带不定权的<inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mi mathvariant="bold-italic">p</mi></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/FB811B8E-85DD-4c90-88FC-605CD1300443-M001.jpg"><?fx-imagestate width="4.23333359" height="6.85799980"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/FB811B8E-85DD-4c90-88FC-605CD1300443-M001c.jpg"><?fx-imagestate width="4.23333359" height="6.85799980"?></graphic></alternatives></inline-formula>-Laplacian方程边值问题正解集的<inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mi mathvariant="bold">S</mi></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/FB811B8E-85DD-4c90-88FC-605CD1300443-M002.jpg"><?fx-imagestate width="4.14866638" height="5.33400011"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/FB811B8E-85DD-4c90-88FC-605CD1300443-M002c.jpg"><?fx-imagestate width="4.14866638" height="5.33400011"?></graphic></alternatives></inline-formula>型连通分支

谭明秋

doi:10.13471/j.cnki.acta.snus.ZR20240302

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带不定权的

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-Laplacian方程边值问题正解集的

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型连通分支

研究论文 | 更新时间：2025-11-14

- 带不定权的 $p$ -Laplacian方程边值问题正解集的 $S$ 型连通分支
- S-shaped component of positive solutions for boundary value problem of $p$ -Laplacian equation with sign-changing weight
- 中山大学学报(自然科学版)(中英文) 2025年64卷第4期页码：156-162
- 作者机构：
  
  西安电子科技大学数学与统计学院，陕西西安 710126
- 作者简介：
  
  谭明秋（1999年生），女；研究方向：常微分方程边值问题；E-mail：23071213324@stu.xidian.edu.cn
- 基金信息：
  
  国家自然科学基金(12061064;12361040)
- DOI：10.13471/j.cnki.acta.snus.ZR20240302
  中图分类号： O175.8
- 收稿：2024-10-14，
  
  录用：2025-02-22，
  
  网络出版：2025-04-27，
  
  纸质出版：2025-07-25
- 稿件说明：
移动端阅览
谭明秋.带不定权的p-Laplacian方程边值问题正解集的S型连通分支[J].中山大学学报(自然科学版)(中英文),2025,64(04):156-162.

TAN Mingqiu.S-shaped component of positive solutions for boundary value problem of p-Laplacian equation with sign-changing weight[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(04):156-162.
谭明秋.带不定权的p-Laplacian方程边值问题正解集的S型连通分支[J].中山大学学报(自然科学版)(中英文),2025,64(04):156-162. DOI： 10.13471/j.cnki.acta.snus.ZR20240302.

TAN Mingqiu.S-shaped component of positive solutions for boundary value problem of p-Laplacian equation with sign-changing weight[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(04):156-162. DOI： 10.13471/j.cnki.acta.snus.ZR20240302.

带不定权的

p

-Laplacian方程边值问题正解集的

S

型连通分支

PDF下载

摘要

研究了带不定权的一维

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type=

2.96333337

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type=

1.77800000

-Laplacian 方程边值问题

</mo><mo> </mo><mo> </mo><mo> </mo><mn mathvariant="normal">0</mn><mo><</mo><mi>t</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>

</mo></mtd></mtr><mtr><mtd><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836478&type=

11.93799973

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836451&type=

73.23667145

正解集的全局结构，其中

</mo><mo> </mo><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836437&type=

4.14866638

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836479&type=

28.27866554

，

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836453&type=

2.28600001

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836465&type=

7.36600018

是一个参数，

<math id="M8"><mi>h</mi><mo>∈</mo><mi>C</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0

1</mn></mrow></mfenced></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836439&type=

3.38666677

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836466&type=

14.90133286

且非负函数

</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836455&type=

3.13266683

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836441&type=

18.11866570

. 当非线性项

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836483&type=

2.96333337

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836457&type=

2.03200006

满足适当的条件时，证得问题存在正解的 S 型连通分支. 主要结果的证明是基于分歧方法.

Abstract

We study the existence for component for positive solutions of

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type=

2.96333337

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type=

1.77800000

-Laplacian equation boundary value problem

</mo><mo> </mo><mo> </mo><mo> </mo><mn mathvariant="normal">0</mn><mo><</mo><mi>t</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836488&type=

11.93799973

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836489&type=

73.23667145

where

</mo><mo> </mo><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836470&type=

4.14866638

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836490&type=

28.27866554

，

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836453&type=

2.28600001

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836465&type=

7.36600018

，

<math id="M15"><mi>h</mi><mo>∈</mo><mi>C</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">0

1</mn></mrow></mfenced></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836471&type=

3.38666677

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836518&type=

14.90133286

and the nonnegative function

</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mfenced></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836505&type=

3.38666677

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836472&type=

17.78000069

. Under some suitable conditions， we show that there exists S-shaped component of positive solutions. The proof of our main result is based upon bifurcation.

关键词

Keywords

references

DAI G W ， 2016 . Bifurcation and one-sign solutions of the <math id="M297"><mi>p</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type= 2.96333337 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type= 1.77800000 -Laplacian involving a nonlinearity with zeros ［J］. Discrete Contin Dyn Syst ， 36 （ 10 ）： 5323 - 5345 .

DAI G W ， MA R Y ， 2012 . Unilateral global bifurcation phenomena and nodal solutions for <math id="M298"><mi>p</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type= 2.96333337 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type= 1.77800000 -Laplacian ［J］. J Differ Equ ， 252 （ 3 ）： 2448 - 2468 .

HAI D D ， 2014 . Nonexistence of positive solutions for a class of <math id="M299"><mi>p</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type= 2.96333337 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type= 1.77800000 -Laplacian boundary value problems ［J］. Appl Math Lett ， 31 ： 12 - 15 .

ITURRIAGA L ， MASSA E ， SÁNCHEZ J ， et al ， 2010 . Positive solutions of the <math id="M300"><mi>p</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type= 2.96333337 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type= 1.77800000 -Laplacian involving a superlinear nonlinearity with zeros ［J］. J Differ Equ ， 248 （ 2 ）： 309 - 327 .

JARǑS J ， KUSANO T ， 1999 . A Picone type identity for second order half-linear differential equations ［J］. Acta Math Univ Comenian ， 68 （ 1 ）： 137 - 151 .

KUSANO T ， NAITO M ， 2001 . Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations ［J］. Rocky Mt J Math ， 31 （ 3 ）： 1039 - 1054 .

LEE Y H ， SIM I ， 2006 . Global bifurcation phenomena for singular one-dimensional <math id="M301"><mi>p</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type= 2.96333337 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type= 1.77800000 -Laplacian ［J］. J Differ Equ ， 229 （ 1 ）： 229 - 256 .

MA R Y ， LIU X L ， XU J ， 2013 . Nodal solutions of the <math id="M302"><mi>p</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type= 2.96333337 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type= 1.77800000 -Laplacian with sign-changing weight ［J］. Abstr Appl Anal ，2013： 248 - 259 .

SIM I ， TANAKA S ， 2015 . Three positive solutions for one-dimensional <math id="M303"><mi>p</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type= 2.96333337 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type= 1.77800000 -Laplacianl problem with sign-changing weight ［J］. Appl Math Lett ， 49 ： 42 - 50 .

WANG W C ， 2021 . Existence of sign-changing solutions for radially symmetric <math id="M304"><mi>p</mi></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836449&type= 2.96333337 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type= 1.77800000 -Laplacian equations with various potentials ［J］. Electron J Differ Equ ， 2021： 40 .

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带不定权的p-Laplacian方程边值问题正解集的S型连通分支

S-shaped component of positive solutions for boundary value problem of p-Laplacian equation with sign-changing weight

DOI：10.13471/j.cnki.acta.snus.ZR20240302

摘要

Abstract

关键词

Keywords

references

带不定权的 $p$ -Laplacian方程边值问题正解集的 $S$ 型连通分支

S-shaped component of positive solutions for boundary value problem of $p$ -Laplacian equation with sign-changing weight