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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2025-04-01
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| author | Khaled Kefi Mohamad M. Al-Shomrani |
| author_facet | Khaled Kefi Mohamad M. Al-Shomrani |
| author_sort | Khaled Kefi |
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| description | 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. |
| format | Article |
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| publishDate | 2025-04-01 |
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| spelling | doaj-art-cde01f6a0faa425f9ad6864c33f7eb212025-08-20T03:52:56ZengMDPI AGMathematics2227-73902025-04-01139144310.3390/math13091443A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy PotentialKhaled Kefi0Mohamad M. Al-Shomrani1Center for Scientific Research and Entrepreneurship, Northern Border University, Arar 73213, Saudi ArabiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaIn 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.https://www.mdpi.com/2227-7390/13/9/1443generalized Sobolev spaceHardy potentialcritical theorem |
| spellingShingle | Khaled Kefi Mohamad M. Al-Shomrani A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential Mathematics generalized Sobolev space Hardy potential critical theorem |
| title | A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential |
| title_full | A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential |
| title_fullStr | A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential |
| title_full_unstemmed | A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential |
| title_short | A Weak Solution for a Nonlinear Fourth-Order Elliptic System with Variable Exponent Operators and Hardy Potential |
| title_sort | weak solution for a nonlinear fourth order elliptic system with variable exponent operators and hardy potential |
| topic | generalized Sobolev space Hardy potential critical theorem |
| url | https://www.mdpi.com/2227-7390/13/9/1443 |
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