExistenceofPositiveRadialSolutions

foraClassofk-HessianEquation

CunyanYue

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Apr.28

,2021;accepted:May31

,2021;published:Jun.8

,2021

Abstract

Basedonthemonotoneiterativemethod,weobtaintheexistenceofpositiveradialso-

©ÙÚ^:•ÿ.˜ak-Hessian•§»•)•35[J].nØêÆ,2021,11(6):1010-1019.

DOI:10.12677/pm.2021.116115

•ÿ

lutionsforaclassofk-Hessianequationbyconstructingamomtoneiterativesequence.

Keywords

k-HessianEquation,PositiveRadialSolution,MonotoneIterativeMethod

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

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

1.0

k-Hessian¯KuAÛÆ,6Nå ÆÚÙ¦A^Æ‰[1].˜„/,·‚½ÂXek-Hessian

Žf:

(λ(D

u)) =

1≤j

<...<j

≤N

...λ

,k= 1,2,···,N,

Ù¥u∈C

),D

u´ëY‡©¼êuHessianÝ,λ

,λ

,...,λ

´D

uA

Š,λ(D

u)=(λ

,λ

,...,λ

)´D

uAŠ•þ,S

(λ(D

u))´1kÐé¡õ‘ª,´

HessianÝD

u¤kk×kÌfªÚ.AO/, k= 1ž,k-HessianŽfòz•Laplace

ŽfS

(λ(D

u))=

i=1

=∆u,•„[2–4];k=Nž,k-HessianŽfòz•Monge-Amp`ere

ŽfS

(λ(D

u)) =

i=1

= det(λ(D

u)),•„[5–7].

Cc5,k-Hessian¯KÚåNõÆö2•'5,¿´L¤J[2–13].Caﬀarelli

[8] <•kïÄk-Hessian •§S

(λ(D

u)) = f1w)•35ÚkO; d,Wei[9] $^

üN©l•{¼k-Hessian•§S

(λ(D

u)) = f(−u)»•)•˜5(J;2015c,ZhangÚ

Zhou[10]$^üNS“•{ÚArzel`a-Ascoli½n¼k-Hessian•§σ

(λ(D

u)) = p(|x|)f(u)

»•)•35(J,¿ïá»•)•˜•3^‡;2020c,Zhang[11]?˜Ú$^ ØÄ

:½n¼ÛÉ‡‚5k-Hessian•§S

(λ(D

u))=λH(x)f(−u)š²…»•)•35,ù

p•Ä´¼êH(x)3>.∂ΩNCÛÉœ¹,fŒ U30:?ÛÉ½ö3∞?k-‡ ‚5

O•.Š5¿´,þã©z¤?ØŽf¥þØ¹FÝ‘η|Ou|I,@ok-HessianŽf¥•¹

FÝ‘ž´Ä•,Œ±)•35,ù´˜‡Š•Ä¯K.

É±þ©zéu,©ò}Á$^üNS“•{•Äk-Hessian•§

(λ(D

u+η|Ou|I)) = h(|x|)f(u),x∈R

(1.1)

DOI:10.12677/pm.2021.1161151011nØêÆ

•ÿ

»•)•35,Ù¥η∈[0,+∞),h∈C([0,+∞),[0,+∞))üN4O.ƒ©z[10]Ú[11]

ó,duŽf¥η|Ou|IÑy¦¯K•\E,.

2.ý•£

•¼•§(1.1)»•)•35,!‰Ñ˜7‡Ún.-r=|x|=

i=1

:= {x∈R

: |x|<R},KkXe5Ÿ¤á.

Ún2.1[13]v(r)∈C

[0,R)´˜‡»•é¡¼ê…v

(0)=0,K¼êu(|x|)=v(r)´

),…

λ(D

u+η|Ou|I) =











(r)+ηv

(r),(

+η)v

(r),...,(

+η)v

(r)),r∈(0,R),

(0),v

(0),...,v

(0)),r= 0;

(λ(D

u+η|Ou|I)) =











k−1

N−1

(r)+ηv

(r))((

+η)v

(r))

k−1

N−1

((

+η)v

(r))

,r∈(0,R),

(0))

,r= 0,

Ù¥C

k!(N−k)!

Ún2.2ev(r) ∈C[0,R]∩C

(0,R)´Cauchy¯K











(r) =



k−1

N−1

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds



,r∈(0,R),

v(0) = γ,γ>0

(2.1)

),Kv(r) ∈C

[0,R)´~‡©•§

k−1

N−1

(r))(v

(r))

k−1

r+r(C

k−1

N−1

η+C

N−1

(

+η))(v

(r))

(1+ηr)

k−1

h(r)f(v(r)),r>0

(2.2)

)…v

(0) = 0.

y²:w,,v(r) ∈C

[0,R),d(2.1)Œ•

(r))

k−1

N−1

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds,r≥0.

(2.3)

k−1

N−1

(r)

(r))

= e

(r)



1+ηr



k−1

h(r)f(v(r)),r≥0.

(2.4)

DOI:10.12677/pm.2021.1161151012nØêÆ

•ÿ

é(2.4)'ur¦,

k−1

N−1

(r))(v

(r))

k−1

r+r(C

k−1

N−1

η+C

N−1

(

+η))(v

(r))

(1+ηr)

k−1

h(r)f(v(r)),r>0

…v

(0) = 0.

T(r) =

k−1

N−1

−ψ

(t)

(s)



1+ηs



k−1

h(s)ds

dt,r≥0,T(∞) :=lim

r→∞

T(r).

F(r) =

f(t)

dt,r≥γ>0,F(∞) :=lim

r→∞

F(r),

Ù¥

(r) =

k−1

N−1



k−1

N−1

ηr+C

N−1

lnr+C

N−1

ηr



3©¥,·‚ob±e^‡¤á:

(H1)F(∞) = ∞,T(∞) <∞;

(H2)f∈C([0,+∞),[0,+∞))´üN4O…t>0ž,f(t) >0.

dÚn(2.2)•,•§(1.1)du~‡©•§(2.2). Ïd,•¼•§(1.1)»•)•

35,•Iy²~‡©•§(2.2)k)=Œ.

3.»•)•35

½n3.1e(H1)Ú(H2)¤á,Kk-Hessian•§(1)kÃ¡õ‡»•)u∈C

[0,+∞).

y²:é•§(2.4)È©,

v(r) = v(0)+

k−1

N−1

−ψ

(t)

(s)



1+ηs



k−1

h(s)f(v(s))ds

dt,r≥0.

(3.1)

(ÜÚn2.2,ÐŠv

(r) = v(0) = γ>0,3[0,+∞)þ½ÂS{v

(r)}

m≥0

,?1XeS“:











(r) = γ,

(r) = γ+



k−1

N−1

−ψ

(t)

(s)



1+ηs



k−1

h(s)f(v

m−1

(s))ds



dt,r≥0.

(3.2)

Äky²{v

(r)}

m≥0

3[0,+∞)þ´š~.w,,m=0ž,v

(r)=v

(r)<v

(r),

bv

m−1

(r)≤v

(r)¤á,‡y{v

(r)}

m≥0

š~,•Iyv

(r)≤v

m+1

(r)=Œ.´•é

∀m≥0,r∈[0,+∞)k

DOI:10.12677/pm.2021.1161151013nØêÆ

•ÿ

(r) = γ+

k−1

N−1

−ψ

(t)

(s)



1+ηs



k−1

h(s)f(v

(s))ds

≤γ+

k−1

N−1

−ψ

(t)

(s)



1+ηs



k−1

h(s)f(v

(s))ds

= v

m+1

(r),

(3.3)

S{v

(r)}

m≥0

3[0,+∞)þ´š~.d(H2)Ú{v

(r)}

m≥0

üN5

k−1

N−1

(r)



(r))



= e

(r)



1+ηr



k−1

h(r)f(v

m−1

(r))

≤e

(r)



1+ηr



k−1

h(r)f(v

(r)),

(3.4)



(r))

≤

k−1

N−1

−ψ

(r)

(s)



1+ηs



k−1

h(s)ds

f(v

(r)).

(3.5)

duv

(r) ≥0,K

(t))

f(v

(t))

dt≤

k−1

N−1

−ψ

(t)

(s)



1+ηs



k−1

h(s)ds

dt= T(r),

?

(r)

f(τ)

dτ≤T(r).

Ïd

F(v

(r)) ≤T(r),∀r≥0.

(3.6)

w,,F´V…F

(r) =

f(v(r))

>0(r>0),¤±F

−1

(F_N)3[0,F(∞))þî‚

4O.qF(∞) = ∞…T(∞) <∞,KF

−1

(∞) = ∞…

(m)

(r) ≤F

−1

T(r) ≤F

−1

T(∞) <∞,∀r≥0.

(3.7)

d,duh∈C([0,+∞),[0,+∞))üN4O,@oé½~êc

>0,∀ε>0,r

∈[0,c

DOI:10.12677/pm.2021.1161151014nØêÆ

•ÿ

)−v

) =



k−1

N−1

−ψ

(t)

(s)



1+ηs



k−1

h(s)f(v

m−1

(s))ds



≤

k−1

N−1





(

1+ηt

)

k−1

h(t)f(v

m−1

(t))





≤

k−1

N−1



(

1+ηc

)

k−1

h(c

)f(v

m−1

))



−r

-D:= (

k−1

N−1

)



(

1+ηc

)

k−1

h(c

)f(v

m−1

))



,δ=

,|r

−r

|<δž,k

)−v

) <ε,

S{v

(r)}

m≥0

3[0,c

]þk.…ÝëY.ŠâArzel`a-Ascoli½n,{v

(r)}

m≥0

3[0,c

]þ

˜—Âñv(r).dr>0•v(r)´~‡©•§(2.2)˜‡).dγ∈(0,∞)?¿5Œ•,

•§(2.2)kÃ¡õ‡).

eyv∈C

[0,+∞).w,,v∈C

(0,+∞),•Iyv

(r)Úv

(r)3r= 0?ëY.¢Sþ,

(0) =lim

r→0

v(r)−v(0)

=lim

r→0



k−1

N−1

−ψ

(t)

(s)



1+ηs



k−1

h(s)f(v(s))ds



=lim

r→0

k−1

N−1

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

k−1

N−1

lim

r→0







(s)



1+ηs



k−1

h(s)f(v(s))ds

(r)







k−1

N−1

lim

r→0









1+ηr



k−1

h(r)f(v(r))

1+rψ







k−1

N−1

lim

r→0



h(r)f(v(r))

(1−k)(1+ηr)

k−1

+N(1+ηr)



= 0

DOI:10.12677/pm.2021.1161151015nØêÆ

•ÿ

…

(r) =

k−1

N−1

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

lim

r→0

(r) =lim

r→0

k−1

N−1

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

= 0,



lim

r→0

(r) = v

(0) = 0.

Ïd,v

(r)3r= 0?ëY.aq/,

(0) =lim

r→0

(r)−v

(0)

=lim

r→0



k−1

N−1

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds



k−1

N−1

lim

r→0







(s)



1+ηs



k−1

h(s)f(v(s))ds

(r)







k−1

N−1

lim

r→0









1+ηr



k−1

h(r)f(v(r))

k−1

(r)







k−1

N−1

lim

r→0



h(r)f(v(r)

N(1+ηr)



k−1

N−1



h(0)f(v(0))



ÏLOŽ,

(r) =

k−1

N−1





−ψ

(r)

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds



1+ηr



k−1

h(r)f(v(r))

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

−1





(r) = −ψ

(r)

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

(r) =



1+ηr



k−1

h(r)f(v(r))

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

−1

DOI:10.12677/pm.2021.1161151016nØêÆ

•ÿ

lim

r→0

(r) = −lim

r→0

(r)

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

= −lim

r→0

(r)lim

r→0

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

=lim

r→0

(−rψ

(r))lim

r→0







(s)



1+ηs



k−1

h(s)f(v(s))ds

(r)







=lim

r→0

(k−N(1+ηr))lim

r→0



h(r)f(v(r))

N(1+ηr)



= (k−N)



h(0)f(v(0))



lim

r→0

(r) =lim

r→0



1+ηr



k−1

h(r)f(v(r))

−ψ

(r)

(s)



1+ηs



k−1

h(s)f(v(s))ds

−1

=lim

r→0



1+ηr



k−1

lim

r→0

h(r)f(r,v(r))lim

r→0







(s)



1+ηs



k−1

h(s)f(v(s))ds

(r)







−1

= N



h(0)f(v(0))





lim

r→0

(r) =

k−1

N−1



lim

r→0

(r)+lim

r→0

(r)



k−1

N−1



h(0)f(v(0))



= v

(0).

Ïdv∈C

[0,+∞).

nþ,•§(2.2)kÃ¡õ‡)v∈C

[0,+∞),=k-Hessian•§(1.1)kÃ¡õ‡»•

)u∈C

[0,+∞).

4.A^Þ~

•Ä5-Hessian•§

(λ(D

u+η|Ou|I)) = h(|x|)f(u),x∈R

(4.1)

Ù¥η=

,H(r) = (

1+r

)

−r−lnr

,f(u) = u+1.w,,f(u)÷v(H2)…

(r) =



lnr+



= r+lnr.

DOI:10.12677/pm.2021.1161151017nØêÆ

•ÿ

d,

T(∞) =

∞

−t−lnt

s+lns



1+s





1+s



−s−lns

∞



−t−lnt

1ds



∞



t+lnt



∞



t+lnt



dt<∞,

F(∞) =

∞

f(t)

dt=

∞

t+1

dt= ∞,

=(H1)¤á.Ïd5-Hessian•§(4.1)kÃ¡õ‡»•)u∈C

[0,+∞).

ë•©z

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