Electron. J. Differential Equations, Vol. 2024 (2024), No. 15, pp. 1-9.

A biharmonic equation with discontinuous nonlinearities

Eduardo Arias, Marco Calahorrano, Alfonso Castro

We study the biharmonic equation with discontinuous nonlinearity and homogeneous Dirichlet type boundary conditions $$\displaylines{ \Delta^2u=H(u-a)q(u) \quad \text{in }\Omega,\cr u=0 \quad \hbox{on }\partial\Omega,\cr \frac{\partial u}{\partial n}=0 \quad \hbox{on }\partial\Omega, }$$ where \(\Delta\) is the Laplace operator, \(a> 0\), \(H\) denotes the Heaviside function, \(q\) is a continuous function, and \(\Omega\) is a bounded domain in \(R^N\) with \(N\geq 3\). Adapting the method introduced by Ambrosetti and Badiale (The Dual Variational Principle), which is a modification of Clarke and Ekeland's Dual Action Principle, we prove the existence of nontrivial solutions. This method provides a differentiable functional whose critical points yield solutions despite the discontinuity of \(H(s-a)q(s)\) at \(s=a\). Considering \(\Omega\) of class \(\mathcal{C}^{4,\gamma}\) for some \(\gamma\in(0,1)\), and the function \(q\) constrained under certain conditions, we show the existence of two non-trivial solutions. Furthermore, we prove that the free boundary set \(\Omega_a=\{x\in\Omega:u(x)=a\}\) has measure zero when \(u\) is a minimizer of the action functional.

Submitted October 15, 2022. Published February 6, 2024.
Math Subject Classifications: 31B30, 35J60, 35J65, 58E05.
Key Words: Biharmonic equation; nonlinear discontinuity; critical point; dual variational principle; free boundary problem.
DOI: 10.58997/ejde.2023.15

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Eduardo Arias
Departamento de Matemática
Escuela Politécnica Nacional
Quito PO-Box 17-01-2759, Ecuador
email: marcelo.arias@epn.edu.ec, eduardo.arias.94@outlook.es
Marco Calahorrano
Departmento de Matemática
Escuela Politécnica Nacional
Quito PO-Box 17-01-2759, Ecuador
email: marco.calahorrano@epn.edu.ec
Alfonso Castro
Department of Mathematics
Harvey Mudd College
Claremont, CA 91711, USA
email: castro@g.hmc.edu

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