The exact boundary blow-up rate of large solutions for semilinear elliptic problems
In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem − △ u = λ u − a ( x ) u p , u | ∂ Ω = + ∞ , where Ω is a bounded smooth domain in R N . The weight function a ( x ) in front of the nonlinearity can vanish on the boundary of the domain...
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| Veröffentlicht in: | Nonlinear analysis 2008-05, Vol.68 (9), p.2791-2800 |
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| creator | Ouyang, Tiancheng Xie, Zhifu |
| description | In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem
−
△
u
=
λ
u
−
a
(
x
)
u
p
,
u
|
∂
Ω
=
+
∞
, where
Ω
is a bounded smooth domain in
R
N
. The weight function
a
(
x
)
in front of the nonlinearity can vanish on the boundary of the domain
Ω
at different rates according to the point
x
0
of the boundary. The decay rate of the weight function
a
(
x
)
may not be approximated by a power function of distance near the boundary
∂
Ω
. We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution
u
(
x
)
must satisfy
lim
x
→
x
0
u
(
x
)
K
(
b
x
0
∗
(
dist
(
x
,
∂
Ω
)
)
)
−
β
=
1
for each
x
0
∈
∂
Ω
,
where
b
x
0
∗
(
r
)
=
∫
0
r
∫
0
s
b
x
0
(
t
)
d
t
d
s
,
K
=
[
β
(
(
β
+
1
)
C
0
−
1
)
]
1
p
−
1
,
β
=
1
p
−
1
,
C
0
=
lim
r
→
0
(
∫
0
r
b
x
0
(
t
)
d
t
)
2
b
x
0
∗
(
r
)
b
x
0
(
r
)
and
b
x
0
(
r
)
is the boundary normal section of
a
(
x
)
at
x
0
∈
∂
Ω
, i.e.,
b
x
0
(
r
)
=
a
(
x
0
−
r
n
x
0
)
,
r
>
0
,
r
∼
0
,
and
n
x
0
stands for the outward unit normal vector at
x
0
∈
∂
Ω
. |
| doi_str_mv | 10.1016/j.na.2007.02.026 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_miscellaneous_32706397</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0362546X07001757</els_id><sourcerecordid>32706397</sourcerecordid><originalsourceid>FETCH-LOGICAL-c355t-50f425db88b90efea74f30fcb0336c0a55beaa421937cc72f29e0a10880490143</originalsourceid><addsrcrecordid>eNp1UE1rGzEQFaWFum7uPerS3NYdSav96K2EfBQCOdSF3MSsPEpk5JUr7Sbpv4-MTW6BB3N5H_MeY98ErASI5sd2NeJKArQrkAXNB7YQXasqLYX-yBagGlnpurn_zL7kvAUA0apmwf6sH4nTC9qJD3EeN5j-8yHE52re84QT8eh4wPRAPMcwTz6OmbuYeKadD34kTJxC8PvJW75PcQi0y1_ZJ4ch09npLtnfq8v1xU11e3f9--LXbWWV1lOlwdVSb4auG3ogR9jWToGzAyjVWECtB0KspehVa20rnewJUEDXQd2DqNWSnR99S_C_mfJkdj7b8g6OFOdslGyhUX1biHAk2hRzTuTMPvldqWoEmMN6ZmtGNIf1DMiCpki-n7wxWwwu4Wh9ftPJMp-G7sD7eeRRKfrkKZlsPY2WNj6Rncwm-vdDXgHbRoQl</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>32706397</pqid></control><display><type>article</type><title>The exact boundary blow-up rate of large solutions for semilinear elliptic problems</title><source>Elsevier ScienceDirect Journals</source><creator>Ouyang, Tiancheng ; Xie, Zhifu</creator><creatorcontrib>Ouyang, Tiancheng ; Xie, Zhifu</creatorcontrib><description>In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem
−
△
u
=
λ
u
−
a
(
x
)
u
p
,
u
|
∂
Ω
=
+
∞
, where
Ω
is a bounded smooth domain in
R
N
. The weight function
a
(
x
)
in front of the nonlinearity can vanish on the boundary of the domain
Ω
at different rates according to the point
x
0
of the boundary. The decay rate of the weight function
a
(
x
)
may not be approximated by a power function of distance near the boundary
∂
Ω
. We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution
u
(
x
)
must satisfy
lim
x
→
x
0
u
(
x
)
K
(
b
x
0
∗
(
dist
(
x
,
∂
Ω
)
)
)
−
β
=
1
for each
x
0
∈
∂
Ω
,
where
b
x
0
∗
(
r
)
=
∫
0
r
∫
0
s
b
x
0
(
t
)
d
t
d
s
,
K
=
[
β
(
(
β
+
1
)
C
0
−
1
)
]
1
p
−
1
,
β
=
1
p
−
1
,
C
0
=
lim
r
→
0
(
∫
0
r
b
x
0
(
t
)
d
t
)
2
b
x
0
∗
(
r
)
b
x
0
(
r
)
and
b
x
0
(
r
)
is the boundary normal section of
a
(
x
)
at
x
0
∈
∂
Ω
, i.e.,
b
x
0
(
r
)
=
a
(
x
0
−
r
n
x
0
)
,
r
>
0
,
r
∼
0
,
and
n
x
0
stands for the outward unit normal vector at
x
0
∈
∂
Ω
.</description><identifier>ISSN: 0362-546X</identifier><identifier>EISSN: 1873-5215</identifier><identifier>DOI: 10.1016/j.na.2007.02.026</identifier><identifier>CODEN: NOANDD</identifier><language>eng</language><publisher>Amsterdam: Elsevier Ltd</publisher><subject>Blow-up rate ; Exact sciences and technology ; Large positive solution ; Mathematical analysis ; Mathematics ; Partial differential equations ; Sciences and techniques of general use ; Semilinear elliptic equation ; Uniqueness</subject><ispartof>Nonlinear analysis, 2008-05, Vol.68 (9), p.2791-2800</ispartof><rights>2007 Elsevier Ltd</rights><rights>2015 INIST-CNRS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c355t-50f425db88b90efea74f30fcb0336c0a55beaa421937cc72f29e0a10880490143</citedby><cites>FETCH-LOGICAL-c355t-50f425db88b90efea74f30fcb0336c0a55beaa421937cc72f29e0a10880490143</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0362546X07001757$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,776,780,3536,27903,27904,65309</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=20175086$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Ouyang, Tiancheng</creatorcontrib><creatorcontrib>Xie, Zhifu</creatorcontrib><title>The exact boundary blow-up rate of large solutions for semilinear elliptic problems</title><title>Nonlinear analysis</title><description>In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem
−
△
u
=
λ
u
−
a
(
x
)
u
p
,
u
|
∂
Ω
=
+
∞
, where
Ω
is a bounded smooth domain in
R
N
. The weight function
a
(
x
)
in front of the nonlinearity can vanish on the boundary of the domain
Ω
at different rates according to the point
x
0
of the boundary. The decay rate of the weight function
a
(
x
)
may not be approximated by a power function of distance near the boundary
∂
Ω
. We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution
u
(
x
)
must satisfy
lim
x
→
x
0
u
(
x
)
K
(
b
x
0
∗
(
dist
(
x
,
∂
Ω
)
)
)
−
β
=
1
for each
x
0
∈
∂
Ω
,
where
b
x
0
∗
(
r
)
=
∫
0
r
∫
0
s
b
x
0
(
t
)
d
t
d
s
,
K
=
[
β
(
(
β
+
1
)
C
0
−
1
)
]
1
p
−
1
,
β
=
1
p
−
1
,
C
0
=
lim
r
→
0
(
∫
0
r
b
x
0
(
t
)
d
t
)
2
b
x
0
∗
(
r
)
b
x
0
(
r
)
and
b
x
0
(
r
)
is the boundary normal section of
a
(
x
)
at
x
0
∈
∂
Ω
, i.e.,
b
x
0
(
r
)
=
a
(
x
0
−
r
n
x
0
)
,
r
>
0
,
r
∼
0
,
and
n
x
0
stands for the outward unit normal vector at
x
0
∈
∂
Ω
.</description><subject>Blow-up rate</subject><subject>Exact sciences and technology</subject><subject>Large positive solution</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Partial differential equations</subject><subject>Sciences and techniques of general use</subject><subject>Semilinear elliptic equation</subject><subject>Uniqueness</subject><issn>0362-546X</issn><issn>1873-5215</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2008</creationdate><recordtype>article</recordtype><recordid>eNp1UE1rGzEQFaWFum7uPerS3NYdSav96K2EfBQCOdSF3MSsPEpk5JUr7Sbpv4-MTW6BB3N5H_MeY98ErASI5sd2NeJKArQrkAXNB7YQXasqLYX-yBagGlnpurn_zL7kvAUA0apmwf6sH4nTC9qJD3EeN5j-8yHE52re84QT8eh4wPRAPMcwTz6OmbuYeKadD34kTJxC8PvJW75PcQi0y1_ZJ4ch09npLtnfq8v1xU11e3f9--LXbWWV1lOlwdVSb4auG3ogR9jWToGzAyjVWECtB0KspehVa20rnewJUEDXQd2DqNWSnR99S_C_mfJkdj7b8g6OFOdslGyhUX1biHAk2hRzTuTMPvldqWoEmMN6ZmtGNIf1DMiCpki-n7wxWwwu4Wh9ftPJMp-G7sD7eeRRKfrkKZlsPY2WNj6Rncwm-vdDXgHbRoQl</recordid><startdate>20080501</startdate><enddate>20080501</enddate><creator>Ouyang, Tiancheng</creator><creator>Xie, Zhifu</creator><general>Elsevier Ltd</general><general>Elsevier</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20080501</creationdate><title>The exact boundary blow-up rate of large solutions for semilinear elliptic problems</title><author>Ouyang, Tiancheng ; Xie, Zhifu</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c355t-50f425db88b90efea74f30fcb0336c0a55beaa421937cc72f29e0a10880490143</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2008</creationdate><topic>Blow-up rate</topic><topic>Exact sciences and technology</topic><topic>Large positive solution</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Partial differential equations</topic><topic>Sciences and techniques of general use</topic><topic>Semilinear elliptic equation</topic><topic>Uniqueness</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ouyang, Tiancheng</creatorcontrib><creatorcontrib>Xie, Zhifu</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Nonlinear analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ouyang, Tiancheng</au><au>Xie, Zhifu</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The exact boundary blow-up rate of large solutions for semilinear elliptic problems</atitle><jtitle>Nonlinear analysis</jtitle><date>2008-05-01</date><risdate>2008</risdate><volume>68</volume><issue>9</issue><spage>2791</spage><epage>2800</epage><pages>2791-2800</pages><issn>0362-546X</issn><eissn>1873-5215</eissn><coden>NOANDD</coden><abstract>In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem
−
△
u
=
λ
u
−
a
(
x
)
u
p
,
u
|
∂
Ω
=
+
∞
, where
Ω
is a bounded smooth domain in
R
N
. The weight function
a
(
x
)
in front of the nonlinearity can vanish on the boundary of the domain
Ω
at different rates according to the point
x
0
of the boundary. The decay rate of the weight function
a
(
x
)
may not be approximated by a power function of distance near the boundary
∂
Ω
. We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution
u
(
x
)
must satisfy
lim
x
→
x
0
u
(
x
)
K
(
b
x
0
∗
(
dist
(
x
,
∂
Ω
)
)
)
−
β
=
1
for each
x
0
∈
∂
Ω
,
where
b
x
0
∗
(
r
)
=
∫
0
r
∫
0
s
b
x
0
(
t
)
d
t
d
s
,
K
=
[
β
(
(
β
+
1
)
C
0
−
1
)
]
1
p
−
1
,
β
=
1
p
−
1
,
C
0
=
lim
r
→
0
(
∫
0
r
b
x
0
(
t
)
d
t
)
2
b
x
0
∗
(
r
)
b
x
0
(
r
)
and
b
x
0
(
r
)
is the boundary normal section of
a
(
x
)
at
x
0
∈
∂
Ω
, i.e.,
b
x
0
(
r
)
=
a
(
x
0
−
r
n
x
0
)
,
r
>
0
,
r
∼
0
,
and
n
x
0
stands for the outward unit normal vector at
x
0
∈
∂
Ω
.</abstract><cop>Amsterdam</cop><pub>Elsevier Ltd</pub><doi>10.1016/j.na.2007.02.026</doi><tpages>10</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0362-546X |
| ispartof | Nonlinear analysis, 2008-05, Vol.68 (9), p.2791-2800 |
| issn | 0362-546X 1873-5215 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_32706397 |
| source | Elsevier ScienceDirect Journals |
| subjects | Blow-up rate Exact sciences and technology Large positive solution Mathematical analysis Mathematics Partial differential equations Sciences and techniques of general use Semilinear elliptic equation Uniqueness |
| title | The exact boundary blow-up rate of large solutions for semilinear elliptic problems |
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