Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms
This paper deals with the existence of bounded and locally Hölder continuous weak solutions of the following nonlinear fourth-order Dirichlet problem: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><ms...
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2024-12-01
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| author | Salvatore D’Asero |
| author_facet | Salvatore D’Asero |
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| description | This paper deals with the existence of bounded and locally Hölder continuous weak solutions of the following nonlinear fourth-order Dirichlet problem: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mo>|</mo><mi>α</mi><mo>|</mo><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></munder></mstyle><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>α</mi><mo>|</mo></mrow></msup><msup><mrow><mi mathvariant="normal">D</mi></mrow><mi>α</mi></msup><mfenced separators="" open="[" close="]"><msub><mi>A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><msup><mrow><mi mathvariant="normal">D</mi></mrow><mn>1</mn></msup><mi>u</mi><mo>,</mo><msup><mrow><mi mathvariant="normal">D</mi></mrow><mn>2</mn></msup><mi>u</mi><mo>)</mo></mrow><mo>−</mo><msub><mi>E</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo> </mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>λ</mi><mo>(</mo><msub><mi>p</mi><mi>α</mi></msub><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo> </mo><mo>sign</mo><mi>u</mi></mfenced><mo>=</mo><mi>f</mi></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>, where the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula> satisfy a strengthened degenerate coercivity condition. |
| format | Article |
| id | doaj-art-12c2ed4838024b60b4f77a8dc9041904 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
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| spelling | doaj-art-12c2ed4838024b60b4f77a8dc90419042025-01-10T13:17:56ZengMDPI AGMathematics2227-73902024-12-01131310.3390/math13010003Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection TermsSalvatore D’Asero0Dipartimento di Matematica e Informatica, University of Catania, Viale A. Doria 6, 95125 Catania, ItalyThis paper deals with the existence of bounded and locally Hölder continuous weak solutions of the following nonlinear fourth-order Dirichlet problem: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mo>|</mo><mi>α</mi><mo>|</mo><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></munder></mstyle><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>α</mi><mo>|</mo></mrow></msup><msup><mrow><mi mathvariant="normal">D</mi></mrow><mi>α</mi></msup><mfenced separators="" open="[" close="]"><msub><mi>A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><msup><mrow><mi mathvariant="normal">D</mi></mrow><mn>1</mn></msup><mi>u</mi><mo>,</mo><msup><mrow><mi mathvariant="normal">D</mi></mrow><mn>2</mn></msup><mi>u</mi><mo>)</mo></mrow><mo>−</mo><msub><mi>E</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo> </mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>λ</mi><mo>(</mo><msub><mi>p</mi><mi>α</mi></msub><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo> </mo><mo>sign</mo><mi>u</mi></mfenced><mo>=</mo><mi>f</mi></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>, where the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula> satisfy a strengthened degenerate coercivity condition.https://www.mdpi.com/2227-7390/13/1/3high-order equationsnonlinear elliptic problemsdegenerate coercivityconvection termboundednessHölder continuity |
| spellingShingle | Salvatore D’Asero Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms Mathematics high-order equations nonlinear elliptic problems degenerate coercivity convection term boundedness Hölder continuity |
| title | Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms |
| title_full | Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms |
| title_fullStr | Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms |
| title_full_unstemmed | Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms |
| title_short | Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms |
| title_sort | existence of bounded solutions for a class of degenerate fourth order elliptic equations with convection terms |
| topic | high-order equations nonlinear elliptic problems degenerate coercivity convection term boundedness Hölder continuity |
| url | https://www.mdpi.com/2227-7390/13/1/3 |
| work_keys_str_mv | AT salvatoredasero existenceofboundedsolutionsforaclassofdegeneratefourthorderellipticequationswithconvectionterms |