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研究论文 | 更新时间：2026-05-22
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The existence of solutions for a class of nonlinear elliptic equations with Hardy potential
- 中山大学学报(自然科学版)(中英文) 2026年65卷第3期 页码：144-150
- 作者机构：
鲁东大学数学与统计科学学院，山东 烟台 264025
- 作者简介：
刘晶晶（2000年生），女；研究方向：微分方程及其应用；E-mail：ljjli1@sina.com
王琳琳（1976年生），女；研究方向：非线性分析及微分方程；E-mail：llwang@ldu.edu.cn
- 基金信息：鲁东大学研究生创新项目(IPGS2024-041);国家自然科学基金(11201213);山东省优质研究生课程建设项目(SDYKC2025172)
DOI：10.11714/acta.snus.ZR20240268
中图分类号： O175.2收稿：2024-09-03，
修回：2025-11-17，
录用：2025-11-18，
网络首发：2026-03-11，
纸质出版：2026-05-25
- 稿件说明：
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刘晶晶,王琳琳.带有Hardy项的一类非线性椭圆型方程解的存在性[J].中山大学学报(自然科学版)(中英文),2026,65(03):144-150.
LIU Jingjing,WANG Linlin.The existence of solutions for a class of nonlinear elliptic equations with Hardy potential[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2026,65(03):144-150.
刘晶晶,王琳琳.带有Hardy项的一类非线性椭圆型方程解的存在性[J].中山大学学报(自然科学版)(中英文),2026,65(03):144-150. DOI： 10.11714/acta.snus.ZR20240268.
LIU Jingjing,WANG Linlin.The existence of solutions for a class of nonlinear elliptic equations with Hardy potential[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2026,65(03):144-150. DOI： 10.11714/acta.snus.ZR20240268.
带有![]()
![]()
摘要
研究了一类非线性椭圆型方程
<math id="M2"><mo>-</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>-</mo><mrow><mrow><mi>λ</mi><mi>u</mi></mrow><mo>/</mo><mrow><mtext> </mtext><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow><mo>+</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>
</mo><mtext> </mtext><mi>x</mi><mo>∈</mo><mi mathvariant="normal">Ω</mi><mo>\</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo><mo>
</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791318&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791304&type=
59.35132980
4.23333359
其中
<math id="M3"><mi mathvariant="normal">Ω</mi><mo>⊆</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mtext> </mtext><mi>N</mi></mrow></msup><mo stretchy="false">(</mo><mi>N</mi><mo>≥</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791329&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791341&type=
21.33600044
3.21733332
是一个包含原点的光滑有界域.当
<math id="M4"><mi>λ</mi><mo>></mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><mn mathvariant="normal">4</mn></mrow></mrow></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791321&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791307&type=
19.98133469
3.47133350
时，通过上下解方法，得到该方程正解的存在性；通过比较原理得到
<math id="M5"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mi>θ</mi></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><mo>≤</mo><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mi>θ</mi></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><mo>
</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791332&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791344&type=
39.62400055
5.41866684
进而得到
<math id="M6"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>→</mo><mn mathvariant="normal">0</mn></mrow></munder><mfrac><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mi>m</mi><mi>r</mi><mo>
</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791356&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791333&type=
23.36800003
7.28133297
其中
<math id="M7"><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mi>θ</mi></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><mo>.</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791348&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791370&type=
26.67000008
5.41866684
Abstract
In this paper， we study a class of nonlinear elliptic equation
<math id="M8"><mo>-</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>-</mo><mrow><mrow><mi>λ</mi><mi>u</mi></mrow><mo>/</mo><mrow><mtext> </mtext><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow><mo>+</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>
</mo><mtext> </mtext><mi>x</mi><mo>∈</mo><mi mathvariant="normal">Ω</mi><mo>\</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791372&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791349&type=
64.93933105
4.23333359
，where
<math id="M9"><mi mathvariant="normal">Ω</mi><mo>⊆</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mtext> </mtext><mi>N</mi></mrow></msup><mo stretchy="false">(</mo><mi>N</mi><mo>≥</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791361&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791360&type=
21.33600044
3.21733332
is a smooth bounded domain and
<math id="M10"><mn mathvariant="normal">0</mn><mo>∈</mo><mi mathvariant="normal">Ω</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791385&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791384&type=
7.87400007
2.37066650
. When
<math id="M11"><mi>λ</mi><mo>></mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><mn mathvariant="normal">4</mn></mrow></mrow></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791399&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791376&type=
20.91266632
3.55599999
， by means of the sub and super solution method， we explore the existence of positive solutions. By the comparison principle， we obtain
<math id="M12"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mi>θ</mi></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><mo>≤</mo><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mi>θ</mi></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791365&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791387&type=
42.41799927
5.92666674
， and then
<math id="M13"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>→</mo><mn mathvariant="normal">0</mn></mrow></munder><mfrac><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mi>m</mi><mi>r</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791380&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791401&type=
25.14599991
8.46666718
has been proven， where
<math id="M14"><msub><mrow><mi>U</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mi>θ</mi></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></mrow></msup></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791413&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108791390&type=
28.95599937
6.09600019
.
关键词
Keywords
references
ADIMURTHI ， ESTEBAN M J ， 2005 . An improved Hardy-Sobolev inequality in W 1， p and its application to Schrödinger operators ［J］. Nonlinear Differ Equ Appl ， 12 ： 243 - 263 .
BREZIS H ， VÁZQUEZ J L ， 1997 . Blow-up solutions of some nonlinear elliptic problems ［J］. Rev Mat Complut ， 10 （ 2 ）： 443 - 469 .
CABRÉ X ， MARTEL Y ， 1998 . Weak eigenfunctions for the linearization of extremal elliptic problems ［J］. J Funct Anal ， 156 （ 1 ）： 30 - 56 .
CARMONA J ， MARTÍNEZ-APARICIO P J ， SUÁREZ A ， 2013 . Existence and non-existence of positive solutions for nonlinear elliptic singular equations with natural growth ［J］. Nonlinear Anal Theory Meth Appl ， 89 ： 157 - 169 .
CÎRSTEA F C ， FĂRCĂŞEANU M ， 2021 . Sharp existence and classification results for nonlinear elliptic equations in <math id="M341"><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>\</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108793461&type= https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=108793452&type= 10.15999985 3.04800010 with Hardy potential ［J］. J Differ Equ ， 292 ： 461 - 500 .
CÎRSTEA F C ， RĂDULESCU V ， 2004 . Extremal singular solutions for degenerate logistic-type equations in anisotropic media ［J］. Comptes Rendus Math ， 339 （ 2 ）： 119 - 124 .
DU Y H ， 2006 . Order Structure and topological methods in nonlinear partial differential equations： Vol . 1 ： Maximum principles and applications ［M］. Singapore ： World Scientific .
GARCÍA AZORERO J P ， PERAL ALONSO I ， 1998 . Hardy inequalities and some critical elliptic and parabolic problems ［J］. J Differ Equ ， 144 （ 2 ）： 441 - 476 .
MONTENEGRO M ， TORDECILLA J A L ， 2021 . Existence of positive solution for elliptic equations with singular terms and combined nonlinearities ［J］. J Math Anal Appl ， 503 （ 2 ）： 125316 .
SAEEDI G ， HUSAINI M A ， 2023 . Existence of solutions to a class of quasilinear elliptic equations ［J］. J Math Anal Appl ， 521 （ 1 ）： 126901 .
VÁZQUEZ J L ， ZUAZUA E ， 2000 . The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential ［J］. J Funct Anal ， 173 （ 1 ）： 103 - 153 .
WANG M C ， ZHANG Q ， 2019 . Existence of solutions for singular critical semilinear elliptic equation ［J］. Appl Math Lett ， 94 ： 217 - 223 .
WEI L ， DU Y H ， 2017 . Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential ［J］. J Differ Equ ， 262 （ 7 ）： 3864 - 3886 .
ZHANG Q Y ， ZHANG M Z ， 2024 . Multiple solutions for perturbed semilinear Schrödinger equations ［J］. J Math Anal Appl ， 536 （ 2 ）： 128220 .
ZHANG X G ， JIANG J Q ， WU Y H ， et al ， 2019 . Existence and asymptotic properties of solutions for a nonlinear Schrödinger elliptic equation from geophysical fluid flows ［J］. Appl Math Lett ， 90 ： 229 - 237 .
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