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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/5/345 |
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| Summary: | 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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| ISSN: | 2075-1680 |