Elliptic equation with singular potential
We consider the following problem of finding a nonnegative function $u(x)$ in a ball $B = B(O, R) ⊂ R^n,\; n ≥ 3:$ $$−Δu=V(x)u,u|∂B=ϕ(x),$$ where $Δ$ is the Laplace operator, $x = (x 1, x 2,…, x n )$, and $∂B$ is the boundary of the ball $B$. It is assumed that $0 ≤ V(x) ∈ L 1(B), 0 ≤ φ(x) ∈ L 1(∂B...
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| Дата: | 2010 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2010
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2994 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509004148506624 |
|---|---|
| author | Hudaigulyev, B. A. Худашулыев, Б. А. Худашулыев, Б. А. |
| author_facet | Hudaigulyev, B. A. Худашулыев, Б. А. Худашулыев, Б. А. |
| author_sort | Hudaigulyev, B. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:41:53Z |
| description | We consider the following problem of finding a nonnegative function $u(x)$ in a ball $B = B(O, R) ⊂ R^n,\; n ≥ 3:$ $$−Δu=V(x)u,u|∂B=ϕ(x),$$
where $Δ$ is the Laplace operator, $x = (x 1, x 2,…, x n )$, and $∂B$ is the boundary of the ball $B$. It is assumed that $0 ≤ V(x) ∈ L 1(B), 0 ≤ φ(x) ∈ L 1(∂B)$, and $φ(x)$ is continuous on $∂B$.
We study the behavior of nonnegative solutions of this problem and prove that there exists a constant $C_{*} (n) = (n − 2)^2/4$ such that if $V_0 (x) = \frac{c}{|x|^2}, then, for $0 ≤ c ≤ $C_{*} (n)$ and $V(x) ≤ V_0 (x)$ in the ball $B$, this problem has a nonnegative solution for any nonnegative continuous boundary function $φ(x) ∈ L_1(∂B)$, whereas, for $c > C_{*} (n)$ and $V(x) ≥ V_0(x)$, the ball $B$ does not contain nonnegative solutions if $φ(x) > 0$. |
| first_indexed | 2026-03-24T02:34:12Z |
| format | Article |
| fulltext |
UDK 517.956
B. A. Xudajhul¥ev (Turkmen. un-t, Aßhabat)
∏LLYPTYÇESKOE URAVNENYE
S SYNHULQRNÁM POTENCYALOM
We consider the problem of finding a nonnegative function u x( ) in the ball B = B O R Rn( , ) ⊂ , n ≥
≥ 3:
−∆u = V x u( ) , u
B∂ = φ( )x ,
where ∆ is the Laplace operator, x = ( , , , )x x xn1 2 … , ∂B is a boundary of the ball B . It is
assumed that 0 ≤ V x( ) ∈ L B1( ) , 0 ≤ φ( )x ∈ L B1( )∂ , and φ( )x is continuous on ∂B.
We study the behavior of nonnegative solutions of this problem and prove that there exists a
constant C n∗( ) = ( )n − 2 2
/ 4 such that if V x0 ( ) =
c
x 2
, then for 0 ≤ c ≤ C n∗( ) and V x( ) ≤
≤ V x0 ( ) in the ball B, this problem has a nonnegative solution for all nonnegative continuous boundary
functions φ( )x ∈ L B1( )∂ and, for c > C n∗( ) and V x( ) ≥ V x0 ( ) in the ball B, this problem has no
nonnegative solutions if φ( )x > 0.
Rozhlqda[t\sq zadaça znaxodΩennq nevid’[mno] funkci] u x( ) u kuli B = B O R Rn( , ) ⊂ , n ≥ 3:
−∆u = V x u( ) , u
B∂ = φ( )x ,
de ∆ — operator Laplasa, x = ( , , , )x x xn1 2 … , ∂B — meΩa kuli B, u prypuwenni, wo 0 ≤
≤ V x( ) ∈ L B1( ) , 0 ≤ φ( )x ∈ L B1( )∂ i φ( )x neperervna na ∂B.
Vyvça[t\sq povedinka nevid’[mnyx rozv’qzkiv ci[] zadaçi i dovedeno, wo isnu[ stala C n∗( ) =
= ( )n − 2 2
/ 4 taka, wo qkwo V x0 ( ) =
c
x 2
, to cq zadaça pry 0 ≤ c ≤ C n∗( ) i V x( ) ≤ V x0 ( ) u
kuli B ma[ nevid’[mnyj rozv’qzok pry bud\-qkij nevid’[mnij neperervnij hranyçnij funkci]
φ( )x ∈ L B1( )∂ , a pry c > C n∗( ) i V x( ) ≥ V x0 ( ) u kuli B ne ma[ nevid’[mnyx rozv’qzkiv, qkwo
φ( )x > 0.
Rassmatryvaetsq zadaça naxoΩdenyq neotrycatel\noj funkcyy u x( ) :
− =∆u V x u( ) , (1)
u xB∂ = φ( ) , (2)
v ßare B = B R Rn( , )0 ⊂ , n ≥ 3, radyusa R, R ≤ 1, s centrom v naçale koordy-
nat, hde x = ( , , , )x x xn1 2 … , ∂B — hranyca ßara B.
V kaçestve reßenyq uravnenyq (1) ponymaetsq obobwennaq funkcyq u x( ) ∈
∈ ′D B( ) takaq, çto u x( ) ≥ 0, Vu L B∈ 1, ( )loc . Predpolahaetsq, çto 0 ≤ V x( ) ∈
∈ L B1( ) , 0 ≤ φ( )x ∈ L B1( )∂ y φ( )x neprer¥vna na ∂B , hde L B1, ( )loc —
prostranstvo lokal\no yntehryruem¥x v B funkcyj, L B1( )∂ — prostranstvo
yntehryruem¥x na ∂B funkcyj. Çerez ′D B( ) oboznaçym prostranstvo obob-
wenn¥x funkcyj.
V rabote yzuçaetsq povedenye neotrycatel\n¥x reßenyj zadaçy (1), (2) y do-
kaz¥vaetsq, çto suwestvuet postoqnnaq C∗ = C n∗( ) =
( )n − 2
4
2
takaq, çto es-
ly V x0( ) = c x/ 2 , to pry 0 ≤ c ≤ C∗ y V x( ) ≤ V x0( ) v ßare B πta zadaça
© B. A. XUDAJHULÁEV, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12 1715
1716 B. A. XUDAJHULÁEV
ymeet reßenye, a pry c > C∗ y V x( ) ≥ V x0( ) ne ymeet reßenyj, esly φ( )x > 0.
Analohyçn¥j vopros yzuçen v rabote [1] dlq pervoj smeßannoj zadaçy dlq
lynejnoho uravnenyq teploprovodnosty y v rabote [2] dlq pervoj smeßannoj
zadaçy dlq nelynejnoho uravnenyq teploprovodnosty.
Pust\ V x0( ) = c x/ 2 , x B∈ . Najdem radyal\nug funkcyg ϕ( )x , udov-
letvorqgwug uravnenyg ∆ ϕ + V x0( )ϕ = −δ( )x v sm¥sle obobwenn¥x funk-
cyj, hde δ( )x — funkcyq Dyraka. Poskol\ku δ( )x = 0 pry x ≠ 0, dostatoçno
reßyt\ uravnenye −∆ ϕ = V x0( )ϕ . PoloΩym ϕ( )x = x −α
. Tohda
∆ ϕ = ϕrr +
n
r
r
− 1
ϕ = α α α( )+ − − −2 2n x ,
tak çto −∆ ϕ ϕ/ = c x −2
, hde c = α(n – 2 – α). V sluçae ϕ( )x = x −α > 0 us-
lovye ∆ ϕ ∈ L B1( ) oznaçaet, çto n – 2 – α > 0. Poslednee v¥polnqetsq, esly
c > 0 (y α > 0). Zametym, çto pry 0 ≤ c ≤ C n∗( ) α opredelqetsq yz ravenstva
α =
n − 2
2
–
( )n
c
−
−
2
4
2
.
Osnovn¥m rezul\tatom rabot¥ qvlqetsq sledugwaq teorema.
Teorema. 1. Esly 0 ≤ c ≤ C∗ y V x( ) ≤ V x0( ) v ßare B, to zadaça (1),
(2) ymeet neotrycatel\noe reßenye pry lgboj neotrycatel\noj neprer¥vnoj
hranyçnoj funkcyy φ( )x ∈ L B1( )∂ .
2. Esly c > C∗ y V x( ) ≥ V x0( ) v ßare B, to pry φ( )x > 0 zadaça (1),
(2) ne ymeet neotrycatel\n¥x reßenyj.
Dokazatel\stvo. Snaçala dokaΩem pervug çast\ teorem¥.
Rassmotrym vspomohatel\nug zadaçu
− =∆u V x um m m( ) , ( )1m
u xm B m∂ = φ ( ) , ( )2m
hde m = 1, 2, … , 0 ≤ V xm ( ) ≤ V x( ) , V xm ( ){ } — posledovatel\nost\ monotonno
vozrastagwyx ohranyçenn¥x yzmerym¥x funkcyj takaq, çto lim ( )m mV x→∞ =
= V x( ) dlq poçty vsex x B∈ , 0 ≤ φm x( ) ≤ φ( )x , φm x( ){ } — posledovatel\-
nost\ monotonno vozrastagwyx neotrycatel\n¥x neprer¥vno dyfferencyrue-
m¥x funkcyj, ravnomerno sxodqwaqsq k neprer¥vnoj hranyçnoj funkcyy
φ( )x ∈ L B1( )∂ . Yz klassyçeskoj teoryy lynejn¥x πllyptyçeskyx zadaç [3]
sleduet, çto zadaça ( )1m , ( )2m ymeet edynstvennoe ohranyçennoe neotrycatel\-
noe reßenye. ∏to reßenye dlq poçty vsex x B∈ udovletvorqet yntehral\-
nomu uravnenyg
u xm ( ) = K x y y dS
B
y( , ) ( )
∂
∫ φ + G x y V y u y dy
B
m m( , ) ( ) ( )∫ ,
hde K x y( , ) — qdro Puassona, G x y( , ) — funkcyq Hryna zadaçy −∆ u = 0,
u B∂ = 0, pryçem G x y( , ) ≥ 0, K x y( , ) > 0 v B. Oçevydno, çto posledovatel\-
nost\ u xm ( ){ } monotonno vozrastaet.
PokaΩem, çto predel u x( ) posledovatel\nosty u xm ( ){ } reßenyj zadaçy
( )1m , ( )2m qvlqetsq reßenyem zadaçy (1), (2). UmnoΩym uravnenye ( )1m na
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
∏LLYPTYÇESKOE URAVNENYE S SYNHULQRNÁM POTENCYALOM 1717
um
p p− −1 2 2ϕ ψ , p ≥ 2, hde ψ = ψ( )x — srezagwaq funkcyq dlq ßara B , ϕ =
= ϕ( )x = x −α
y α opredelqetsq yz ravenstva α(n – 2 – α) = c, y proyntehry-
ruem po B :
−∫ − −∆ u u dxm
B
m
p p1 2 2ϕ ψ = V x u u dxm
B
m m
p p( )∫ − −1 2 2ϕ ψ .
PoloΩym km = um/ϕ . Posle prqm¥x v¥çyslenyj budem ymet\
4 1
2
2 2 2 2( )p
p
k dxm
p
B
−
∇ ( )∫ / ϕ ψ + k dxm
p
B
∫ −ϕ ϕ ψ( )∆ 2 +
+ 2 1 2∇ ∇∫ −k k dxm
B
m
p ϕ ψ ψ = V x k dxm
B
m
p∫ ( ) ϕ ψ2 2 .
Poskol\ku po predpoloΩenyg −∆ ϕϕ = V x0
2( )ϕ ≥ V xm ( )ϕ2
, yz posledneho
ravenstva poluçaem
4 1
2
2 2 2 2( )p
p
k dxm
p
B
−
∇ ( )∫ / ϕ ψ ≤ 2 1 2∇ ∇−∫ k k dxm m
p
B
ϕ ψ ψ . (3)
Dlq yntehrala v pravoj çasty yspol\zuem ob¥çnoe neravenstvo Koßy:
2 1 2∇ ∇∫ −k k dxm
B
m
p ϕ ψ ψ ≤
2
2
2 2 2 2
p
k dxm
p
B
∇ ( )∫ / ϕ ψ + 2 2 2k dxm
p
B
∫ ∇ϕ ψ .
Teper\ neravenstvo (3) moΩno zapysat\ v vyde
∇ ( )∫ k dxm
p
B
/2 2 2 2ϕ ψ ≤
p
p
k dxm
p
B
2
2 2
2 3 2( )−
∇∫/
ϕ ψ . (4)
Yspol\zuq yntehryrovanye po çastqm y neravenstvo Hel\dera, moΩno dokazat\
sledugwyj fakt.
Pust\ 0 ≤ h s( ) ∈ C r1 0,[ ] y h r( ) = 0, 0 < r ≤ 1. Tohda pry p ≥ 2 v¥pol-
nqetsq neravenstvo
h s dsnp n n
r n n
/
/
( )
( )
− − −
−
∫
2 2 1
0
2
α ≤ K h s s dsp
r
n/2
0
2
2 1( )( )′∫ − −α , (5)
hde postoqnnaq K = K(n, α) > 0 y α opredelqetsq yz ravenstva α(n – 2 – α) =
= c.
Pust\ Br = B r( , )0 ⊂⊂ B. V¥berem funkcyg ψ( )x sledugwym obrazom:
0 ≤ ψ( )x ≤ 1, ψ( )x = 1 pry x ∈ Br−δ , ψ( )x = 0 pry x ∈ B Br\ , δ > 0. Predpo-
loΩym, çto ∇ ψ 2 ≤ C1
2δ−
. Tohda neravenstvo (4) prymet vyd
∇( )
−
∫ k dxm
p
Br
/2 2 2
δ
ϕ ≤
C p
p
k dxm
p
Br
1
2 2
2
2 3 2
δ
ϕ
−
− ∫( )/
. (6)
Yz neravenstva (5) sleduet, çto dlq lgboj neotrycatel\noj funkcyy v( )x ∈
∈ C Br0
1( ) v¥polnqetsq neravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
1718 B. A. XUDAJHULÁEV
vnp n
B
n n
r
x dx/( )
( )/
( )−
−
∫
2 2
2
ϕ ≤ C x dxp
Br
2
2 2 2∇( )∫ v / ( ) ϕ . (7)
Poπtomu, yspol\zuq neravenstvo (6) y neravenstvo Koßy, poluçaem
k x dxm
np n
B
n n
r
/( )
( )/
( )−
−
−
∫
2 2
2
δ
ϕ ≤ ( ) ( )/( )
( )/
k x dxm
np n
B
n n
r
ψ ϕ−
−
∫
2 2
2
≤
≤ C k x dxm
p
Br
2
2 2 2∇( )∫ / ( ) ψ ϕ ≤
C p
p
k x dxm
p
Br
3
2 2
2
3 2
δ
ϕ
−
− ∫/
( ) .
Takym obrazom,
k x dxm
np n
B
n pn
r
/( )
( )
( )−
−
−
∫
2 2
2
δ
ϕ
/
≤
C p
p
k x dx
p
m
p
B
p
r
3
2 2 1
2
1
3 2
δ
ϕ
−
−
∫/
/ /
( ) . (8)
Pust\ ε > 0 — dostatoçno maloe çyslo. PoloΩym
δ
ε
=
2 j
, r r1 = , r rj j j+ = −1
2
ε
,
p
n
n
j
j
=
−
−
2
2
1
, j = 1, 2, … .
Tohda v πtyx oboznaçenyqx neravenstvo (8) prymet vyd
k x dxm
p
B
p
j
rj
j
+
+
+
∫
1
1
1
2
1
( )
( )
ϕ
/
≤
C p
p
k x dx
j
j
j
p
m
p
B
j
j
rj
3
2 2
2
1
22
3 2ε
ϕ
( )
( )
/
−
∫
/
1/ pj
, (9)
otkuda po yndukcyy naxodym
k x dxm
p
B
p
j
rj
j
+
+
+
∫
1
1
1
2
1
( )
( )
ϕ
/
≤
C p
p
j
j
p j
3
2
2
1
3 2ε ( )
/
−
/
…
…
C p
p
p
i pii
j
3 1
2
2
1
1
2
3 2
2
1
1
ε ( )
/
/
−
⋅ ∑ =
/
. (10)
No rqd
1
3 21
3
2
2p
C p
pj
j
j
j
=
∞∑
−
ln
( )ε /
= α jj=
∞∑ 1
sxodytsq, tak kak α j ∼
∼ µ µ− j jCln ( ) pry µ > 1. Krome toho, sxodytsq y rqd
2
1
j
p j
j=
∞∑ =
= j
n
nj
j−
=
∞
−
∑ 2
1
1
. Otsgda dlq vsex j ≥ 1 ymeem k x dxm
p
B
p
j
rj
j
( )∫
ϕ2
1/
≤ A,
pryçem postoqnnaq A > 0 ne zavysyt ot p. Poπtomu, perexodq v neravenst-
veO(10) k predelu pry j → ∞, poluçaem sup ( )k xm ≤ A dlq poçty vsex x B∈ .
∏to oznaçaet, çto k xm ( ) ≤ A dlq poçty vsex x B∈ y u xm ( ) ≤ A xϕ( ) dlq poç-
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
∏LLYPTYÇESKOE URAVNENYE S SYNHULQRNÁM POTENCYALOM 1719
ty vsex x B∈ . V sylu monotonnosty posledovatel\nosty u xm ( ){ } moΩem po-
loΩyt\ k = limm mk→∞ , u = limm mu→∞ dlq poçty vsex x B∈ .
Teper\ pokaΩem, çto Vu L B∈ 1, ( )loc . Zafyksyruem toçku x B0 ∈ takug,
çto u x( )0 koneçna. Tohda dlq lgboho m y dlq lgboj kompaktnoj podoblas-
ty Ω ⊂ B yz yntehral\noho uravnenyq dlq funkcyy u xm ( ) ymeem
u xm ( )0 ≥ G x y V y u y dym m( , ) ( ) ( )0
Ω
∫ .
No v sylu strohoho pryncypa maksymuma suwestvuet postoqnnaq C0 0> ta-
kaq, çto G x y( , )0 ≥ C0 v Ω . Poπtomu
V y u y dym m( ) ( )
Ω
∫ ≤ C u xm0
1
0
− ( ) ≤ C u x0
1
0
− ( ) . (11)
∏to y pokaz¥vaet, çto Vu L B∈ 1, ( )loc .
Poskol\ku um — reßenye zadaçy ( )1m , ( )2m , dlq lgboj funkcyy ξ( )x ∈
∈ C0
∞ ( )Ω ymeem
u dxm
Ω
∆∫ ξ + V u dxm m
Ω
∫ ξ = 0.
Otsgda, uçyt¥vaq (11) y perexodq k predelu pry m → ∞, poluçaem
u dx
Ω
∆∫ ξ + V u dx
Ω
∫ ξ = 0.
∏to y pokaz¥vaet, çto u x( ) qvlqetsq reßenyem zadaçy (1), (2). Pervaq çast\
teorem¥ dokazana.
Teper\ pokaΩem, çto esly V x( ) ≥ V x0( ) , to pry φ > 0 dlq lgboj ohrany-
çennoj podoblasty ′Ω ⊂⊂ B = B R( , )0 takoj, çto 0 ∈ ′Ω , suwestvuet po-
stoqnnaq C = C( , )ε ′Ω > 0 takaq, çto
u x( ) ≥ C x⋅ ϕ( ) (12)
dlq poçty vsex x ∈ ′Ω , hde ϕ( )x = x −α , α opredelqetsq yz ravenstva α(n –
– 2 – α) = c.
Pust\ B0 = B r( , )0 0 ⊂ ′Ω — ßar radyusa r0 s centrom v naçale koordy-
nat. Dalee, pust\ v — reßenye zadaçy
− =∆ v vV0 , v ∂ =B x
0
φ( ) .
Zdes\ v — predel posledovatel\nosty vm{ } edynstvenn¥x neotrycatel\n¥x
reßenyj zadaçy
− =∆ v vm m mV , vm B m x∂ =
0
φ ( ) , (13)
hde Vm = inf ,V m0{ } y φm x( ){ } — posledovatel\nost\ monotonno vozrastag-
wyx neotrycatel\n¥x neprer¥vno dyfferencyruem¥x funkcyj, ravnomerno
sxodqwaqsq k neprer¥vnoj hranyçnoj funkcyy φ( ) ( )x L B∈ ∂1 . Oçevydno, çto
u ≥ v ≥ vm .
PokaΩem, çto dlq poçty vsex x B∈
1
2
0 = Br0 2/ v¥polnqetsq neravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
1720 B. A. XUDAJHULÁEV
v( )x ≥ Const ⋅ ϕ( )x . (14)
Pust\ g : 0, ∞[ [ → 0, ∞[ [ — neotrycatel\naq v¥puklaq funkcyq yz C2
.
UmnoΩym uravnenye yz (13) na ′g k g km m( ) ( )ϕψ2
, hde km =
vm
ϕ
, ψ — srezag-
waq funkcyq dlq ßara B0 , y proyntehryruem po B0 :
− ′∫ ∆ vm m
B
mg k g k dx( ) ( )
0
2ϕψ = V g k g k dxm m m
B
mv ′∫ ( ) ( )
0
2ϕψ .
Otsgda posle prqm¥x v¥çyslenyj poluçaem
∇∫ g k dxm
B
( )
0
2 2 2ϕ ψ + ′′ ∇∫ g k k g k dxm
B
m m( ) ( )
0
2 2 2ϕ ψ +
+ ∇ ∇∫ g k g k dxm m
B
( ) ( )
0
2 2ϕ ψ = ( ) ( ) ( )∆ ϕ ϕ ϕψ+ ′∫ V g k g k k dxm
B
m m m
0
2
.
No vtoroj çlen v levoj çasty neotrycatelen, tak kak funkcyq g v¥pukla y
neotrycatel\na. Dlq tret\eho çlena v levoj çasty yspol\zuem ob¥çnoe nera-
venstvo Koßy:
2
0
2∇ ∇∫ g k g k dxm
B
m( ) ( )ϕ ψ ψ ≤
1
2
2 2 2
0
∇∫ g k dxm
B
( ) ϕ ψ +
+ 2 2 2 2
0
g k dxm
B
( )∫ ∇ϕ ψ .
Poπtomu
1
2
2 2 2
0
∇∫ g k dxm
B
( ) ϕ ψ ≤ 2 2 2 2
0
g k dxm
B
( )∫ ∇ϕ ψ +
+ ( ) ( ) ( )∆ ϕ ϕ ϕψ+ ′∫ V g k g k k dxm
B
m m m
0
2 .
Pust\ B B rr = ( , )0 — ßar dostatoçno maloho radyusa. Teper\ predpoloΩym,
çto ϕ ϕ∆ ∈ L Br1( ) (çto πkvyvalentno uslovyg α < (n – 2) / 2). Poskol\ku po
predpoloΩenyg funkcyq g v¥pukla y neotrycatel\na, V xm ( ) ≤ V x0( ) =
= −∆ ϕ ϕ/ y, krome toho, km ∞ ohranyçena, po teoreme Lebeha o predel\nom
perexode pod znakom yntehrala vtoroj çlen v pravoj çasty posledneho nera-
venstva stremytsq k nulg pry m → ∞ . Perexodq k predelu pry m → ∞ , po-
luçaem
∇∫ g k dx
Br
( ) 2 2 2ϕ ψ ≤ 4 2 2 2g k dx
Br
( )∫ ∇ϕ ψ . (15)
V¥berem funkcyg ψ( )x sledugwym obrazom: 0 ≤ ψ( )x ≤ 1, ψ( )x = 1 v Br−δ ,
ψ( )x = 0 v B Br0 \ , δ > 0. PredpoloΩym, çto ∇ ψ 2 ≤ C4
2δ−
, hde postoqnnaq
C4 0> ne zavysyt ot δ. Tohda neravenstvo (15) prymet vyd
∇
−
∫ g k dx
Br
( ) 2 2
δ
ϕ ≤ 4 4
2 2 2C g k dx
Br
δ ϕ− ∫ ( ) . (16)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
∏LLYPTYÇESKOE URAVNENYE S SYNHULQRNÁM POTENCYALOM 1721
Teper\ nam ponadobytsq sledugwyj fakt yz rabot¥ [1, s. 129]:
esly 0 < ′r ≤ r ≤ 1 y h s( ) ∈ C r1 0,[ ] , to v¥polnqetsq neravenstvo
h s s dsq
r
n
q
( )
0
2 1
2
∫ − −
α
/
≤ C h s h s s ds
r
n
5
2 2
0
2 1′ +
∫ − −( ) ( ) α , (17)
hde
1
q
>
1
2
–
1
2n − α
(y q < ∞, esly n – 2α = 2). Postoqnnaq C5 zavysyt ot
′r , no ne zavysyt ot r.
Zameçanye 1. Esly Q zadano y 2 ≤ q ≤ min Q{ , 2 2 2 2 1( ) ( )n n− − − }−α α ,
to postoqnnaq C5 v neravenstve (17) ravnomerno ohranyçena pry α ∈ 0[ ,
( )n − ]2 2/ .
Opredelym β formuloj β +
2
q
= 1, hde 1 >
2
q
≥
1
2
–
1
2n − α
. Yspol\zuq
neravenstvo Hel\dera y neravenstvo (17) dlq neotrycatel\noj funkcyy h,
ymeem
h dx
Br
2 2 2+∫ βϕ ≤ h dx h dxq
B
q
Br r
ϕ ϕ
β
2
2
2 2∫ ∫
/
≤
≤ C h dx h dx h dx
B B Br r r
5
2 2 2 2 2 2∇ +
∫ ∫ ∫ϕ ϕ ϕ
ββ
, (18)
otkuda, zamenqq h na g k( ) y Br na Br−δ , poluçaem
g k dx
Br
2 2 2+
−
∫ β
δ
ϕ( ) ≤
≤ C g k dx g k dx g k
B Br r
5
2 2 2 2 2 2∇ +
−
∫ ∫( ) ( ) ( )
δ
ϕ ϕ ϕ
BBr
dx∫
β
.
Yspol\zuq neravenstvo (16), yz posledneho neravenstva naxodym
g k dx
Br
2 2 2+
−
∫ β
δ
ϕ( ) ≤ C C g k dx g k dx
B Br r
5 4
2 2 2 2 24 1( ) ( ) ( )δ ϕ ϕ
β
− +
∫ ∫ ≤
≤ C g k dx
Br
6
2 2 2
1
δ ϕ
β
−
+
∫
( )
yly
g k dx
Br
2 2 2
1 2 2
+
+
−
∫
β
β
ϕ
δ
( )
/( )
≤ ( ) ( )/( )C g k dx
Br
6
2 1 2 2 2 2
1 2
δ ϕβ− + ∫
/
≤
≤ C g k dx
Br
7
1 2 2
1 2
δ ϕ− ∫
( )
/
. (19)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
1722 B. A. XUDAJHULÁEV
Pust\ ε > 0 — dostatoçno maloe çyslo. PoloΩym
δ
ε
=
2 j
, r r1 = , r rj j j+ = −1
2
ε
, g j+1 = g j
1+β ,
H j = g k dxj
Brj
2 2
1 2
( )
/
ϕ∫
, j = 1, 2, 3, … , g g1 = .
V πtyx oboznaçenyqx neravenstvo (19) prymet vyd
H j+
+
1
1 1/( )β ≤ C Hj
j7
12 ε− ,
otkuda po yndukcyy
H j
1 1/( )+β ≤ C H
j j
j
7
1
1
12
2
ε
α γ β− +( ) ( ) –
,
hde α j = ( ) ( )1 12
0
2+ +− −
=
−∑β β ν
ν
j j
, γ j = ( ) ( )1 1 2
0
1 + + − −
=
−∑ ν β ν
ν
jj
. Poskol\ku
g j = g
j( )1 1+ −β
, perexodq v poslednem neravenstve k predelu pry j → ∞ , po-
luçaem
sup ( )
Br
g k x
−
( )
ε
≤ C g k dx
Br
7
1 1 1 2 2
1 2
2ε ϕβ β β β− + +( )
∫( ) ( )
/
( )/ /
.
Zamenym g posledovatel\nost\g gl{ } , hde gl te Ωe, çto y g, g kl ( ) → k−γ
pry l → ∞. Tohda
sup ( )
Br
k x
−
−
ε
γ ≤ C k dx
Br
7
1 1 1 2 2
1 2
2ε ϕβ β β β γ− + + −( )
∫( ) ( )
/
/ /
.
Tak kak pry r r< 0 dlq poçty vsex x Br∈ : k x( ) = v( ) ( )x x/ϕ ≥ C x8
1ϕ− ( ) ,
ymeem
sup ( )
Br
k x
−
−
ε
γ ≤ C dx
Br
9
1 1 2 2
1 2
ε ϕβ γ− − +∫
/
/
,
otkuda
k x( ) ≥ C dx
Br
10
1 1 2 2
1 2
ε ϕβ γ γ
γ
( )
/
+ +
−
∫
/ /
(20)
dlq poçty vsex x Br∈ −ε , Zdes\ postoqnnaq C10 0> ne zavysyt ot r y ε. Ne-
ravenstvo (14), a sledovatel\no, y neravenstvo (12) dokazan¥ dlq sluçaq α <
< ( )/n − 2 2 ; netrudno pokazat\, çto πto neravenstvo ymeet mesto y v predel\nom
sluçae α = ( )/n − 2 2 .
Perejdem k dokazatel\stvu vtoroj çasty teorem¥.
Pust\ c C n> ∗( ) , V x V x( ) ( )≥ 0 y φ( )x > 0 . Esly zadaça (1), (2) ymeet reße-
nye, to ono udovletvorqet uravnenyg
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
∏LLYPTYÇESKOE URAVNENYE S SYNHULQRNÁM POTENCYALOM 1723
−∆ u = C n x u∗
−( ) 2 + c C n x u−( )∗
−( ) 2
v ′D B( ) . UmnoΩaq πto uravnenye na x n− −( )/2 2
y yntehryruq po B, poluçaem
− − −∫ ∆ u x dxn
B
( )/2 2 = C u x dxn
B
∗
− +∫ ( )/2 2 + ( ) ( )/c C u x dxn
B
− ∗
− +∫ 2 2
.
No yz neravenstva (12) sleduet, çto dlq lgboho ßara B Br ⊂⊂ : u x( ) ≥
≥ Const ⋅ϕ( )x = Const ⋅ − −x n( )/2 2
, poπtomu
u x x dxn
Br
( ) ( )/− +∫ 2 2 ≥ Const x dxn
Br
−∫ = ∞.
∏to y dokaz¥vaet vtorug çast\ teorem¥.
Teorema dokazana.
Zameçanye 2. Teorema moΩet b¥t\ dokazana dlq potencyalov V x( ) =
= −∆ ϕ ϕ/ , hde ϕ > 0, ∆ ϕ ∈L B1( ) y dlq kotor¥x pry nekotorom q > 2 v¥pol-
nqetsq neravenstvo
h dxq
B
q
ϕ2
1
∫
/
≤ Const ∇ +
∫∫ h dx h dx
BB
2 2 2 2
1 2
ϕ ϕ
/
.
1. Baras P., Goldstein J. A. The heat equation with a singular potential // Trans. Amer. Math. Soc. –
1984. – 284, # 1. – P. 121 – 139.
2. Garsia Azorero J., Peral I. Hardy inequalities and some critical elliptic and parabolic problems //
Different. Equat. – 1998. – 144. – P. 441 – 476.
3. Hylbarh D., Trudynher N. ∏llyptyçeskye dyfferencyal\n¥e uravnenyq s çastn¥my proyz-
vodn¥my vtoroho porqdka – M.: Nauka, 1989.
Poluçeno 23.10.09,
posle dorabotky — 27.07.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
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| id | umjimathkievua-article-2994 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:34:12Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fd/e0a2d830d190845f740281a399f7fbfd.pdf |
| spelling | umjimathkievua-article-29942020-03-18T19:41:53Z Elliptic equation with singular potential Эллиптическое уравнение с сингулярным потенциалом Hudaigulyev, B. A. Худашулыев, Б. А. Худашулыев, Б. А. We consider the following problem of finding a nonnegative function $u(x)$ in a ball $B = B(O, R) ⊂ R^n,\; n ≥ 3:$ $$−Δu=V(x)u,u|∂B=ϕ(x),$$ where $Δ$ is the Laplace operator, $x = (x 1, x 2,…, x n )$, and $∂B$ is the boundary of the ball $B$. It is assumed that $0 ≤ V(x) ∈ L 1(B), 0 ≤ φ(x) ∈ L 1(∂B)$, and $φ(x)$ is continuous on $∂B$. We study the behavior of nonnegative solutions of this problem and prove that there exists a constant $C_{*} (n) = (n − 2)^2/4$ such that if $V_0 (x) = \frac{c}{|x|^2}, then, for $0 ≤ c ≤ $C_{*} (n)$ and $V(x) ≤ V_0 (x)$ in the ball $B$, this problem has a nonnegative solution for any nonnegative continuous boundary function $φ(x) ∈ L_1(∂B)$, whereas, for $c > C_{*} (n)$ and $V(x) ≥ V_0(x)$, the ball $B$ does not contain nonnegative solutions if $φ(x) > 0$. Розглядається задача знаходження невід'ємної функції $u(x)$ у кулі $B = B(O, R) ⊂ R^n,\; n ≥ 3:$ $$−Δu=V(x)u,u|∂B=ϕ(x),$$ де $Δ$ — оператор Лапласа, $x = (x 1, x 2,…, x n )$, $∂B$ —межа кулі $B$, у припущенні, що $0 ≤ V(x) ∈ L 1(B), 0 ≤ φ(x) ∈ L 1(∂B)$ і $φ(x)$ неперервна на ЭВ. Вивчається поведінка невід'ємних розв'язків цієї задачі і доведено, що існує стала $C_{*} (n) = (n − 2)^2/4$ така, що якщо $V_0 (x) = \frac{c}{|x|^2}, то ця задача при $0 ≤ c ≤ $C_{*} (n)$ і $V(x) ≤ V_0(x)$ кулі $В$ має невід'ємний розв'язок при будь-якій невід'ємній неперервній граничній функції $φ(x) ∈ L_1(∂B)$, а при $0 ≤ c ≤ C_{*} (n)$ і $V(x) ≥ V_0(x)$ у кулі $В$ не має невід'ємних розв'язків, якщо $φ(x) > 0$. Institute of Mathematics, NAS of Ukraine 2010-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2994 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 12 (2010); 1715 – 1723 Український математичний журнал; Том 62 № 12 (2010); 1715 – 1723 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2994/2738 https://umj.imath.kiev.ua/index.php/umj/article/view/2994/2739 Copyright (c) 2010 Hudaigulyev B. A. |
| spellingShingle | Hudaigulyev, B. A. Худашулыев, Б. А. Худашулыев, Б. А. Elliptic equation with singular potential |
| title | Elliptic equation with singular potential |
| title_alt | Эллиптическое уравнение с сингулярным потенциалом |
| title_full | Elliptic equation with singular potential |
| title_fullStr | Elliptic equation with singular potential |
| title_full_unstemmed | Elliptic equation with singular potential |
| title_short | Elliptic equation with singular potential |
| title_sort | elliptic equation with singular potential |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2994 |
| work_keys_str_mv | AT hudaigulyevba ellipticequationwithsingularpotential AT hudašulyevba ellipticequationwithsingularpotential AT hudašulyevba ellipticequationwithsingularpotential AT hudaigulyevba élliptičeskoeuravneniessingulârnympotencialom AT hudašulyevba élliptičeskoeuravneniessingulârnympotencialom AT hudašulyevba élliptičeskoeuravneniessingulârnympotencialom |