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AdvancesinAppliedMathematicsA^êÆ?Ð,2019,8(8),1418-1431
PublishedOnlineAugust2019inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2019.88166
RadialPositiveSolutionsofp-Laplacian
ProblemwithNonnegativeLocalTerms
YongZeng
GuangxiUniversity,NanningGuangxi
Received:Aug.1
st
,2019;accepted:Aug.15
th
,2019;published:Aug.22
nd
,2019
Abstract
Inthispaper,wemainlystudytheexistenceofradialsolutionsforthefollowingp-
Laplacianproblem:









−m(kuk
p
)div(|Ou|
p−2
Ou) = f(|x|,u),x∈B;
u(x) >0,x∈B;
u(0) = 0
wherefandmsatisfycertainconditions.Weprovethattheabovep-Laplacianproblem
has a radial solution through the origin mainly by means of upper and lower solutions.
Firstly,wemaketheauxiliaryproblemsequenceoftheoriginalproblem.Thenwe
getamonotoneboundedsolutionsequencebysolvingtheproblemsequence.Then
wecangetthatwhenntendstoinfinity,thereexistsau,whichmakesthissolution
sequencetendtou.Finally,weprovethatuistheradialsolutionoftheoriginal
problem.Theconcreteproofisgiveninthethirdpart.
Keywords
RadialSolution,UpperandLowerSolutions,p-LaplacianProblem
‘kšKÛÜ‘p-Laplacian¯K
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©ÙÚ^:Q].‘kšKÛÜ‘p-Laplacian¯K»•)[J].A^êÆ?Ð,2019,8(8):1418-1431.
DOI:10.12677/aam.2019.88166
Q]
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u(0) = 0
Ù ¥fÚm÷v˜½^‡§·‚̇ÏLþe)•{5y²þãp-Laplacian¯K kL:»
•)§Äk·‚‰Ñ¯K9ϯKS§ ,ÏL¦)d¯K Sј‡üNk.)S§
lŒ±Ñn→∞ž§•3˜‡u∈C
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n
→u§•y²u=•¤¦
»•)§äNy²31nÜ©‰Ñ"
'…c
»•)§þe)§p-Laplacian¯K
Copyright
c
2019byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
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
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)div(|Ou|
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•ëY¼ê§f:[0,1) ×(0,+∞)→R´ŒCÒëY¼ê§…f3|x|=1½u=0?ŒUÛ
DOI:10.12677/aam.2019.881661419A^êÆ?Ð
Q]
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DOI:10.12677/aam.2019.881661420A^êÆ?Ð
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DOI:10.12677/aam.2019.881661421A^êÆ?Ð
Q]
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DOI:10.12677/aam.2019.881661422A^êÆ?Ð
Q]
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
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1
[0,1)…éur∈[0,1]§u≤u≤¯u.
DOI:10.12677/aam.2019.881661423A^êÆ?Ð
Q]
y²µÄk½Â˜‡9ϼêµ
f
∗
(r,u) = f(r,β(r,u))+β(r,u)
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0
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1−N
Z
1
r
s
N−1
f
∗
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m(φ(N,p))
ds)
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r
0
ϕ
−1
p
(t
1−N
Z
1
t
s
N−1
f
∗
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m(φ(N,p))
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^XL«C[0,1]§½Â˜‡ŽfA: X→XXeµ
(Au)(r) = a−
Z
r
0
ϕ
−1
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(t
1−N
Z
1
t
s
N−1
f
∗
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m(φ(N,p))
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1
(0,1)Ú(H
3
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²u´(13))=†¤y²u´AuØÄ:"
|
d(Au)(r)
dr
|=|ϕ
−1
p
(r
1−N
Z
1
r
s
N−1
f
∗
(s,u(s))
m(φ(N,p))
ds)|
≤|ϕ
−1
p
(r
1−N
Z
1
r
(h(s)+M)s
N−1
ds|
≤|ϕ
−1
p
(
Z
1
r
h(s)
κ
2
ds+
M
κ
2
)|
DOI:10.12677/aam.2019.881661424A^êÆ?Ð
Q]
¤±§Au´ÝëY§dArzela−Ascoli½n•§A(X)´˜‡ƒé;8§2dSchauderØÄ:
½nµŽfA•3ØÄ:§¤±u∈C[0,1]∩C
1
[0,1)´¯K(13))"
e5·‚y²u≤u≤¯u§Äk§y²u≤u§·‚-d(r)=u(r) −u(r)§d‡y{§b
•3r
∗
∈[0,1)÷vu(r
∗
)>u(r
∗
)§,de)½Â§·‚k:d(0)≤09d
0
(1)≤0§y3·‚Š
âd(1)ÎÒ·‚©eü«œ¹µ
œ¹˜µd(1) >0
bXé?¿r∈[0,1]§d(r) ≥0ð¤á"Šâu(r)´e)Ú(11)ªµ
−m(φ(N,p))r
1−N
(r
N−1
|u
0
|
p−2
u
0
)
0
≤f(r,u)
=f
∗
(r,u)−β(r,u)
Ï•m(φ(N,p)) >0§¤±Œµ
(r
N−1
|u
0
|
p−2
u
0
)
0
<(r
N−1
|u
0
|
p−2
u
0
)
0
(16)
òþª˜ lr1È©2dd
0
(1) ≤0µ
r
N−1
|u
0
|
p−2
u
0
<r
N−1
|u
0
|
p−2
u
0
=
u
0
(r) <u
0
(r)
2òþªl0rþÈ©µ
u(r) <u(r)
=d(r)<0gñ§,˜•¡§XJd(r)≥0Øð¤á§=•3r
2
∈(0,1),¦d(r)<0"qÏ
•d(1) >0§¤±d":•3½nµ•3r
3
∈(r
2
,1)§¦









d(r) ≥0,r∈[r
3
,1);
d(r
3
) = 0
d
0
(r
3
) ≥0
(17)
3«m(r
3
,1)þ‰†þaq?ØŒ(16)ªE,¤á",23r
3
1þÈ©Ú˜ µ
u
0
(r
3
) <u
0
(r
3
)
=
d
0
(r
3
) <0
†(17)ªgñ§¤±œ/˜´ØŒU"
œ/µd(1) ≤0
DOI:10.12677/aam.2019.881661425A^êÆ?Ð
Q]
Ï•d(1)≤0…d(r
∗
)>0§¤±•3λ
1
!λ
2
∈[0,1]§¦d(λ
1
)=d(λ
2
)=0§…é?¿
r∈(λ
1
,λ
2
)§d(r)>0.¤±d¥Š½n•µ•3ξ∈(λ
1
,λ
2
)¦d
0
(ξ)=0éN´y3r∈
[λ
1
,λ
2
]þ(16)ªE,¤á"é?¿r∈[ξ,λ
2
),ò(16)ª3ξrþÈ©µ
r
N−1
|u
0
|
p−2
u
0
>r
N−1
|u
0
|
p−2
u
0
=é?¿r∈[ξ,λ
2
)k
d
0
(r) >0
2òþª3rλ
2
þÈ©µ
d(r) <0,r∈(λ
1
,λ
2
)
ù†é?¿r∈[λ
1
λ
2
],d(r)>0gñ"¤±§dþã?ØŒ•3[0,1]þu(r)≤u(r)§²Lþ¡a
q?ØŒ§u(r) ≤¯u(r)§¤±u(r) ≤u(r) ≤¯u(r)§y."
3.̇(Ø9Ùy²
•y²½n1.1·‚I‡˜Ún"y‰Xe½§é?¿n∈N§
e
n
(r) = max{r,
1
2
n
},r∈[0,1]
˜
f
n
(r,u) = max{f(r,u),f(e
n
(r),u)}
é²w§
˜
f
n
: [0,1)×(0,∞) →R´ëY…
˜
f
n
(r,u) ≥f(r,u),(r,u) ∈[0,1)×(0,∞)
˜
f
n
(r,u) = f(r,u),(r,u) ∈E
n
×(0,∞)
…éu?¿;f8K⊂[0,1)§éN´3K×(0,∞)þ
˜
f
n
→f",·‚½ÂS{f
n
}
∞
n=1
X
eµ
f
1
(r,u) =
˜
f
1
(r,u)
f
2
(r,u) = min{f
1
(r,u),
˜
f
2
(r,u)}
.
.
.
f
n
(r,u) = min{f
n−1
(r,u),
˜
f
n
(r,u)}
ddŒ•§{f
n
}
∞
n=1
•½Â3[0,1)×(0,∞)þëY4~ê§=
f
1
(r,u) ≥f
2
(r,u) ≥···≥f
n
(r,u) ≥f
n+1
(r,u) ≥···≥f(r,u)
,§·‚5¿(r,u) ∈E
n
×(0,∞)
DOI:10.12677/aam.2019.881661426A^êÆ?Ð
Q]
f(r,u) ≤f
n
(r,u) ≤
˜
f
n
(r,u) = f(r,u)
=
f
n
(r,u) = f(r,u),(r,u) ∈E
n
×(0,∞)
¤±§é?¿;f8K⊂[0,1)§3K×(0,∞)þf
n
→f"y3·‚?Øe¡¯K(2)9Ï>Š
¯KSµ



−m(φ(N,p))r
1−N
(r
N−1
|u
0
|u
0
)
0
= f
n
(r,u),r∈(0,1);
u
0
(1) = 0,u(0) = ε
n
(17)
n
é²w§(17)
n
)´(17)
n+1
þ)§e¡·‚2‰Ñü^Únµ
Ún3.1µé?¿c∈(0,ε
n
]§u
n
= c´¯K(17)
n
e)"
y²µÏ•u(r)•~ꧤ±(17)
n
†>•0§=‡yÚn3.1•Iyé?¿c∈(0,ε
n
]§f
n
(r,c)>
0"3ùp·‚^êÆ8B{µ
n= 1ž§f
1
(r,c) =
e
f
1
(r,u) = max{f(r,u),f(e
1
,u)}≥f(e
1
(r),c) ≥L>0.
bn= k>1ž(ؤá"Kn= k+1ž§
f
k+1
(r,c) = min{f
k
(r,c),
e
f
k+1
(r,u)}
qϕ
e
f
k+1
(r,u) = max{f(r,u),f(e
k+1
,u)}≥f(e
k+1
(r),c) ≥L
dþ¡ü‡ªfŒµ
f
k+1
(r,c) >0
u´é?¿c∈(0,ε
n
],f
n
(r,u) >0§¤±(ؤá"
Ún3.2µ¯K(17)
n
(n=1,2,3···)–•3˜‡)u
n
∈C
1
[0,1) ∩C[0,1]…÷vn≥2ž§•
3η
n
≤u
n
(r) ≤u
n−1
(r)"
y²µ^êÆ8B{§n=1ž§M>0•~ê§db(H
2
)§•3¼êh
M
∈L(0,1)∩
C([0,1);(0,+∞))¦µ
|f(r,u)|≤h
M
(r),(r,u) ∈[0,1)×[M,∞)
…éu,‡~êM
0
kµ
|f(e
1
(r),u)|≤h
M
(e
1
(r)) ≤M
0
,(r,u) ∈[0,1)×[M,∞)
·‚-G(r) = h
M
(r)+M
0
KkµG(r) ∈L(0,1)∩C([0,1);(0,+∞))§…÷vµ
|f
1
(r,u)|≤G(r),(r,u) ∈[0,1)×[M,∞)
e5§4·‚•Äe¡¯Kµ
DOI:10.12677/aam.2019.881661427A^êÆ?Ð
Q]



−m(φ(N,p))r
1−N
(r
N−1
|u
0
|
p−2
u
0
)
0
= G(r),r∈(0,1);
u
0
(1) = 0,u(0) = M
(18)
ØJwÑþªäk±e/ª•˜)µ
u
G
(r) = M−
Z
r
0
ϕ
−1
p
(t
1−N
Z
1
t
s
N−1
G(s)
m(φ(N,p))
ds)dt<+∞
Ï•G(r)≥f
1
(r,u
G
(r))…u
0
G
(r)<0min
r∈[0,1]
u
G
(r)=u
G
(1)§ Ï•u
G
(r)>0§¤±•3η
1
∈(0,ε
1
]§
¦µη
1
≤u
G
(1)§2dÚn3.1•§η
1
•¯K(17)
1
e)§¤±N´µ
0 <η
1
≤u
G
(1) ≤u
G
(r) <+∞
,db(H
2
)•§•3¼êh
η
1
∈L(0,1)∩C([0,1);(0,+∞))¦eª¤á"
|f(r,u)|≤h
η
1
(r),(r,u) ∈[0,1)×hη
1
,u
G
(r)i
2d·K2.3§¯K(17)
1
•3)u
1
∈C
1
[0,1)∩C[0,1]…÷vη
1
≤u
1
(r) ≤u
G
(r)"¤±(ؤá"
bn=kž§¯K(17)
k
•3)u
k
Úη
k
§¦η
k
≤u
k
≤u
k−1
K²Lþ¡Ó?Øµ•
3G
k
(r)¦µ
|f
k
(r,u)|≤G
k
(r),(r,u) ∈[0,1)×[M,∞)
n= k+1ž§dþŒ•§-G
k+1
(r) = G
k
(r)+M
k
Ù¥M÷vµ
|f(e
k+1
(r),u)|≤h
M
(e
k+1
(r)) ≤M
k
,(r,u) ∈[0,1)×[M,∞)
KG
k+1
(r)÷vµ
|f
k+1
(r,u)|≤G
k+1
(r)
,§²L†n=1žƒq?ØŒµ¯K(17)
k+1
•3)u
k
∈C
1
[0,1) ∩C[0,1]…÷vη
k+1
≤
u
k+1
(r) ≤u
k
(r)"¤±(ؤá"
–dŒ½n1.1y²Xeµ
y²µdÚn3.2•§¯K(17)
1
•3)u
1
(r)∈hη
1
,u
G
(r)i§Ï•f
1
(r,u)≥f(r,u)§¤±d·K2.2
·‚µ
u
1
(r) ≥K
1
p−1
0
γ(r),r∈[0,1]
bu
n
´(17)
n
)§ÏL8B{Œ•éu?¿r∈[0,1]÷vµ
u
n
≥η
n
,u
n
(r) ≥K
1
p−1
0
γ(r)
dÚn3.1·‚•η
n+1
u
n
©O´¯K(17)
n+1
e)Úþ)§…η
n+1
≤η
n
≤u
n
§Kd(H
2
)§•3
¼êh
η
n+1
∈L(0,1)∩C([0,1);(0,+∞))¦µ
|f(r,u)|≤h
η
n+1
(r),(r,u) ∈[0,1)×hη
n+1
,u
n
i
DOI:10.12677/aam.2019.881661428A^êÆ?Ð
Q]
2d·K2.3•§¯K(17)
n+1
•3)u
n+1
…éur∈[0,1]kη
n+1
≤u
n+1
(r) ≤u
n
(r)",§df
n+1
(r,u) ≥
f(r,u)Ú·K2.2•µ
u
n+1
(r) ≥K
1
p−1
0
γ(r),r∈[0,1]
¤±§ÏL8B§·‚Œ±¯K(17)
n
)S{u
n
}
∞
n=1
÷vµ









u
n
(r) ≥K
1
p−1
0
γ(r),r∈[0,1];
ε
n
≤u
n
≤u
n−1
,r∈[0,1]
u
0
n
(1) = 0,u
n
(0) = η
n
(19)
w,§{u
n
}
∞
n=1
´üN4~k.S§¤±•3¼êu¦é?¿r∈[0,1]§n→∞ž§
u
n
→u"…§
K
1
p−1
0
γ(r) ≤u(r) ≤u
n
(r),r∈[0,1](20)
e5§·‚y²u´(2))"Ï•u
n
´¯K(17)
n
)§¤±§Œ±¤e¡/ªµ
u
n
(r) = ε
n
−
Z
r
0
ϕ
−1
p
(t
1−N
Z
1
t
s
N−1
f
n
(s,u
n
(s))
m(φ
n
(N,p))
ds)dt
Ù¥§m(φ
n
(N,p)) = m(Nα(N)
R
1
0
s
N−1
|u
n
(s)|
p
ds)…
u
0
n
(r) = −ϕ
−1
p
(r
1−N
Z
1
r
s
N−1
f
n
(s,u
n
(s))
m(φ
n
(N,p))
ds)
é?¿;f8K= [σ
k
,1] ⊂[0,1)§•3˜‡êN
∗
>0§¦K⊂E
N
∗
§Ké?¿n≥N
∗
,
f
n
(r,u
n
) = f(r,u
n
),r∈K
¤±§·‚kµ
u
0
n
(r) = −ϕ
−1
p
(r
1−N
Z
1
r
s
N−1
f(s,u
n
(s))
m(φ
n
(N,p))
ds),r∈K
duγ(r)3[0,1]þî‚4~…γ(r)>0§¤±• 3δ
0
>0§¦3KþkK
1
p−1
0
γ(r)>δ
0
"qÏ
•é?¿r∈[0,1]u
n
(r) ≥K
1
p−1
0
γ(r)§,·‚Œ±µ
u
n
(r) ≥δ
0
,r∈K
db(H
2
)§•3¼êh
δ
0
(r) ∈C([0,1);(0,+∞))∩L
1
(0,1)¦
|f(r,u
n
)|≤h
δ
0
(r),r∈K
5¿h
δ
0
(r)3Kþk.§¤±éu,‡••6K~êM
K
§·‚kµ
|u
0
n
(r)|≤|ϕ
−1
p
(r
1−N
Z
1
r
s
N−1
h
δ
0
(s)
m(φ
n
(N,p))
ds)|≤M
K
,r∈K
DOI:10.12677/aam.2019.881661429A^êÆ?Ð
Q]
dArzela-Ascoli½n§·‚Œ±µ3K¥u
n
→u…u∈C(K)"Ï•u
n
´¯K(17)
n
)§¤
±§äk±e/ªµ
u
n
(r) = u
n
(0)−
Z
r
0
ϕ
−1
p
(t
1−N
Z
1
t
s
N−1
f(s,u
n
(s))
m(φ
n
(N,p))
ds)dt,r∈K
,·‚éþª4•§2d››Âñ½nµ
u(r) = u(0)−
Z
r
0
ϕ
−1
p
(t
1−N
Z
1
t
s
N−1
f(s,u(s))
m(φ
n
(N,p))
ds)dt,r∈K(21)
lþªéN´Œ±wѧu(r)∈C
1
(K)…÷v(2)1˜‡•§§2dK?¿5§Œu∈
C
1
[0,1)§ÏL(21)ªŒ•u
0
(0) = 0§,§
u(0) =lim
n→∞
u
n
(0) =lim
n→∞
ε
n
= 0
y3·‚y²u(r)3r=0?´ëY§Ï•{ε
n
}4~…ªu0§¤±é?‰θ>0§•
3
¯
N>0§é¤kn≥
¯
Nk
ε
n
<θ
·‚n=
¯
N§u
¯
N
(0) = ε
¯
N
<θ"2du
¯
N
(r) ∈C
1
[0,1)∩C[0,1]§•3vδ>0¦µ
0 ≤u
¯
N
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2dê{u
n
}
∞
n=1
üN59(20)ªŒµ
K
1
p−1
0
γ(r) ≤u(r) ≤u
¯
N
(r) <θ,r∈[0,δ)
dθ?¿5Úlim
r→0
K
1
p−1
0
γ(r)=0Œ§lim
r→0
u(r)=0=u(0)§=u(r)3r=0?ëY§¤±u∈
C
1
[0,1)∩C[0,1]"l•u´¯K(2)˜‡)"
—
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DOI:10.12677/aam.2019.881661430A^êÆ?Ð
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ë•©z
[1]SantosJr.,J.R.andSiciliano,G.(2018)PositiveSolutionsforaKirchhoffProblemwith
VanishingNonlocalTerm.JournalofDifferentialEquations,265,2034-2043.
https://doi.org/10.1016/j.jde.2018.04.027
[2]Le,P.,Huynh,N.andHo,V.(2019)PositiveSolutionsofthep-KirchhoffProblemwith
DegenerateandSign-ChangingNonlocalTerm.Zeitschriftf¨urAngewandteMathematikund
Physik,70,68.https://doi.org/10.1007/s00033-019-1114-2
[3]Chu,K.,Hai,D.andShivaji,R.(2019)UniquenessofPositiveRadialSolutionsforInfinite
Semipositonep-LaplacianProblemsinExteriorDomains.JournalofMathematicalAnalysis
andApplications,472,510-525.https://doi.org/10.1016/j.jmaa.2018.11.037
[4]L¨u,H.andBai,Z.(2006)PositiveRadialSolutionsofaSingularEllipticEquationwithSign
ChangingNonlinearities.AppliedMathematicsLetters,19,555-567.
https://doi.org/10.1016/j.aml.2005.08.002
[5]Jin,C.H.,Yin, J. and Wang,Z. (2007) Positive Radial Solutions ofp-LaplacianEquationwith
SignChangingNonlinearSources.MathematicalMethodsintheAppliedSciences,30,1-14.
https://doi.org/10.1002/mma.771
[6]g1,îm,ÓÊ,.•¼©Û1§[M].®:p˜Ñ‡,2008.
[7]Habets,P. andZanolin,F. (1994)Upper and LowerSolutions fora Generalized Emden-Fowler
Equation.JournalofMathematicalAnalysisandApplications,181,684-700.
https://doi.org/10.1006/jmaa.1994.1052
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DOI:10.12677/aam.2019.881661431A^êÆ?Ð

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