A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential

A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential

In this paper, we investigate the existence of at least one weak solution for a nonlinear fourth-order elliptic system involving variable exponent biharmonic and Laplacian operators. The problem is set in a bounded domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&...

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Bibliographic Details
Main Authors: Khaled Kefi, Mohamad M. Al-Shomrani
Format: Article
Language:English
Published: MDPI AG 2025-04-01
Series:Mathematics
Subjects:
generalized Sobolev space
Hardy potential
critical theorem
Online Access:https://www.mdpi.com/2227-7390/13/9/1443
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Summary:In this paper, we investigate the existence of at least one weak solution for a nonlinear fourth-order elliptic system involving variable exponent biharmonic and Laplacian operators. The problem is set in a bounded domain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">D</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>) with homogeneous Dirichlet boundary conditions. A key feature of the system is the presence of a Hardy-type singular term with a variable exponent, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> represents the distance from <i>x</i> to the boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∂</mo><mi mathvariant="script">D</mi></mrow></semantics></math></inline-formula>. By employing a critical point theorem in the framework of variable exponent Sobolev spaces, we establish the existence of a weak solution whose norm vanishes at zero.
ISSN:2227-7390

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