Stable Solutions of a Class of Degenerate Elliptic Equations
This paper deals with the second-order semi-linear degenerate elliptic equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><msub><mi>u</mi><mrow><mi>...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/12/856 |
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| Summary: | This paper deals with the second-order semi-linear degenerate elliptic equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><msub><mi>u</mi><mrow><mi>y</mi><mi>y</mi></mrow></msub><mo>+</mo><mi>b</mi><msub><mi>u</mi><mi>y</mi></msub><mo>+</mo><msub><mo>Δ</mo><mi>x</mi></msub><mi>u</mi><mo>+</mo><msup><mrow><mo stretchy="false">|</mo><mi>u</mi><mo stretchy="false">|</mo></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>×</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>α</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. We establish a Liouville theorem of stable solution of the degenerate equation mentioned above by using the energy method. The classification results for stable solutions belonging to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>2</mn></msup></semantics></math></inline-formula> can be thought of as an analogue of the recent results of Farina for the Lane–Emden equation. |
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| ISSN: | 2075-1680 |