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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2024-12-01
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| author | Yin Lang Hairong Liu |
| author_facet | Yin Lang Hairong Liu |
| author_sort | Yin Lang |
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| description | 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. |
| format | Article |
| id | doaj-art-a3e26a12f11f439bb75dba0cb5049bdb |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
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| spelling | doaj-art-a3e26a12f11f439bb75dba0cb5049bdb2025-08-20T02:00:51ZengMDPI AGAxioms2075-16802024-12-01131285610.3390/axioms13120856Stable Solutions of a Class of Degenerate Elliptic EquationsYin Lang0Hairong Liu1School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, ChinaSchool of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, ChinaThis 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.https://www.mdpi.com/2075-1680/13/12/856stable solutiondegenerate elliptic equationsLiouville-type theorem |
| spellingShingle | Yin Lang Hairong Liu Stable Solutions of a Class of Degenerate Elliptic Equations Axioms stable solution degenerate elliptic equations Liouville-type theorem |
| title | Stable Solutions of a Class of Degenerate Elliptic Equations |
| title_full | Stable Solutions of a Class of Degenerate Elliptic Equations |
| title_fullStr | Stable Solutions of a Class of Degenerate Elliptic Equations |
| title_full_unstemmed | Stable Solutions of a Class of Degenerate Elliptic Equations |
| title_short | Stable Solutions of a Class of Degenerate Elliptic Equations |
| title_sort | stable solutions of a class of degenerate elliptic equations |
| topic | stable solution degenerate elliptic equations Liouville-type theorem |
| url | https://www.mdpi.com/2075-1680/13/12/856 |
| work_keys_str_mv | AT yinlang stablesolutionsofaclassofdegenerateellipticequations AT hairongliu stablesolutionsofaclassofdegenerateellipticequations |