Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs
This work establishes the multiplicity of solutions for the fractional Kazdan–Warner equation on finite graphs for the negative case. Our main focus lies in analyzing the nonlinear equation defined on a finite graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML"...
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2025-04-01
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| author | Liang Shan Yang Liu |
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| description | This work establishes the multiplicity of solutions for the fractional Kazdan–Warner equation on finite graphs for the negative case. Our main focus lies in analyzing the nonlinear equation defined on a finite graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>=</mo><mrow><mo>(</mo><mi>K</mi><mo>+</mo><mi>λ</mi><mo>)</mo></mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>u</mi></mrow></msup><mo>−</mo><mi>κ</mi><mspace width="1.em"></mspace><mi>in</mi><mspace width="4pt"></mspace><mi>V</mi><mo>,</mo></mrow></semantics></math></inline-formula> where the fraction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and real parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> are given, and the graph functions <i>K</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula> satisfy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>max</mi><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow></msub><mi>K</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>≢</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∫</mo><mi>V</mi></msub><mi>κ</mi><mi>d</mi><mi>μ</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>. We derive the solvability characteristics of the above equation with the help of variational theory and the upper and lower solutions method. |
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| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-4c5570ae8b574207b8e768cfeb6cae2b2025-08-20T03:14:32ZengMDPI AGAxioms2075-16802025-04-0114534510.3390/axioms14050345Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite GraphsLiang Shan0Yang Liu1Gaoling School of Artificial Intelligence, Renmin University of China, Beijing 100872, ChinaSchool of Mathematics, Renmin University of China, Beijing 100872, ChinaThis work establishes the multiplicity of solutions for the fractional Kazdan–Warner equation on finite graphs for the negative case. Our main focus lies in analyzing the nonlinear equation defined on a finite graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>=</mo><mrow><mo>(</mo><mi>K</mi><mo>+</mo><mi>λ</mi><mo>)</mo></mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>u</mi></mrow></msup><mo>−</mo><mi>κ</mi><mspace width="1.em"></mspace><mi>in</mi><mspace width="4pt"></mspace><mi>V</mi><mo>,</mo></mrow></semantics></math></inline-formula> where the fraction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and real parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> are given, and the graph functions <i>K</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula> satisfy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>max</mi><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow></msub><mi>K</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>≢</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∫</mo><mi>V</mi></msub><mi>κ</mi><mi>d</mi><mi>μ</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>. We derive the solvability characteristics of the above equation with the help of variational theory and the upper and lower solutions method.https://www.mdpi.com/2075-1680/14/5/345finite graphsvariational theoryfractional Kazdan–Warner equation |
| spellingShingle | Liang Shan Yang Liu Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs Axioms finite graphs variational theory fractional Kazdan–Warner equation |
| title | Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs |
| title_full | Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs |
| title_fullStr | Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs |
| title_full_unstemmed | Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs |
| title_short | Multiple Solutions of Fractional Kazdan–Warner Equation for Negative Case on Finite Graphs |
| title_sort | multiple solutions of fractional kazdan warner equation for negative case on finite graphs |
| topic | finite graphs variational theory fractional Kazdan–Warner equation |
| url | https://www.mdpi.com/2075-1680/14/5/345 |
| work_keys_str_mv | AT liangshan multiplesolutionsoffractionalkazdanwarnerequationfornegativecaseonfinitegraphs AT yangliu multiplesolutionsoffractionalkazdanwarnerequationfornegativecaseonfinitegraphs |