Electron. J. Differential Equations, Vol. 2026 (2026), No. 02, pp. 1-15.
Multiplicity of solutions for biharmonic equations with critical exponent and prescribed singularity
Nadjet Yagoub, Mohammed El Mokhtar Ould El Mokhtar, Atika Matallah, Safia Benmansour
Abstract:
In this article, we study the singular critical biharmonic problem
$$\displaylines{
\Delta^2u-\mu V(x)u=|u|^{2^*-2}u+\lambda f(x) \quad \text{in } \Omega, \cr
u= \frac{\partial u}{\partial n}=0 \quad \text{on } \partial \Omega,
}$$
where \( \Delta^2\) is the biharmonic operator, \(\Omega\) is an open bounded domain
in \(\mathbb{R}^N\) \((N \geq 5)\) with smooth boundary
\(\partial \Omega, 2^* = \frac{2N}{N-4}\),
\(0 < \mu < \bar{\mu} :=\big( \frac{N(N-4)}{4} \big)^2\), \(f(x)\) and \(V(x)\) are given functions.
By using variational method and Nehari-type constraint, we establish the existence of
multiple solutions for this problem when \(0<\lambda<\lambda^*\), for some
\(\lambda^* > 0\).
Submitted September 8, 2025. Published January 8, 2026.}
Math Subject Classifications: 35J20, 35IJ60, 47J30.
Key Words: Variational methods; singular potential; multiplicity of solutions; critical Sobolev exponent; biharmonic equation.
DOI: 10.58997/ejde.2026.02
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Nadjet Yagoub
Laboratory of Analysis and Control of Partial Differential Equations
University of Sidi Bel Abbes, Algeria
email: nadjet.yagoub@univ-sba.dz
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Mohammed El Mokhtar Ould El Mokhtar
Department of Mathematics
College of Science
Qassim University, BO 6644
Buraidah 51452, Saudi Arabia
email: med.mokhtar66@yahoo.fr
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Atika Matallah
Laboratory of Analysis and Control of Partial Differential Equations
Higher School of Management, Tlemcen, Algeria
email: atika_matallah@yahoo.fr
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Safia Benmansour
Laboratory of Analysis and Control of Partial Differential Equations
Higher School of Management, Tlemcen, Algeria
email: safiabenmansour@hotmail.fr
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