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S-shaped component of positive solutions for boundary value problem of
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 64, Issue 4, Pages: 156-162(2025)
- 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：
- 基金信息：
DOI：10.13471/j.cnki.acta.snus.ZR20240302
CLC： O175.8Received：14 October 2024，
Accepted：22 February 2025，
Published Online：27 April 2025，
Published：25 July 2025
- 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.
![PDF Download](/rc-pub/front/front-article/download/90745847/lowqualitypdf/S-shaped component of positive solutions for boundary value problem of <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><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-M003.jpg"><?fx-imagestate width="2.79399991" height="4.57200003"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/FB811B8E-85DD-4c90-88FC-605CD1300443-M003c.jpg"><?fx-imagestate width="2.79399991" height="4.57200003"?></graphic></alternatives></inline-formula>-Laplacian equation with sign-changing weight.pdf){kind=link}
摘要
研究了带不定权的一维
<math id="M4"><mi>p</mi></math>
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2.96333337
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type=
1.77800000
-Laplacian 方程边值问题
<math id="M5"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced><mi>'</mi><mo>+</mo><mi>λ</mi><mi>h</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi>f</mi><mfenced separators="|"><mrow><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>
</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
<math id="M6"><msub><mrow><mi>φ</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>s</mi><mo>
</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
，
<math id="M7"><mi>λ</mi><mo>></mo><mn mathvariant="normal">0</mn></math>
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
<math id="M9"><mi>f</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>
</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
<math id="M10"><mi>f</mi></math>
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
<math id="M11"><mi>p</mi></math>
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2.96333337
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836435&type=
1.77800000
-Laplacian equation boundary value problem
<math id="M12"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>u</mi><mi>'</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced><mi>'</mi><mo>+</mo><mi>λ</mi><mi>h</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mi>f</mi><mfenced separators="|"><mrow><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>
</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=84836488&type=
11.93799973
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84836489&type=
73.23667145
where
<math id="M13"><msub><mrow><mi>φ</mi></mrow><mrow><mi>p</mi></mrow></msub><mfenced separators="|"><mrow><mi>s</mi></mrow></mfenced><mo>=</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>s</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>s</mi><mo>
</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
，
<math id="M14"><mi>λ</mi><mo>></mo><mn mathvariant="normal">0</mn></math>
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
<math id="M16"><mi>f</mi><mo>∈</mo><mi>C</mi><mfenced open="[" separators="|"><mrow><mn mathvariant="normal">0</mn><mo>
</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
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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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