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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
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