Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms

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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Bibliographic Details
Main Author: Salvatore D’Asero
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Mathematics
Subjects:
high-order equations
nonlinear elliptic problems
degenerate coercivity
convection term
boundedness
Hölder continuity
Online Access:https://www.mdpi.com/2227-7390/13/1/3
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Summary: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.
ISSN:2227-7390

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