Electron. J. Differential Equations, Vol. 2024 (2024), No. 15, pp. 19.
A biharmonic equation with discontinuous nonlinearities
Eduardo Arias, Marco Calahorrano, Alfonso Castro
Abstract:
We study the biharmonic equation with discontinuous nonlinearity and
homogeneous Dirichlet type boundary conditions
$$\displaylines{
\Delta^2u=H(ua)q(u) \quad \text{in }\Omega,\cr
u=0 \quad \hbox{on }\partial\Omega,\cr
\frac{\partial u}{\partial n}=0 \quad \hbox{on }\partial\Omega,
}$$
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 \(H(sa)q(s)\) 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 nontrivial 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.2024.15
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Eduardo Arias
Departamento de Matemática
Escuela Politécnica Nacional
Quito POBox 17012759, Ecuador
email: marcelo.arias@epn.edu.ec, eduardo.arias.94@outlook.es


Marco Calahorrano
Departmento de Matemática
Escuela Politécnica Nacional
Quito POBox 17012759, 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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