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

Khaled Kefi () and Mohamad M. Al-Shomrani
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Khaled Kefi: Center for Scientific Research and Entrepreneurship, Northern Border University, Arar 73213, Saudi Arabia
Mohamad M. Al-Shomrani: Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Mathematics, 2025, vol. 13, issue 9, 1-12

Abstract: 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 D ⊂ R N ( N ≥ 3 ) 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 δ ( x ) represents the distance from x to the boundary ∂ D . 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.

Keywords: generalized Sobolev space; Hardy potential; critical theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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